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+arXiv:2301.08681v1  [math.RT]  20 Jan 2023
+A STRUCTURAL VIEW OF
+MAXIMAL GREEN SEQUENCES
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Abstract. We initiate a new approach to maximal green sequences by con-
+sidering them up to an equivalence relation.
+This reveals extra structure,
+since the set of equivalence classes of maximal green sequences of an algebra
+carries interesting partial orders. We show that the equivalence relation may
+be defined in several equivalent ways. We likewise define three conjecturally
+equivalent partial orders on the set of equivalence classes, and prove some of
+the implications between them. In the case of Nakayama algebras we prove
+that these three partial orders indeed coincide.
+Contents
+1.
+Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+2.
+Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.
+Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2.
+Maximal green sequences
+. . . . . . . . . . . . . . . . . . . .
+6
+2.3.
+Relative projectives in torsion classes: τ-tilting
+. . . . . . . .
+8
+2.4.
+Relative simples in torsion classes: bricks
+. . . . . . . . . . .
+10
+3.
+Equivalence relations on maximal green sequences . . . . . . . . . .
+14
+3.1.
+Preliminary lemmas
+. . . . . . . . . . . . . . . . . . . . . . .
+14
+3.2.
+Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+3.3.
+Deformations across squares . . . . . . . . . . . . . . . . . . .
+19
+3.4.
+Characterising equivalent maximal green sequences . . . . . .
+25
+4.
+Partial orders on equivalence classes
+. . . . . . . . . . . . . . . . .
+33
+4.1.
+Deformations across oriented polygons . . . . . . . . . . . . .
+33
+4.2.
+Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+4.3.
+Exchange pairs . . . . . . . . . . . . . . . . . . . . . . . . . .
+49
+4.4.
+An example from the twice-punctured torus . . . . . . . . . .
+52
+5.
+Nakayama algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
+57
+5.1.
+Equivalence using bricks . . . . . . . . . . . . . . . . . . . . .
+57
+5.2.
+Partial order using bricks
+. . . . . . . . . . . . . . . . . . . .
+62
+2020 Mathematics Subject Classification. Primary:
+16G20; Secondary:
+13F60, 16G10,
+18E40.
+Key words and phrases. Maximal green sequences, τ-tilting, silting, torsion classes, Harder–
+Narasimhan filtrations, cluster algebras.
+1
+
+2
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+5.3.
+Bricks versus summands . . . . . . . . . . . . . . . . . . . . .
+64
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+66
+1. Introduction
+Maximal green sequences were introduced by Keller in [Kel11], but were al-
+ready implicit in the physics literature in the context of BPS spectra of particles
+in string theory [CCV11]; see also [GMN13; Ali+14; Xie16]. The origin of maxi-
+mal green sequences lies in the theory of cluster algebras, which were introduced
+by Fomin and Zelevinsky [FZ02]. Cluster algebras are commutative rings with
+distinguished generating sets known as ‘clusters’ which are related by a process
+called ‘mutation’. Every cluster has a skew-symmetrisable matrix associated to
+it, with mutation both transforming the matrix and the variables of the cluster.
+If the matrix is in fact skew-symmetric, it gives a quiver. The clusters of a clus-
+ter algebra form the vertices of a graph whose edges connect clusters related by
+mutation; this graph is known as the ‘exchange graph’.
+Subsequent to their introduction, deep connections were found between cluster
+algebras and the representation theory of finite-dimensional algebras [MRZ03;
+CCS06; Bua+06]. Here, the clusters of a cluster algebra correspond to objects
+in a certain category, thereby “categorifying” the cluster algebra. Some cate-
+gorifications of cluster algebras — in particular, those related to τ-tilting theory
+[AIR14] — give an orientation to the edges of the exchange graph, thereby pro-
+ducing a partial order. For a helpful survey of this phenomenon, see [BY13].
+A maximal green sequence is then simply a maximal chain of finite length in
+the resulting poset. The name comes from a combinatorial construction due to
+Keller [Kel11], in which some vertices of the quiver of a cluster are green and
+some are red, with a maximal green sequence given by a sequence of mutations
+at green vertices which turns the quiver from all green to all red.
+The original motivation for introducing maximal green sequence comes from
+the theory of Donaldson–Thomas invariants, which, roughly speaking, are count-
+ing functions of aforementioned BPS states. A maximal green sequence for a
+quiver gives an explicit formula for the refined Donaldson–Thomas invariant as-
+sociated to the quiver by Kontsevich and Soibelman [KS08]. Consequently, max-
+imal green sequences give quantum dilogarithm identities [FV93; FK94; Rei10].
+The existence of a maximal green sequence also gives a formula for the twist au-
+tomorphism of a cluster algebra [GLS12], as well as guaranteeing the existence of
+a theta basis [Gro+18] and a generic basis [Qin22] in the upper cluster algebra.
+Maximal green sequences are intimately related to Bridgeland stability condi-
+tions on module categories [Bri07], where a chamber of type I with finitely many
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+3
+isomorphism classes of stable objects gives a maximal green sequence [Bri07]. A
+useful survey of maximal green sequences is given in [KD20].
+It was shown in [Wil22b] that the set of maximal green sequences of an al-
+gebra could be given more structure by subjecting it to an equivalence relation.
+Indeed, the set of equivalence classes forms a partially ordered set. Posets of
+maximal green sequences were also studied in [Gor14a; Gor14b] using the lan-
+guage of [BKT14]. Considering maximal green sequences modulo an equivalence
+relation leads to nice results. Equivalence classes of maximal green sequences of
+linearly oriented An are in bijection with triangulations of the three-dimensional
+cyclic polytope with n + 3 vertices [Wil22b]. Moreover, the partial order on the
+equivalence classes of these maximal green sequences is a lattice and coincides
+with the higher Stasheff–Tamari order [Wil22b], which is a higher-dimensional
+version of the Tamari lattice [KV91; ER96].
+In [Wil22b], only one equivalence relation on maximal green sequences is con-
+sidered and the idea of equivalence of maximal green sequences is not studied
+as a subject in its own right. Indeed, there are several other appealing ways
+of defining equivalence relations on maximal green sequences. The first main
+theorem of this paper shows that the equivalence relation on maximal green se-
+quences admits the following six equivalent characterisations. For the detail on
+the terminology in this theorem, see Section 2.
+Theorem A (Theorem 3.20, Lemma 3.22, Example 3.26). Let Λ be a finite-
+dimensional algebra over a field K with G and G′ maximal green sequences of Λ.
+Then the following are equivalent.
+(1) G and G′ can be deformed into each other across squares.
+(2) G and G′ have the same set of exchange pairs.
+(3) G and G′ have the same set of indecomposable direct summands of two-
+term silting complexes.
+(4) G and G′ have the same set of indecomposable direct summands of support
+τ-tilting modules.
+(5) For any Λ-module M, the set of semistable factors of M is the same for
+the respective Harder–Narasimhan filtrations given by G and G′.
+(6) For any Λ-module M, the multiset of stable factors of M is the same for
+the respective Harder–Narasimhan filtrations given by G and G′.
+Moreover, if G and G′ satisfy any of the equivalent statements above, then G and
+G′ have the same set of bricks. However, G and G′ may have the same set of
+bricks without any of the above holding.
+The fact that equivalence of maximal green sequences admits several different
+characterisations indicates that the notion is a natural one. On the other hand,
+it is rather surprising that two maximal green sequences may have the same set
+
+4
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+of bricks while having different sets of τ-rigid modules, since bricks are dual to
+indecomposable τ-rigid modules [DIJ19]. This is related to the “tropical duality”
+that exists between c-vectors and g-vectors for cluster algebras [NZ12]. However,
+we show that two maximal green sequences with the same bricks are equivalent
+in the case of Nakayama algebras (Theorem 5.8). More generally, as we explain,
+having the same factors of Harder–Narasimhan filtrations can be considered an
+augmentation of the condition of having the same bricks. An intriguing impli-
+cation of Theorem A is that the set of indecomposable summands of support
+τ-tilting modules of a maximal green sequence determines the semistable fac-
+tors of every module; it would be interesting if there were a direct construction
+relating the two.
+Having introduced equivalence relations on maximal green sequences, we can
+define partial orders on the equivalence classes, as again in [Wil22b]. Hence,
+considering maximal green sequences subject to an equivalence relation reveals
+extra structure. These partial orders can also be seen in the context of partial
+orders on chambers of stability conditions, although this is not the language
+we choose to use. There is a partial order on the chambers of King stability
+conditions [Kin94] given by those from τ-tilting theory [BST19; Asa21], and a
+partial order on chambers of type II for Bridgeland stability conditions given by
+inclusion of aisles of t-structures [Bri07]. Roughly speaking, we study partial
+orders on equivalence classes of certain type I chambers.
+As an aside, it is
+curious to note that while all two-term silting complexes give chambers of King
+stability conditions [Bri17; BST19; Asa21], not all maximal green sequences give
+chambers of Bridgeland stability conditions, see [Qiu15, Counterexample 7.16]
+and [AI20, Theorem L3].
+We introduce three partial orders on equivalence classes of maximal green
+sequences: one in terms of certain deformations, one in terms of indecompos-
+able summands of two-term silting complexes, and one in terms of Harder–
+Narasimhan filtrations.
+We conjecture that these partial orders coincide, to-
+wards which we show the following result. See Definition 4.1 for the definition
+of an increasing elementary polygonal deformation and Definition 4.5(3) for the
+definition of refinement of Harder–Narasimhan filtrations.
+Theorem B (Theorem 4.14 and Theorem 4.15). Let Λ be a finite-dimensional
+algebra over a field K with G and G′ maximal green sequences of Λ. Further-
+more, suppose that [G′] is the result of a series of increasing elementary polygonal
+deformations of [G]. Then we have the following.
+(1) Every indecomposable direct summand of a two-term silting complex of
+G′ is a direct summand of a two-term silting complex of G.
+(2) The Harder–Narasimhan filtrations induced by G′ refine those induced
+by G.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+5
+The partial order given by deformations is very natural in terms of stability
+conditions, since increasing elementary polygonal deformations correspond to
+crossing walls of type I which decrease the number of stables. We show that in
+certain cases these orders have unique maxima and minima (Proposition 4.20
+and Proposition 4.21).
+For Nakayama algebras, or algebras with two simple modules up to isomor-
+phism, we prove the converse implications from Theorem B. Namely, we show
+that the three partial orders in fact coincide, along with an additional order
+given by inclusion of bricks. The latter is not well-defined for an arbitrary finite-
+dimensional algebra Λ, since in general non-equivalent maximal green sequences
+may have the same bricks.
+Theorem C (Theorem 4.19 and Corollary 5.17). Let Λ be a finite-dimensional
+algebra over a field K which is either a Nakayama algebra or an algebra with two
+isomorphism classes of simple modules. Further, let G and G′ maximal green
+sequences of Λ. Then the following are equivalent.
+(1) [G′] is the result of a series of increasing elementary polygonal deforma-
+tions of [G].
+(2) Every indecomposable direct summand of a support τ-tilting module of G′
+is a direct summand of a support τ-tilting module of G.
+(3) The Harder–Narasimhan filtrations induced by G′ refine those induced
+by G.
+(4) Every brick of G′ is a brick of G.
+Organisation of the paper. The structure of this paper is as follows. We
+begin in Section 2 by giving background to the paper. We introduce maximal
+green sequences in Section 2.2, and then give their description in terms of τ-tilting
+theory in Section 2.3, followed by their description in terms of sequences of bricks
+in Section 2.4. We introduce equivalence relations on maximal green sequences
+in Section 3, and prove our first main result Theorem A showing the different
+ways of characterising the equivalence relation. In Section 4, we introduce three
+partial orders on equivalence classes of maximal green sequences and prove our
+second main result Theorem B, showing how these are related. We then go on
+to consider the interaction between one of the partial orders and the exchange
+pairs of a maximal green sequence. We then discuss an interesting example of a
+poset of equivalence classes of maximal green sequences of an algebra related to
+a triangulation of the twice-punctured torus. Finally, in Section 5 we study the
+posets for Nakayama algebras in detail and prove Theorem C.
+
+6
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Acknowledgements. We would like to thank Aran Tattar for help with the
+proof of Lemma 3.16, and Haruhisa Enomoto, Bernhard Keller, Hipolito Treffin-
+ger, Osamu Iyama, Aaron Chan, Daniel Labardini-Fragoso, and H˚avard Terland
+for useful discussions. This work is part of a project that has received funding
+from the European Research Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation programme (grant agreement No. 101001159).
+Parts of this work were done during stays of MG at the University of Stuttgart,
+and he is very grateful to Steffen Koenig for the hospitality. NJW is currently
+supported by EPSRC grant EP/V050524/1, and part of the work on this paper
+was done while a JSPS short-term postdoctoral research fellow at the University
+of Tokyo.
+Notation. Throughout this paper, we let Λ be a finite-dimensional algebra over
+a field K, with mod Λ the category of finitely generated right Λ-modules. In this
+paper we use the symbols ‘⊂’ and ‘⊃’ to denote strict inclusion of sets, that is,
+inclusion but not equality. This is more commonly denoted with the symbols ‘⊊’
+and ‘⊋’ respectively.
+2. Background
+2.1. Partially ordered sets. Given a set P, a partial order on P is a relation
+R ⊆ P × P which is reflexive, symmetric, and transitive. Partially ordered sets
+are referred to as posets. We usually write partial orders with the symbol ⩽,
+so that if (x, y) ∈ R, where R is a partial order on a set P, we write x ⩽ y. A
+covering relation in a partial order ⩽ is a relation x < z such that if x ⩽ y ⩽ z,
+then y = x or y = z. One can also say that z covers x. It is usual to write x ⋖ z
+when x < z is a covering relation. An interval of a poset P is a subset of the
+form {y ∈ P | x ⩽ y ⩽ z} for some x, z ∈ P.
+The Hasse diagram of a partial order ⩽ on a set P is the quiver with the
+elements of P as vertices, with arrows z → x whenever x ⋖ z is a covering
+relation. In this paper, we illustrate posets using their Hasse diagrams. Recall
+that a Hasse diagram is n-regular if every vertex is incident to precisely n arrows.
+2.2. Maximal green sequences. Maximal green sequences were introduced
+by Keller in the context of Donaldson–Thomas theory using a combinatorial
+definition in terms of quivers [Kel11]. It follows from work of Nagao that this is
+equivalent to having a maximal chain of torsion classes [Nag13]. This is the first
+notion of a maximal green sequence that we will cover.
+2.2.1. Torsion classes. Torsion pairs were introduced by Dickson to generalise
+the structure given by torsion and torsion-free abelian groups to arbitrary abelian
+categories [Dic66]. A torsion pair is a pair of full subcategories (T , F) of mod Λ
+such that
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+7
+(1) HomΛ(T , F) = 0;
+(2) if HomΛ(T, F) = 0, then T ∈ T ;
+(3) if HomΛ(T , F) = 0, then F ∈ F.
+Here T is called the torsion class and F is called the torsion-free class. More
+generally, a full subcategory T is called a torsion class if it is a torsion class in
+some torsion pair, and likewise for torsion-free classes. It is well-known that a full
+subcategory T of mod Λ is a torsion class if and only if it is closed under factor
+modules and extensions [Dic66, Theorem 2.3]. Given a set X of Λ-modules, we
+write Tors(X) for the smallest torsion class containing X and Torf(X) for the
+smallest torsion-free class containing X.
+Given a torsion pair (T , F) in mod Λ and a Λ-module M, there is an exact
+sequence
+0 → L → M → N → 0
+such that L ∈ T and N ∈ F, which is unique up to isomorphism. Here L is
+called the torsion submodule of M and N is called the torsion-free factor module.
+The torsion classes of mod Λ form a complete lattice under inclusion, denoted
+tors Λ. We call the covering relations of this lattice minimal inclusions. Hence,
+T ⊂ T ′ is a minimal inclusion if and only if whenever we have T ⊆ T ′′ ⊆ T ′, we
+must either have T ′′ = T or T ′′ = T ′.
+We will be particularly interested in the subposet ff-tors Λ of functorially finite
+torsion classes of Λ, where ‘functorially finite’ is defined as follows.
+Given a
+subcategory X ⊆ mod Λ and a map f : X → M, where X ∈ X and M ∈ mod Λ,
+we say that f is a right X-approximation if for any X′ ∈ X, the sequence
+HomΛ(X′, X) → HomΛ(X′, M) → 0
+is exact, following [AS80]. Left X-approximations are defined dually. The sub-
+category X is said to be contravariantly finite if every M ∈ mod Λ admits a
+right X-approximation, and covariantly finite if every M ∈ mod Λ admits a left
+X-approximation. If X is both contravariantly finite and covariantly finite, then
+X is functorially finite.
+Certain sorts of approximations are of particular note. A morphism f : X → Y
+is right minimal if any morphism g: X → X such that fg = f is an isomor-
+phism. Left minimal morphisms are defined dually. A right X-approximation
+is a minimal right X-approximation if it is also right minimal, and minimal left
+approximations are defined analogously.
+2.2.2. First notion of maximal green sequence. A maximal green sequence is a
+maximal chain in tors Λ of finite length. More explicitly, a maximal green se-
+quence is a chain of minimal inclusions of torsion classes
+mod Λ = T0 ⊃ T1 ⊃ · · · ⊃ Tr−1 ⊃ Tr = {0}.
+
+8
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+This is the first definition of a maximal green sequence that we shall see, but we
+shall see two further notions as well. We shall regard the three notions as being
+cryptomorphic to each other.
+We note at this point that a result of Demonet, Iyama, and Jasso [DIJ19,
+Theorem 3.1] implies that every torsion class in a finite maximal chain is func-
+torially finite, so that maximal green sequences are in fact finite maximal chains
+in ff-tors Λ.
+2.3. Relative projectives in torsion classes: τ-tilting. Our second notion
+of maximal green sequences will operate in terms of the relative projectives of
+the torsion classes in the maximal chain. Relative projectives in torsion classes
+were studied by Adachi, Iyama, and Reiten in [AIR14] in terms of what is called
+‘τ-tilting theory’.
+2.3.1. Relative projectives in torsion classes. Given a torsion class T ∈ tors Λ,
+a module X ∈ T is a relative projective if Ext1
+Λ(X, M) = 0 for all M ∈ T .
+We write P(T ) for the direct sum of one copy of each indecomposable relative
+projective in T , up to isomorphism.
+2.3.2. Support τ-tilting pairs. Relative projectives in torsion classes correspond
+to what are called ‘support τ-tilting modules’, where τ denotes the Auslander–
+Reiten translate in mod Λ. A Λ-module M is called τ-rigid if HomΛ(M, τM) = 0.
+A pair (M, P) of Λ-modules where P is projective is called τ-rigid if M is τ-
+rigid and HomΛ(P, M) = 0. A τ-rigid pair (M, P) is called support τ-tilting if
+|M|+|P| = |Λ|, where |X| denotes the number of non-isomorphic indecomposable
+direct summands of X. In this case M is called a support τ-tilting module. We
+write sτ-tilt Λ for the set of isomorphism-class representatives of basic support
+τ-tilting modules over Λ.
+Theorem 2.1 ([AIR14, Theorem 2.7]). There is a bijection
+ff-tors Λ ←→ sτ-tilt Λ,
+T �−→ P(T ),
+Fac M ←−� M.
+Here Fac M := {X ∈ mod Λ | ∃ an epimorphism M⊕m ։ X for some m}.
+2.3.3. Two-term silting. An equivalent framework to support τ-tilting modules
+is given by two-term silting complexes. We will often prefer to work with these
+objects instead, for reasons that we will explain.
+We denote by Kb(proj Λ) the homotopy category of bounded complexes of
+projective right Λ-modules. We will often want to consider K[−1,0](proj Λ), the
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+9
+subcategory of Kb(proj Λ) consisting of two-term complexes, that is, complexes
+concentrated in degrees −1 and 0:
+P −1 → P 0.
+An object T of Kb(proj Λ) is pre-silting if HomKb(proj Λ)(T, T[i]) = 0 for all i >
+0. A pre-silting complex T is silting if, additionally, thick T = Kb(proj Λ). Here
+thick T denotes the smallest full subcategory of Kb(proj Λ) which contains P and
+is closed under cones, [±1], direct summands, and isomorphisms. For a two-term
+complex T to be pre-silting, it suffices that HomKb(proj Λ)(T, T[1]) = 0. Moreover,
+for a pre-silting two-term complex T to be silting, it suffices that |T| = |Λ| by
+[AIR14, Proposition 3.3(b)]. We write 2-silt Λ for the set of isomorphism-class
+representatives of basic two-term silting complexes of Λ.
+Theorem 2.2 ([AIR14, Theorem 3.2]). There is a bijection
+2-silt Λ ←→ sτ-tilt Λ,
+T �−→ H0(T).
+Hence, support τ-tilting and two-term silting are essentially equivalent. Func-
+torially finite torsion classes of Λ are therefore also in bijection with two-term
+silting complexes over Λ. The advantage of support τ-tilting is that it is easier
+to describe the relation with torsion classes. The advantage of two-term silting
+is that it is easier to talk about mutation, which is why we often work in this
+framework.
+2.3.4. Mutation. Given a silting complex T = E ⊕ X in Kb(proj Λ) where X is
+indecomposable, let
+X
+f−→ E′ g−→ Y → X[1]
+be a triangle in Kb(proj Λ) such that f is a minimal left add E-approximation
+of X. This triangle is known as the exchange triangle. Then, by [AI12, The-
+orem 2.35], Y is indecomposable with g a minimal right add E-approximation
+of Y and, by [AI12, Theorem 2.31], T ′ = E ⊕ Y is a silting complex. In this
+situation, we say that T ′ is a green mutation (or left mutation) of T and T is a
+red mutation of T ′ (or right mutation). The opposite convention for green and
+red mutations is used by some authors. Such choice would not make any differ-
+ence to our considerations of maximal chains in the lattice of two-term silting
+complexes, since the chain remains the same whichever direction one traverses
+it in. We call (X, Y ) the exchange pair of the mutation.
+We say that a pair of two-term silting complexes T, T ′ ∈ K[−1,0](proj Λ) are
+mutations of each other if and only if T = E ⊕ X and T ′ = E ⊕ Y where X
+and Y are indecomposable. By [AIR14, Corollary 3.8(b)], we have that T and
+T ′ are mutations of each other if and only if either T ′ is a green mutation of T,
+
+10
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+or T is a green mutation of T ′. The set of basic two-term silting complexes of
+Λ forms a poset denoted 2-silt Λ where the covering relations are that T ′ ⋖ T
+if and only if T ′ is a green mutation of T. The partial order itself is then the
+transitive–reflexive closure of these covering relations.
+Theorem 2.3 ([AIR14, Corollary 2.34, Corollary 3.9]). The bijection between
+2-silt Λ and ff-tors Λ induces an isomorphism between the Hasse diagrams of these
+posets.
+In particular, if functorially finite torsion classes T and T ′ correspond to two-
+term silting complexes T and T ′ respectively, then there is a minimal inclusion
+T ⊃ T ′ if and only if T ′ is a green mutation of T.
+2.3.5. Second notion of maximal green sequence. We can use Theorem 2.3 to
+obtain the second notion of maximal green sequence. Since a maximal green
+sequence is a maximal chain of minimal inclusions in ff-tors Λ, by Theorem 2.3,
+we have that a maximal green sequence is a maximal sequence of green mutations
+of two-term silting complexes. More explicitly, a maximal green sequence is a
+sequence of two-term silting complexes Λ = T0, T1, . . . , Tr = Λ[1] such that for
+each i ∈ {1, . . . , r}, we have that Ti is a green mutation of Ti−1. This was first
+observed by Br¨ustle, Smith, and Treffinger [BST19, Proposition 4.9].
+Such a maximal green sequence can be specified by labelling each of the sum-
+mands of Λ from 1 to n where n = |Λ| and then giving a list of numbers specifying
+the sequence of summands to be mutated. (When summand i is mutated, the
+summand that replaces it is then also labelled i.) In terms of the original notion
+of maximal green sequence from [Kel11] using quiver mutation, this can be seen
+as a sequence of vertices of the quiver to mutate. We will use this perspective a
+couple of times.
+2.4. Relative simples in torsion classes: bricks. We now give background
+leading up to the third notion of maximal green sequences, which comes from
+looking at the relative simples in torsion classes.
+2.4.1. Relative simples in torsion classes. Relatively simple objects in exact cat-
+egories were first studied in [BG16] and were considered in the specific case of
+torsion-free classes in [Eno21]. A Λ-module B in a torsion class T is a relative
+simple if there is no short exact sequence
+0 → A → B → C → 0
+such that A and C are both non-zero modules in T . Since torsion classes are
+closed under factor modules, it is in fact necessary and sufficient for B to have no
+proper submodules which are in T . We write simp(T ) for the set of isomorphism-
+class representatives of relative simples of T .
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+11
+2.4.2. Brick labelling. In a somewhat, but not entirely, analogous way to how rel-
+ative projectives of torsion classes correspond to support τ-tilting modules, rel-
+ative simples of torsion classes correspond to certain modules known as ‘bricks’.
+An object B of mod Λ is a brick if EndΛ B is a division ring. Equivalently,
+B is a brick if every non-zero endomorphism of B is an isomorphism. This is
+the case if and only if B has no proper factor module which is isomorphic to a
+proper submodule.
+Theorem 2.4 ([BCZ19; Dem+18]). We have that T ⊇ U is a minimal inclusion
+if and only if T ∩ U⊥0 = Filt(B) for a brick B. Moreover, this brick B is unique
+up to isomorphism.
+Here Filt and U⊥0 are defined as follows. We have
+U⊥0 := {M ∈ mod Λ | HomΛ(U, M) = 0 for all U ∈ U}.
+Given a full subcategory C of mod Λ, Filt(C) is the full subcategory of mod Λ
+consisting of modules M with a finite filtration
+M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0
+such that Mi−1/Mi ∈ C for all 1 ⩽ i ⩽ l. It is known that Filt(B) for a brick
+B is a wide subcategory of mod Λ [Rin76, 1.2], meaning that it is closed under
+extensions, kernels and cokernels. It is therefore also closed under images.
+In this way, the covering relations of tors Λ can be labelled by bricks. We often
+write the brick labels of the inclusions by
+T
+B⊃ U.
+For two torsion classes T ⊇ U, we denote [T , U] := T ∩U⊥0. The relation between
+brick labels and intervals in the lattice of torsion classes extends beyond intervals
+given by covering relations.
+Theorem 2.5 ([Tat21, Theorem 6.8], [Eno21, Theorem 3.5]). Let
+Ti
+Bi+1
+⊃ . . .
+Bj⊃ Tj
+be a chain of minimal inclusions in tors Λ with brick labels. Then
+[Ti, Tj] = Filt(Bi+1, Bi+2, . . . , Bj).
+The following proposition is a slight generalisation of [Eno21, Proposition 3.8].
+Proposition 2.6. Given torsion classes Ti ⊇ Tj in mod Λ, every relatively sim-
+ple object in [Ti, Tj] must occur as a brick label in every finite maximal chain in
+the interval of tors Λ between Ti and Tj.
+
+12
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Proof. To see this, take a relatively simple object S in [Ti, Tj] and sup-
+pose that there is a maximal chain connecting Ti and Tj with brick labels
+Bi+1, Bi+2, . . . , Bj. Then, since [Ti, Tj] = Filt(Bi+1, Bi+2, . . . , Bj), we must have
+that S has a filtration with factors in {Bi+1, Bi+2, . . . , Bj}.
+But, since S is
+relatively simple in [Ti, Tj], this filtration can only have one factor, and so we
+must have that S = Bk for some k, as desired.
+□
+Relatively simple objects in [Ti, Tj] were studied in [AP22] in the case where
+this category is abelian. Finally, the relationship between the relative simples of
+torsion classes in a maximal green sequence and the brick labels is as follows.
+Theorem 2.7 ([Eno21, Proposition 3.8]). Let G be a maximal green sequence
+mod Λ = T0
+B1
+⊃ T1
+B2
+⊃ . . .
+Br−1
+⊃
+Tr−1
+Br
+⊃ Tr = {0}.
+Then
+r�
+i=0
+simp(Ti) = {B1, B2, . . . , Br}.
+More precisely, [Eno21, Proposition 3.8] proves the inclusion
+r�
+i=0
+simp(Ti) ⊆ {B1, B2, . . . , Br},
+and the converse inclusion follows from the fact that Bi is a relative simple in
+Ti−1, for 1 ≤ i ≤ r.
+2.4.3. Third notion of maximal green sequence. We can now give the third notion
+of maximal green sequences. In the appendix to [KD20], Demonet shows that
+(not necessarily finite) maximal chains of torsion classes may be characterised
+in terms of bricks, generalising [Igu19, Theorem 1.1]. Another related result to
+this is [Tre20, Theorem 5.3].
+A backwards Hom-orthogonal sequence of bricks is a sequence of bricks
+B1, B2, . . . , Br
+such that if i < j then HomΛ(Bj, Bi) = 0. A backwards Hom-orthogonal se-
+quence of bricks is maximal if one cannot insert a brick at any point in the
+sequence without losing the backwards Hom-orthogonality property. By [KD20,
+Theorem A.3], there is a bijection between maximal green sequences and max-
+imal backwards Hom-orthogonal sequences of bricks. Given a maximal green
+sequence
+mod Λ = T0 ⊃ T1 ⊃ · · · ⊃ Tr = {0},
+one obtains the maximal backwards Hom-orthogonal sequence of bricks by taking
+Bi as the brick label of the minimal inclusion Ti−1 ⊃ Ti. Conversely, given a
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+13
+maximal backwards Hom-orthogonal sequence of bricks
+B1, B2, . . . , Br,
+one obtains the corresponding maximal green sequence by taking Ti to be the
+smallest torsion class Tors(Bi+1, . . . , Br) containing Bi+1, . . . , Br.
+2.4.4. Harder–Narasimhan filtrations. It was shown in [Igu20, Theorem 3.8]
+that maximal green sequences induce so-called “Harder–Narasimhan filtra-
+tions”. This was then shown for arbitrary chains of torsion classes in [Tre20,
+Theorem 2.10], see also [BKT14, Section 3.3]. Such filtrations were originally
+studied in [HN75] and are well known from their role in the theory of stability
+conditions, for example [Rud97; Bri07]. Let
+B1, B2, . . . , Br
+be a maximal green sequence G of a finite-dimensional algebra Λ given as a
+maximal backwards Hom-orthogonal sequence of bricks. Then, every non-zero
+Λ-module M has a unique filtration
+M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0
+such that Fj := Mj−1/Mj ∈ Filt(Bij) for some ij, with
+i1 < i2 < · · · < il.
+This filtration is called the Harder–Narasimhan filtration (HN filtration) or the
+G-Harder–Narasimhan filtration.
+This is precisely the filtration of M by the
+torsion submodules associated to it by the torsion classes in G, with duplicated
+torsion submodules removed.
+We call Fj the semistable factors here, or the
+G-semistable factors. We write
+SSFG(M) = {F1, F2, . . . , Fl}
+for the set of G-semistable factors of M.
+Since the module Fj lies in Filt(Bij), it has a filtration where all of the factors
+are isomorphic to Bij. (In terms of stability conditions, this is the Jordan–H¨older
+filtration of a semistable module in terms of stable modules [Rud97, Theorem 3].)
+We write SFG(Fj) for the multiset of Bij factors in this filtration of Fj. That
+is, SFG(Fj) is s copies of Bij, where s is the number of factors in this filtration
+of Fj. We furthermore write
+SFG(M) =
+l�
+i=1
+SFG(Fj).
+We refer to SFG(M) as the multiset of stable factors or G-stable factors of M.
+
+14
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+3. Equivalence relations on maximal green sequences
+In this section, we define equivalence of maximal green sequences and give six
+different criteria for a pair of maximal green sequences to be equivalent. We also
+show that two non-equivalent maximal green sequences can have the same sets
+of bricks. Hence, having the same set of bricks is not a sufficient criterion for
+two maximal green sequences to be equivalent.
+3.1. Preliminary lemmas. Before we start characterising equivalent maximal
+green sequences, we must prove some preliminary lemmas.
+In the following
+lemma, we collect some useful facts. Here, for T ∈ K[−1,0](proj Λ), we denote
+T ⊥1 = {X ∈ K[−1,0](proj Λ) | HomKb(proj Λ)(T, X[1]) = 0},
+⊥1T = {X ∈ K[−1,0](proj Λ) | HomKb(proj Λ)(X, T[1]) = 0}.
+Lemma 3.1. Let T = U ⊕ X and T ′ = U ⊕ Y be two two-term silting complexes
+in K[−1,0](proj Λ) such that T ′ is a green mutation of T. Then:
+(1) ⊥1T ⊂ ⊥1T ′; T ⊥1 ⊃ T ′⊥1;
+(2) X ∈ ⊥1T ′, Y ∈ T ⊥1;
+(3) X /∈ T ′⊥1, Y /∈ ⊥1T.
+Proof. (1) follows from [AI12, Theorem 2.35]. (2) then follows immediately from
+this, since clearly X ∈ ⊥1T and Y ∈ T ′⊥1. (3) then follows, since, by assumption,
+T ′ ⊕ X and T ⊕ Y cannot be silting complexes.
+□
+By iterating Lemma 3.1, we obtain the following.
+Corollary 3.2. Let G be a maximal green sequence of Λ containing two-term
+silting complexes T and T ′ where T ′ occurs after T. Furthermore, let X be an
+indecomposable summand of T which is not a summand of T ′ and let Y be an
+indecomposable summand of T ′ which is not a summand of T. Then:
+(1) ⊥1T ⊂ ⊥1T ′; T ⊥1 ⊃ T ′⊥1;
+(2) X ∈ ⊥1T ′, Y ∈ T ⊥1;
+(3) X /∈ T ′⊥1, Y /∈ ⊥1T.
+We now introduce the following notation.
+Definition 3.3. Let G be a maximal green sequence of Λ. We consider G to be
+a sequence T0, T1, . . . , Tr of green mutations of two-term silting complexes with
+T0 = Λ and Tr = Λ[1] in K[−1,0](proj Λ).
+(1) We denote by
+Ss(G) :=
+r�
+i=0
+{Indecomposable summands of Ti}/ ∼=
+the set of isomorphism classes of indecomposable complexes which occur
+as direct summands of two-term silting complexes in G.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+15
+(2) We denote by
+E(G) :=
+r�
+i=1
+{(X, Y ) | Ti−1 = E ⊕ X, Ti = E ⊕ Y }
+the set of exchange pairs of G.
+Furthermore, now consider G to be a sequence of support τ-tilting modules
+M0, M1, . . . , Mr corresponding to the two-term silting complexes Ti.
+(3) We denote by
+Sτ(G) :=
+r�
+i=0
+{Indecomposable summands of Mi}/ ∼=
+the set of isomorphism classes of indecomposable modules which occur
+as direct summands of support τ-tilting modules in G.
+Finally, consider G to be a maximal backwards Hom-orthogonal sequence of
+bricks B1, B2, . . . , Br.
+(4) We denote the set of bricks in the sequence by
+B(G) := {B1, B2, . . . , Br}.
+The following fact is well-known, but we do not know whether a proof has ap-
+peared in the literature. We use ind K[−1,0](proj Λ) to denote a set of representa-
+tives of the isomorphism classes of indecomposable complexes in K[−1,0](proj Λ).
+Lemma 3.4. Let A ∈ ind K[−1,0](proj Λ) and let G be a maximal green sequence
+of Λ. Then there is at most one exchange pair (X, Y ) ∈ E(G) such that X ∼= A
+and at most one exchange pair (X, Y ) ∈ E(G) such that Y ∼= A. Moreover, the
+exchange pair (X, A) must occur before the exchange pair (A, Y ) in G.
+Proof. Suppose for contradiction that G possesses two exchange pairs (A, X1)
+and (A, X2) which respectively correspond to green mutations from T1 to T ′
+1
+and from T2 to T ′
+2. Suppose without loss of generality that T1 occurs before T2
+in G. Then, by Corollary 3.2, we obtain that A /∈ T ′
+1
+⊥1 ⊇ T2⊥1 ∋ A, which is
+a contradiction. The case where A is the second half of the exchange pair is
+similar.
+For the final statement it suffices to observe that if (A, Y ) occurs without
+(X, A) before it, then A must be a projective, and if (X, A) occurs without (A, Y )
+after it, then A must be a shifted projective. But both cannot simultaneously
+be true.
+□
+We finish this subsection by proving the following useful result on exchange
+pairs.
+Lemma 3.5. Exchange pairs have the following properties.
+
+16
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+(1) Suppose that (X, Y ) is an exchange pair for a green mutation from T to
+T ′ in a maximal green sequence G. Then X is the unique indecomposable
+in Ss(G) ∩ T ⊥1 such that
+HomKb(proj Λ)(Y, X[1]) ̸= 0.
+(2) Conversely, let T be a two-term silting complex in a maximal green se-
+quence G. Suppose that, for Y ∈ Ss(G) ∩ T ⊥1, there exists an indecom-
+posable X ∈ Ss(G) ∩ T ⊥1, unique up to isomorphism, such that
+HomKb(proj Λ)(Y, X[1]) ̸= 0.
+Then (X, Y ) ∈ E(G).
+Proof.
+(1) Suppose that there exists an indecomposable two-term complex
+Z ∈ Ss(G) ∩ T ⊥1 such that HomKb(proj Λ)(Y, Z[1]) ̸= 0. Then Z /∈ T ′⊥1, since
+Y is a summand of T ′. Hence, by Corollary 3.2, we cannot have that Z is a
+summand of T ′ or any silting complex which occurs later in G than T ′. If Z
+leaves G before T, then Z /∈ T ⊥1, which is contrary to our assumption. We
+conclude that Z must leave G between T and T ′, and so Z ∼= X.
+(2) Now suppose that (X, Y ) is such that Y ∈ Ss(G) ∩ T ⊥1 and X is the
+unique indecomposable in Ss(G) ∩ T ⊥1 such that HomKb(proj Λ)(Y, X[1]) ̸= 0.
+The fact that X ∈ T ⊥1 means that we cannot have Y as a summand of T.
+Therefore, Y cannot occur in G before T by Corollary 3.2(3), otherwise Y /∈ T ⊥1.
+Thus, Y occurs in G after T as the second half of an exchange pair (Z, Y ).
+But then we must have that Z ∈ Ss(G) ∩ T ⊥1 by Corollary 3.2(2) and that
+HomKb(proj Λ)(Y, Z[1]) ̸= 0. We obtain that Z ∼= X, since X is the unique such
+indecomposable up to isomorphism.
+□
+3.2. Polygons. One of the equivalence relations we introduce on maximal green
+sequences corresponds to deformations across squares in 2-silt Λ. Squares are a
+special case of the larger class of “polygons” in 2-silt Λ, so we introduce all
+polygons at this juncture. Several authors have studied the notion of a polygon
+in the poset of torsion classes or two-term silting objects. A lattice-theoretic
+notion is used by Reading [Rea16], and Garver and McConville [GM19]. Our
+notion is instead based on that of Hermes and Igusa [HI19].
+Definition 3.6. A polygon in the poset 2-silt Λ is a subposet consisting of all
+two-term silting complexes possessing some presilting complex E as a direct
+summand, where |E| = |Λ| − 2. A polygon in ff-tors Λ is the image of a polygon
+in 2-silt Λ under the bijection between the two posets.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+17
+Proposition 3.7. A polygon in the poset 2-silt Λ falls under one of the four
+different types shown in Figure 1.
+Proof. The two-regularity of the Hasse diagram of the polygon follows from the
+fact that all but two summands are fixed. The existence of a unique maximum
+and minimum follows from the existence of the Bongartz and co-Bongartz com-
+pletions [AIR14, Theorem 2.10]. The cases displayed are then clearly exhaustive,
+which are as follows.
+(a) Square: the two paths from the maximum to the minimum are both of
+length two.
+(b) Oriented polygon: one path from the maximum to the minimum is of
+length two and the other is finite of length greater than two.
+(c) Unoriented polygon: both paths from the maximum to the minimum are
+finite of length greater than two.
+(d) Infinite polygon: the polygon contains infinitely many elements. Equiv-
+alently, there is at most one path of finite length from the maximum to
+the minimum.
+□
+Definition 3.8. A finite polygon in 2-silt Λ is a polygon which is not an infinite
+polygon. A finite polygon therefore has two finite paths from its maximum to
+its minimum.
+Two maximal green sequences G and G′ are related by deformation across a
+(finite) polygon if we have
+G = (T0, . . . , Ti, U1, . . . , Uk, Ti+1, . . . , Tr),
+G′ = (T0, . . . , Ti, V1, . . . , Vl, Ti+1, . . . , Tr),
+as sequences of green mutations of silting complexes, with
+Ti
+U1
+Uk
+Ti+1
+V1
+Vl
+
+18
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Figure 1. Different types of polygons
+(a) Square
+•
+•
+•
+•
+(b) Oriented polygon
+•
+•
+•
+•
+•
+2 < length < ∞
+(c) Unoriented polygon
+•
+•
+•
+•
+•
+•
+2 < length < ∞
+2 < length < ∞
+(d) Infinite polygon
+•
+•
+•
+•
+•
+•
+length = ∞
+a finite polygon.
+In the case of a finite polygon, the two paths around the polygon from the
+maximum to the minimum are the only paths between these silting complexes
+in the poset 2-silt Λ.
+Lemma 3.9. Let E ⊕ X ⊕ X′ and E ⊕ Y ⊕ Y ′ be the respective maximum and
+minimum of a polygon in 2-silt Λ. Then the only paths from E ⊕ X ⊕ X′ to
+E ⊕ Y ⊕ Y ′ in 2-silt Λ are the two paths around the polygon.
+Proof. Each vertex in the Hasse diagram of the polygon has degree two, corre-
+sponding to the two indecomposable summands of the two-term silting complex
+at the vertex which are not summands of E. Having another path from E⊕X⊕X′
+to E ⊕ Y ⊕ Y ′ would require mutating an indecomposable direct summand of E.
+However, Lemma 3.4 then precludes the path reaching E ⊕ Y ⊕ Y ′.
+□
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+19
+Any convex subposet which looks like a square or an oriented polygon must
+in fact be a square or an oriented polygon. Recall that a subposet P′ of a poset
+P is convex if whenever p ⩽ q ⩽ r with p, r ∈ P′, we have that q ∈ P′ too.
+Lemma 3.10. Any convex subposet of 2-silt Λ isomorphic to Figure 1(a) or 1(b)
+is a polygon.
+Proof. This follows from the fact that in each of these cases there is a path from
+the maximum of the poset to the minimum of the poset of length two. Since
+the covering relations from Figure 1 correspond to covering relations in 2-silt Λ,
+we must have that the maximum of the poset is two mutations away from the
+bottom of the poset, and so shares all but two summands with it. Hence, there
+exists E such that |E| = |Λ| − 2 such that E is a summand of all two-term
+silting complexes along the length-two path from maximum to minimum.
+It
+then follows from Lemma 3.4 that E must be a summand of all the two-term
+silting complexes along the other path too. We conclude that the subposet is
+indeed a polygon.
+□
+Remark 3.11. The different types of polygons correspond to the two-term silting
+theory of different algebras Γ with |Γ| = 2. Indeed, let E be a two-term presilting
+complex with |E| = |Λ| − 2. Then the complexes which complete E to a silting
+complex must lie in
+Z = (⊥0E[> 0]) ∩ (E[< 0]⊥0).
+Here
+⊥0E[> 0] = {X ∈ Kb(proj Λ) | HomKb(proj Λ)(X, E[i]) = 0 ∀i > 0},
+E[< 0]⊥0 = {X ∈ Kb(proj Λ) | HomKb(proj Λ)(E[i], X) = 0 ∀i < 0}.
+Let �
+(−): Z → Z/[E] be the ideal quotient of Z by the ideal of morphisms
+factoring through add E. We then have that Z/[E] is a triangulated category
+by [IY08, Theorem 4.2]. Further, let E ⊕ X ⊕ Y be the Bongartz completion
+of E. Then, it follows from [Jas15; IY18] that the polygon determined by E
+is isomorphic as a poset to 2-silt Γ where Γ = EndZ/[E](X ⊕ Y ) via sending a
+two-term silting complex E ⊕ X′ ⊕ Y ′ in K[−1,0](proj Λ) to the two-term silting
+complex �
+X′ ⊕ �
+Y ′ of Γ.
+This process is known as silting reduction and was
+introduced in [AI12] and given this description in [IY18].
+3.3. Deformations across squares. A particular instance of deformation
+across a polygon is deformation across a square. This will be used to define one
+notion of equivalence for maximal green sequences. In this section, we study
+deformations across squares in terms of silting complexes and bricks.
+We first note the following straightforward characterisation of deformations
+across squares in terms of chains of torsion classes.
+
+20
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Lemma 3.12. Given maximal green sequences G
+T0 ⊃ T1 ⊃ · · · ⊃ Tr−1 ⊃ Tr
+and G′
+T ′
+0 ⊃ T ′
+1 ⊃ · · · ⊃ T ′
+r′−1 ⊃ T ′
+r′
+of Λ, we have that G and G′ are related by deformation across a square if and
+only if r = r′ and there is some j such that Ti = T ′
+i for all i ̸= j, and Tj ̸= T ′
+j .
+3.3.1. Deformations across squares in terms of silting. We prove some results
+on deformations across squares from the perspective of silting.
+Lemma 3.13. If the exchange pairs in one path around the square are (X, Y )
+and then (X′, Y ′), then the exchange pairs in the path around the other side of
+the square are (X′, Y ′) and then (X, Y ).
+Proof. Let T be the two-term silting complex at the top of the square. Then
+X is certainly an indecomposable summand of T. We must also have that X′
+is an indecomposable summand of T, since we cannot have X′ ∼= Y . Hence, let
+T = E ⊕ X ⊕ X′. This gives the path around the square we know as
+E ⊕ X ⊕ X′ → E ⊕ Y ⊕ X′ → E ⊕ Y ⊕ Y ′.
+Let T ′ be the two-term silting complex in the middle of the path around the
+other side of the square. We must have that T ′ has all but one indecomposable
+summand in common with E ⊕ X ⊕ X′ and E ⊕ Y ⊕ Y ′, since it is related to
+each of these silting complexes by mutation. It is then immediate to see that
+T ′ ∼= E ⊕ X ⊕ Y ′, which gives us the result.
+□
+The following criterion for when one can swap the order of two exchange pairs
+will be useful later.
+Lemma 3.14. Let G be a maximal green sequence of Λ with an exchange pair
+(X, Y ) immediately succeeded by an exchange pair (X′, Y ′). Then one can deform
+G across a square with sides (X, Y ) and (X′, Y ′) to obtain another maximal green
+sequence G′ if and only if HomKb(proj Λ)(Y ′, X[1]) = 0.
+Proof. Let
+Ti−1
+Ti
+Ti+1
+(X,Y )
+(X′,Y ′)
+be the relevant part of G. Suppose that we can deform G across a square with
+these sides to give a maximal green sequence G′, which, by Lemma 3.13, instead
+has the sequence
+Ti−1
+T ′
+i
+Ti+1.
+(X′,Y ′)
+(X,Y )
+Then T ′
+i contains both Y ′ and X as summands, and so we must have that
+HomKb(proj Λ)(Y ′, X[1]) = 0, since T ′
+i is silting.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+21
+Now we show the reverse direction, maintaining our labelling of G as above,
+and supposing that HomKb(proj Λ)(Y ′, X[1]) = 0.
+We then have that Ti−1 =
+E ⊕ X ⊕ X′ for some E ∈ K[−1,0](proj Λ). Moreover, if we let T ′
+i = E ⊕ X ⊕ Y ′,
+then HomKb(proj Λ)(T ′
+i, T ′
+i[1]) = 0. Indeed, we have the following:
+• HomKb(proj Λ)(E ⊕ Y ′, (E ⊕ Y ′)[1]) = 0, as E ⊕ Y ′ is a summand of Ti+1;
+• HomKb(proj Λ)(E ⊕ X, (E ⊕ X)[1]) = 0, as E ⊕ X is a summand of Ti−1;
+• HomKb(proj Λ)(Y ′, X[1]) = 0 by assumption;
+• HomKb(proj Λ)(X, Y ′[1]) = 0, since Y ′ ∈ Ti⊥1 ⊂ Ti−1⊥1 by Lemma 3.1.
+By Lemma 3.4, we have Y ′ ̸∼= X. Therefore, T ′
+i has the maximal number of
+isomorphism classes of indecomposable summands. Hence T ′
+i is a silting complex
+and we obtain a maximal green sequence G by replacing the relevant portion of
+G with
+Ti−1
+T ′
+i
+Ti+1.
+(X′,Y ′)
+(X,Y )
+□
+3.3.2. Deformations across squares in terms of bricks. We now consider defor-
+mations across squares from the point of view of bricks.
+Lemma 3.15. Given a maximal green sequence G of Λ as the maximal backwards
+Hom-orthogonal sequence of bricks
+B1, B2, . . . , Br,
+a maximal green sequence G′ is related to G by deformation across a square if
+and only if G′ is given by
+B0, . . . , Bi−1, Bi+1, Bi, Bi+2, . . . , Br
+as a maximal backwards Hom-orthogonal sequence of bricks for some i.
+Proof. The maximal green sequences G and G′ are related by deformation across
+a square if and only if there is a square
+Ti−1
+Ti
+T ′
+i
+Ti+1
+such that G and G′ differ only in that G contains the left-hand path around
+the square, whilst G′ contains the right-hand path around the square. Let Bi
+and Bi+1 be the respective brick labels of the minimal inclusions Ti−1 ⊃ Ti and
+Ti ⊃ Ti+1. Then we have that [Ti−1, Ti+1] = Filt(Bi, Bi+1). Moreover, Bi and
+Bi+1 must be precisely the relatively simple objects in Filt(Bi, Bi+1) since if
+either Bi ∈ Filt(Bi+1) or Bi+1 ∈ Filt(Bi), then backwards Hom-orthogonality
+
+22
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+is violated, and there cannot be more relatively simple objects in Filt(Bi, Bi+1)
+by Proposition 2.6. By applying Proposition 2.6 to the other path around the
+square, we see that the brick labels of the other two minimal inclusions must also
+be Bi and Bi+1. We then must have that Bi+1 labels Ti−1 ⊃ T ′
+i and Bi labels
+T ′
+i ⊃ Ti+1, since Ti ̸= T ′
+i .
+□
+In order to prove the analogue of Lemma 3.14 for bricks, we need the following
+lemma.
+Lemma 3.16. Suppose that L and N are bricks over Λ such that
+HomΛ(L, N) = HomΛ(N, L) = 0.
+Then every non-split extension of L and N is a brick.
+Proof. Suppose that
+0 → L
+f−→ M
+g−→ N → 0
+is a non-split extension of L and N.
+We want to show that M is a brick,
+that is, that every non-zero morphism h: M → M is an isomorphism. Since
+HomΛ(L, N) = 0, we have that ghf = 0. Hence, by the universal properties of
+kernels and cokernels, we have a commutative diagram
+0
+L
+M
+N
+0
+0
+L
+M
+N
+0.
+f
+a
+g
+h
+b
+f
+g
+Since L and N are both bricks, we have that a and b are both either isomorphisms
+or zero. If they are both isomorphisms, then h is also an isomorphism by the
+Five Lemma.
+Hence, we suppose that at least one of a and b is not an isomorphism. Suppose
+first that a is zero and b is an isomorphism. Thus, hf = fa = 0. By the universal
+property of cokernels, we have a map s: N → M such that sg = h.
+0
+L
+M
+N
+0
+0
+L
+M
+N
+0
+f
+0
+g
+h
+b
+s
+f
+g
+We then have that gsg = gh = bg, which implies that gs = b, since g is epic. Since
+b is an isomorphism, it has an inverse b−1. We then have that gsb−1 = id, so that
+sb−1 is a section of g. But this means that the extension 0 → L → M → N → 0
+is split, which is a contradiction. The case where a is an isomorphism and b is
+zero is similar to this.
+The final case to consider is where a and b are both zero. This gives that
+hf = fa = 0, and so we have a map s: N → M such that h = sg.
+Then
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+23
+gsg = gh = bg = 0, which implies that gs = 0, since g is epic. By the universal
+property of kernels, we have that there is a map t: N → L such that s = ft.
+0
+L
+M
+N
+0
+0
+L
+M
+N
+0
+f
+0
+g
+h
+0
+s
+t
+f
+g
+However, HomΛ(N, L) = 0, so t = 0. Consequently, s = ft = 0 and, in turn, h =
+sg = 0. We conclude that every endomorphism of M is either an isomorphism
+or zero, as desired.
+□
+We apply this to show the following.
+Lemma 3.17. Let G be a maximal green sequence of Λ given by a maximal
+backwards Hom-orthogonal sequence of bricks
+B1, B2, . . . , Bi, Bi+1, . . . , Br.
+If HomΛ(Bi, Bi+1) = 0, then Ext1
+Λ(Bi+1, Bi) = 0.
+Proof. If Ext1
+Λ(Bi+1, Bi) ̸= 0, then there is a non-split short exact sequence
+0 → Bi → B → Bi+1 → 0.
+The module B here must then be a brick by Lemma 3.16.
+We claim that the sequence of bricks given by
+B1, B2, . . . , Bi−1, Bi, B, Bi+1, Bi+2, . . . , Br
+is also backwards Hom-orthogonal.
+Suppose there exists Bj with j > i + 1
+such that there is a non-zero homomorphism Bj → B. The composition Bj →
+B → Bi+1 must then be zero, by backwards Hom-orthogonality of the original
+sequence. But this gives a non-zero map Bj → Bi by the universal property of
+the kernel, which is a contradiction. One can similarly argue that there is no
+non-zero map B → Bj for j < i.
+We must finally show that there cannot be any non-zero maps Bi+1 → B
+or B → Bi. In the first case, if the composition Bi+1 → B → Bi+1 is zero,
+then there is a contradictory non-zero map Bi+1 → Bi. Hence the composition
+Bi+1 → B → Bi+1 is non-zero and cannot be an isomorphism since B is inde-
+composable. This contradicts the fact that Bi+1 is a brick. The existence of a
+non-zero map B → Bi is likewise contradictory.
+□
+The following lemma is the analogue of Lemma 3.14 for bricks: it tells us when
+we can exchange two consecutive bricks in order to deform across a square.
+
+24
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Lemma 3.18. Let G be a maximal green sequence of Λ given by a maximal
+backwards Hom-orthogonal sequence of bricks
+B1, B2, . . . , Bi, Bi+1, . . . , Br.
+Then
+B1, B2, . . . , Bi−1, Bi+1, Bi, Bi+2, . . . , Br
+is a maximal backwards Hom-orthogonal sequence of bricks if and only if
+HomΛ(Bi, Bi+1) = Ext1
+Λ(Bi, Bi+1) = 0.
+Proof. We first show that the conditions are necessary. It is immediate that we
+must have HomΛ(Bi, Bi+1) = 0 if the sequence is to remain backwards Hom-
+orthogonal. By Lemma 3.17, we then also have that Ext1
+Λ(Bi, Bi+1) = 0.
+We now show that the conditions are sufficient. We have that the new sequence
+is backwards Hom-orthogonal as HomΛ(Bi, Bi+1) = 0. Suppose now that the
+new sequence is not maximal. Since the original sequence is maximal, the only
+place where a new brick B could be added to the new sequence is between Bi+1
+and Bi. We have then that B ∈ Filt(Bi, Bi+1). Since, by Hom-orthogonality,
+we have that HomΛ(Bi, B) = HomΛ(B, Bi+1) = 0, B cannot contain Bi as a
+submodule or Bi+1 as a factor module. We let
+B = L0 ⊃ L1 ⊃ · · · ⊃ Ls = 0
+be the composition series of B in terms of Bi and Bi+1. We let Fi = Li−1/Li.
+Then we must have that F1 ∼= Bi and Fs ∼= Bi+1 by backwards Hom-
+orthogonality. Hence, there must be some j such that Fj ∼= Bi and Fj+1 ∼= Bi+1.
+We can swap the order of these two factors in the composition series if the
+subquotient Lj−1/Lj+1 ∼= Bi ⊕ Bi+1. If we continue making such swaps where
+possible, we must eventually reach a case where Lj−1/Lj+1 is not isomorphic to
+such a direct sum, since we cannot have a composition series with F1 ∼= Bi+1 or
+Fs ∼= Bi.
+Thus, there exists a module B′ = Lj−1/Lj+1 with a non-split short exact
+sequence
+0 → Bi+1 → B′ → Bi → 0.
+This gives that Ext1
+Λ(Bi, Bi+1) ̸= 0, which is a contradiction.
+□
+The following observation will be used in several proofs.
+Lemma 3.19. Suppose that G and G′ are distinct maximal green sequences such
+that B(G) ⊇ B(G′). Then there exists a pair of bricks B, B′ which are adjacent
+with B < B′ in G′, but which have B′ < B in G.
+Proof. There must be a pair of bricks of G′ which are ordered differently under
+G, since we are assuming that G and G′ are distinct. Thus, choose a pair of
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+25
+bricks B and B′ of G′ which are ordered differently in G, such that B and B′
+are as close as possible in G′. If there is a brick B′′ of G′ between B and B′,
+then one of the pairs (B, B′′) and (B′′, B′) must be ordered differently in G. But
+this contradicts the choice of B and B′ as the closest bricks which are ordered
+differently between the two sequences.
+□
+3.4. Characterising equivalent maximal green sequences. Our prelimi-
+nary work now puts us in a position to give different characterisations of equiv-
+alent maximal green sequences. The following theorem gives us six equivalent
+criteria for when a pair of maximal green sequences are equivalent.
+Theorem 3.20. Let Λ be a finite-dimensional algebra over a field K. Let G1
+and G2 be two maximal green sequences of Λ. Then the following are equivalent.
+(1) G1 and G2 are related by a finite sequence of deformations across squares.
+(2) E(G1) = E(G2).
+(3) Ss(G1) = Ss(G2).
+(4) Sτ(G1) = Sτ(G2).
+(5) For any Λ-module M, SSFG1(M) = SSFG2(M).
+(6) For any Λ-module M, SFG1(M) = SFG2(M).
+The fact that these six notions of equivalence of maximal green sequences are
+the same indicates that this is the “correct” equivalence relation to impose upon
+maximal green sequences. See also Remarks 3.23 and 3.24 for further explanation
+of the intuition behind this relation.
+Definition 3.21. Given a finite-dimensional algebra Λ over a field K with G
+and G′ two maximal green sequences of Λ, we say that G and G′ are equivalent
+if any one of the six interchangeable conditions from Theorem 3.20 holds. If G
+and G′ are equivalent, then we write G ∼ G′.
+We first show that, under one of the conditions in the statement of Theo-
+rem 3.20, maximal green sequences have the same bricks.
+Lemma 3.22. If G and G′ are maximal green sequences such that for any Λ-
+module M we have SFG(M) = SFG′(M), then we have that B(G) = B(G′).
+Proof. Let B ∈ B(G). Then {B} = SFG(B) = SFG′(B), by assumption. We must
+then have B ∈ B(G′).
+□
+We now prove our first main theorem.
+Proof of Theorem 3.20. We will prove the implications
+(1) ⇒ (2) ⇒ (3) ⇒ (1),
+(1) ⇒ (5) ⇒ (6) ⇒ (1),
+
+26
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+and the equivalence
+(3) ⇔ (4).
+It is immediate from [AIR14, Theorem 3.2] that (3) and (4) are equivalent. That
+(2) implies (3) is evident. It is also immediate that (1) implies (2) since deforming
+across a square does not change the set of exchange pairs by Lemma 3.13. Finally,
+it is clear that (5) implies (6), since the set of semistable factors determines the
+set of stable factors.
+We now show that (3) implies (1).
+Suppose that we have maximal green
+sequences G1, G2 such that Ss(G1) = Ss(G2). We will prove that G1 and G2 can be
+deformed into each other across squares by induction on the distance between
+the first point where they diverge and the end of the maximal green sequences at
+Λ[1]. The base case is when this distance is zero, so that G1 = G2, in which case
+it is trivial that the two maximal green sequences are related by deformations
+across squares.
+Now suppose that G1 ̸= G2. Consider the first exchange pairs where G1 and
+G2 diverge. Let this exchange pair be (X1, Y1) for G1 and (X2, Y2) for G2 and
+let T be the last two-term silting complex they share before they diverge. By
+Lemma 3.5, we must actually also have (X1, Y1) ∈ E(G2) and (X2, Y2) ∈ E(G1).
+Indeed, we know from the fact that (X1, Y1) is an exchange pair for G1 that
+X1 is the unique indecomposable object (up to isomorphism) in Ss(G1) ∩ T ⊥1 =
+Ss(G2)∩T ⊥1 such that HomKb(proj Λ)(Y1, X1[1]) ̸= 0, using Lemma 3.5(1). Then,
+using Lemma 3.5(2) on G2, we obtain that (X1, Y1) ∈ E(G2). The mirror-image
+of this argument shows that (X2, Y2) ∈ E(G1).
+We claim that, by deforming across squares, we can make (X2, Y2) the ex-
+change pair before (X1, Y1) in G1.
+Suppose that we cannot deform (X2, Y2)
+back past some exchange pair (A, B) in G1 which occurs after T.
+Then, by
+Lemma 3.14, we have that HomKb(proj Λ)(Y2, A[1]) ̸= 0.
+We then know that
+we cannot have A as a summand of T, since this would mean we cannot ex-
+change X2 for Y2 in T as part of G2, by Lemma 3.1(3). We further know that
+A ∈ Ss(G1) ∩ T ⊥1 by Corollary 3.2, since A occurs in G1 after T. By Lemma 3.5,
+we know that X2 is the unique indecomposable in Ss(G2) ∩ T ⊥1 = Ss(G1) ∩ T ⊥1
+such that HomKb(proj Λ)(Y2, X2[1]) ̸= 0. But then A ∼= X2, which contradicts
+Lemma 3.4.
+Hence we may deform G1 across squares to obtain a maximal green sequence
+G′
+1 where (X1, Y1) is the exchange pair immediately following (X2, Y2). But then,
+G′
+1 and G2 agree on a longer initial segment than G1 and G2, so, by the induction
+hypothesis, we can deform G′
+1 into G2 across squares. This then gives that we
+can deform G1 into G2 across squares, which establishes the claim.
+We now show that (1) implies (5) by showing that the factors of Harder–
+Narasimhan filtrations are preserved by deformations across squares. We start
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+27
+with a maximal backwards Hom-orthogonal sequence of bricks G1
+B1, B2, . . . , Bi, Bi+1, . . . , Br
+and deform across a square to obtain a sequence G2 given by
+B1, B2, . . . , Bi−1, Bi+1, Bi, Bi+2, . . . , Br.
+By Lemma 3.18, we have that
+HomΛ(Bi, Bi+1) = HomΛ(Bi+1, Bi)
+= Ext1
+Λ(Bi, Bi+1) = Ext1
+Λ(Bi+1, Bi) = 0.
+We let M be a Λ-module and show that SSFG1(M) = SSFG2(M). If Filt(Bi) ∩
+SSFG1(M) = ∅ or Filt(Bi+1) ∩ SSFG1(M) = ∅, then the same filtration is also
+the Harder–Narasimhan filtration of M by G2. Suppose then that the Harder–
+Narasimhan filtration of M by G1 is
+M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0
+where Fk = Mk−1/Mk ∈ Filt(Bjk) for some jk for all 1 ⩽ k ⩽ l, with
+j1 < j2 < · · · < jl.
+Suppose that jk = i and jk+1 = i+1. Since Ext1
+Λ(Bi, Bi+1) = 0, the subquotient
+Mk−1/Mk+1 is isomorphic to Fk ⊕ Fk+1. Hence, we may replace Mk by M′
+k such
+that Mk−1/M′
+k = Fk+1 and M′
+k/Mk+1 = Fk. The filtration obtained is then
+the Harder–Narasimhan filtration of M by G2, since we have Fk+1 ∈ Filt(Bi+1),
+Fk ∈ Filt(Bi) with Fk+1 occurring before Fk in the filtration. This has the same
+factors as the Harder–Narasimhan filtration of M by G1, so that SSFG1(M) =
+SSFG2(M), as desired.
+Finally, we show that (6) implies (1). Note first that this implies that B(G1) =
+B(G2) by Lemma 3.22. We now show that one can deform G1 across squares to
+obtain G2 by swapping adjacent bricks. By Lemma 3.19, we have adjacent bricks
+B < B′ in G1 which are ordered B′ < B in G2.
+Since both sequences are
+backwards Hom-orthogonal, we have
+HomΛ(B, B′) = HomΛ(B′, B) = 0.
+If we have Ext1
+Λ(B, B′) ̸= 0, then, by Lemma 3.16, we have that there is a brick
+M which is given by a non-split extension 0 → B′ → M → B → 0. But then, by
+uniqueness of Harder–Narasimhan filtrations, this short exact sequence must give
+the Harder–Narasimhan filtration of M with respect to G1. However, the Harder–
+Narasimhan filtration of M with respect to G2 cannot have the same bricks,
+otherwise there is a non-split extension 0 → B → M → B′ → 0. This gives a
+endomorphism M → B′ → M of M which is neither zero nor an isomorphism,
+contradicting the fact that M is a brick. This contradicts the assumption that
+
+28
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+the Harder–Narasimhan filtrations with respect to G1 and G2 must have the same
+multisets of bricks.
+Hence, we have that Ext1
+Λ(B, B′) = 0, so by Lemma 3.18, we can deform G1
+across a square by swapping B and B′. By repeating this process, we eventually
+deform G1 into G2 across a sequence of squares.
+□
+Remark 3.23. Note that, if Λ is a Jacobian algebra of a quiver with potential,
+there is a cluster algebra associated to Λ. Its cluster variables will be in bijec-
+tion with the reachable indecomposable presilting complexes in K[−1,0](proj Λ)
+[AIR14; BY13; CK06].
+Condition (3) in Theorem 3.20 means that maximal
+green sequences are equivalent in the sense of Definition 3.21 if and only if the
+corresponding sets of cluster variables appearing in the clusters in the mutation
+sequences coincide.
+Remark 3.24. It is shown in [Rei10] that, given a path algebra of a simply-laced
+Dynkin diagram, maximal green sequences correspond to products of quantum
+dilogarithms and that the value of this product is in fact independent of the
+maximal green sequence chosen. These products of quantum dilogarithms give
+so-called “refined Donaldson–Thomas invariants”. See also [Kel11; KD20] for
+more general statements and further references. Maximal green sequences which
+are related by deformation across a square correspond to products of quantum
+dilogarithms which differ by swapping two adjacent commuting terms in the
+product. Hence, maximal green sequences which are equivalent are related by
+finitely many such swaps. It is natural to consider such products as the same.
+This is analogous to considering reduced words of longest elements of Weyl groups
+up to commutation, as we will explore in a sequel paper [GW].
+Thanks to Theorem 3.20, Lemma 3.22 implies the following.
+Corollary 3.25. If G ∼ G′ are equivalent maximal green sequences, then we
+have that B(G) = B(G′).
+The converse is false, as is shown in the following example. Maximal green
+sequences which have the same set of bricks are not necessarily equivalent. Hence,
+our notion of equivalence does not coincide with the notion of “weak equivalence”
+from [Qiu15, p.257].
+Example 3.26. Consider the path algebra of the quiver
+1 ← 2 → 3,
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+29
+where we use the convention of composing arrows as if they were functions, so
+that α−→
+β−→= βα. The Auslander–Reiten quiver of this algebra is
+2
+2
+3
+2
+1
+2
+1 3
+1.
+3
+The lattice of its two-term silting complexes is shown in Figure 2 and the lattice
+of its torsion classes is shown in Figure 3.
+Consider the two maximal green
+sequences given by the maximal backward Hom-orthogonal sequences of bricks
+2, 2
+1, 1, 2
+3, 3
+and
+2, 2
+3, 3, 2
+1, 1.
+These two maximal green sequences have the same set of bricks, but inspection
+shows that one cannot transform one into the other by deforming across squares.
+Indeed, these two maximal green sequences have different sets of summands,
+namely
+�
+2
+1, 2, 2
+3, 2
+1 3, 3, 2
+1[1], 2[1], 2
+3[1]
+�
+and
+�
+2
+1, 2, 2
+3, 2
+1 3, 1, 2
+1[1], 2[1], 2
+3[1]
+�
+.
+Because of Lemma 3.22, one can regard having the same stable factors as an
+augmentation of the condition of having the same bricks. This augmentation is
+required to determine the equivalence class of the maximal green sequence.
+In some ways it is surprising that maximal green sequences may have the same
+set of bricks whilst having different sets of τ-rigids. Theorem 2.1 shows that the
+τ-rigids form the relative projectives of the torsion classes, whilst Theorem 2.7
+shows that the bricks form the relative simples of the torsion classes.
+Hence
+there is no duality between simples and projectives on the level of maximal
+green sequences. It is known from [Eno22, Example 6.17] that, in general, there
+is no duality between simples and projectives within torsion classes — see also
+[Eno22, Theorem 5.10]. But it is still not obvious why the bricks of a maximal
+green sequence should give less information than the τ-rigids.
+As shown in [Eno22, Corollary 5.15], the existence of the duality between
+simples and projectives — at least in the sense of having the same number of
+simples as indecomposable projectives — in a functorially finite torsion class T
+
+30
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Figure 2. The lattice of two-term silting complexes from Exam-
+ple 3.26
+2
+1 ⊕ 2 ⊕ 2
+3
+2
+1[1] ⊕ 2 ⊕ 2
+3
+2
+1 ⊕ 2
+1 3 ⊕ 2
+3
+2
+1 ⊕ 2 ⊕ 2
+3[1]
+3 ⊕ 2
+1 3 ⊕ 2
+3
+2
+1 ⊕ 2
+1 3 ⊕ 1
+2
+1[1] ⊕ 3 ⊕ 2
+3
+3 ⊕ 2
+1 3 ⊕ 1
+2
+3[1] ⊕ 2
+1 ⊕ 1
+3 ⊕ 2[1] ⊕ 1
+2
+1[1] ⊕ 3 ⊕ 2[1]
+2
+1[1] ⊕ 2 ⊕ 2
+3[1]
+2
+3[1] ⊕ 2[1] ⊕ 1
+2
+1[1] ⊕ 2[1] ⊕ 2
+3[1].
+in mod Λ, for an Artin algebra Λ, is equivalent to the Jordan–H¨older property
+(JHP) for this torsion class considered as an exact category.
+Here the exact
+structure is the one induced by the embedding T ֒→ mod Λ. Enomoto shows
+that functorially finite torsion classes of Nakayama algebras possess the JHP
+[Eno22, Corollary 5.19]; we show in Section 5 that for these algebras the set of
+bricks does determine the equivalence class of the maximal green sequence.
+One may wonder whether the set of bricks determines the equivalence class of
+the maximal green sequence if and only if every functorially finite torsion class
+in mod Λ satisfies the JHP. [Eno21, Example 2.9, Example 4.17] shows that the
+“only if” part cannot be true. Namely, the example in [Eno21] gives a torsion
+class in the preprojective algebra of type D4 which does not satisfy the JHP. On
+the other hand, in a sequel to this paper [GW], we will show that the converse of
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+31
+Figure 3. The lattice of torsion classes from Example 3.26 with
+brick labelling
+add
+�
+2
+1, 2
+1 3, 2
+3, 1, 2, 3
+�
+add
+�
+2, 2
+3, 3
+�
+add
+�
+2
+1, 2
+1 3, 2
+3, 1, 3
+�
+add
+�
+2, 2
+1, 1
+�
+add
+�
+2
+1 3, 2
+3, 1, 3
+�
+add
+�
+2
+1, 2
+1 3, 1, 3
+�
+add
+�
+2
+3, 3
+�
+add
+�
+2
+1 3, 1, 3
+�
+add
+�
+2
+1, 1
+�
+add
+�
+1, 3
+�
+add
+�
+3
+�
+add
+�
+2
+�
+add
+�
+1
+�
+{0}.
+1
+2
+3
+2
+1
+2
+3
+2
+2
+3
+1
+2
+3
+1
+2
+1
+3
+2
+1 3
+2
+3
+2
+1
+1
+3
+3
+2
+1
+
+32
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Lemma 3.22 holds for all preprojective algebras of Dynkin type. Hence, there are
+examples where the converse of Lemma 3.22 holds whilst there exist functorially
+finite torsion classes for which the JHP fails. We conjecture that the “if” part
+is true, motivated by the case of Nakayama algebras.
+Conjecture 3.27. Let Λ be an Artin algebra such that every functorially finite
+torsion class in mod Λ satisfies the JHP. Then two maximal green sequences G
+and G′ have the same set of bricks if and only they are equivalent.
+Note that the JHP does not hold for some functorially finite torsion classes of
+the non-linearly oriented A3 algebra considered in Example 3.26. Indeed, in the
+torsion class
+add
+�
+2
+1,
+2
+1 3, 2
+3, 1, 3
+�
+the brick
+2
+1 3 has two different composition series: one with factors
+3 and 2
+1,
+and the other with factors
+1 and 2
+3.
+Consequently, there is no bijection between relative projectives and relative sim-
+ples. There are three relative projectives
+2
+1,
+2
+1 3, and 2
+3,
+while there are four relative simples
+2
+1, 2
+3, 1, and 3.
+Remark 3.28. Another plausible definition of equivalence of maximal green se-
+quences which fails to coincide with those of Theorem 3.20 is as follows. Recall
+from Section 2.3.5 that a maximal green sequence can be specified as the sequence
+of vertices mutated at. One might conjecture that two maximal green sequences
+are equivalent if and only if the underlying multisets of these sequences coincide.
+It is clear that these multisets are preserved by deformation across squares. How-
+ever, these multisets may coincide for inequivalent maximal green sequences. For
+instance, the maximal green sequence down the left-hand side of Figure 2 has
+sequence
+1, 2, 3, 2,
+while the sequence down the right-hand side has sequence
+3, 2, 1, 2.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+33
+Figure 4. An increasing elementary polygonal deformation,
+where X and A are indecomposable
+. . .
+E ⊕ X ⊕ A
+E ⊕ Z ⊕ A
+E ⊕ Z ⊕ C
+E ⊕ X ⊕ C′
+E ⊕ X′ ⊕ C
+. . .
+. . .
+The underlying multisets are the same here, but the maximal green sequences
+are not equivalent.
+4. Partial orders on equivalence classes
+The equivalence relation on maximal green sequences defined in the previous
+section reveals more structure on the set of maximal sequences. Indeed, this
+equivalence relation allows one to define partial orders on the equivalence classes.
+In this section, we define three such orders. The first uses deformations across
+oriented polygons; the second uses reverse-inclusion of summands; the third
+uses refinement of Harder–Narasimhan filtrations. We show that the first order
+implies the second and the third, but we conjecture that the three actually
+coincide.
+4.1. Deformations across oriented polygons. The first of these partial or-
+ders has covering relations given by deformations across oriented polygons.
+Definition 4.1. Let G and G′ be maximal green sequences of Λ. If G and G′
+only differ in that G contains the path of length greater than two around an
+oriented polygon, whilst G′ contains the length-two path, then we say that G′ is
+an increasing elementary polygonal deformation of G. By extension, we also say
+that [G′] is an increasing elementary polygonal deformation of [G]. Similarly, we
+say that G is a decreasing elementary polygonal deformation of G′ and that [G]
+is a decreasing elementary polygonal deformation of [G′].
+Note that an increasing elementary polygonal deformation decreases the length
+of the maximal green sequence. One can think of it instead as increasing the
+speed of the maximal green sequence.
+4.1.1. Deformations across oriented polygons in terms of bricks. Just as we in-
+terpreted deformations across squares in terms of maximal backwards Hom-
+orthogonal sequences of bricks, we also wish to do the same for increasing ele-
+mentary polygonal deformations.
+
+34
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Lemma 4.2. Let Λ be a finite-dimensional algebra over a field K. Suppose that
+V
+T
+U
+V′
+1
+. . .
+V′
+r
+M
+L
+B1
+B2
+Br−1
+Br
+is an oriented polygon in ff-tors Λ with its brick labels.
+Then B1 ∼= M and
+Br ∼= L.
+Proof. We have that [T , U] = Filt(L, M) = Filt(B1, B2, . . . , Br), recalling the
+notation [T , U] from Section 2.4.2. We then have that L and M must be precisely
+the relatively simple objects of [T , U], since neither brick can admit a filtration
+with the other as factors without violating the brick condition or the backwards
+Hom-orthogonality condition.
+It then follows from Proposition 2.6 that L and M must occur as elements
+of the set {B1, B2, . . . , Br}. It is clear that L cannot occur before M amongst
+these bricks, otherwise L, M cannot be a maximal backwards Hom-orthogonal
+sequence of bricks in [T , U]. If we do not have B1 ∼= M and Br ∼= L, then we also
+get a contradiction to backwards Hom-orthogonality, since all of B1, B2, . . . , Br
+have filtrations with L and M as factors.
+□
+One can view an increasing elementary polygonal deformation as swapping
+maximal green sequences in an abelian category with two simple objects. Such
+a category has at most two maximal green sequences up to equivalence, and in
+our case it has precisely two, where one has length two and the other is longer.
+Lemma 4.3. In the situation of the polygon from Lemma 4.2, we have the
+following.
+(1) Filt(L, M) is an abelian category.
+(2) The two paths around the polygon give two maximal green sequences of
+Filt(L, M).
+(3) The two paths around the polygon give two different sets of Harder–
+Narasimhan filtrations of Filt(L, M).
+Proof. We have HomΛ(L, M) = HomΛ(M, L) = 0 by backwards Hom-orthogon-
+ality of maximal green sequences going through the sides of the polygon. Then
+(1) follows from [Rin76, 1.2], which states that Filt(−) of a set of Hom-orthogonal
+bricks is an abelian category.
+For (2), it follows from the definition of brick
+labels that the sequences of bricks given by the two paths around the polygon
+must be backwards Hom-orthogonal in Filt(L, M). For (3), the fact that the
+two paths around the polygon give Harder–Narasimhan filtrations on Filt(L, M)
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+35
+then follows from (2) by [Igu20; Tre20]. The fact that the two sets of Harder–
+Narasimhan filtrations must be different then follows from the fact that in the
+longer path around the polygon, B2 is the only factor in its filtration, whilst this
+cannot be the case for the shorter path around the polygon.
+□
+The interpretation of increasing elementary polygonal deformations is then as
+follows.
+Lemma 4.4. A maximal green sequence G′ is an increasing elementary polygonal
+deformation of a maximal green sequence G if and only if, as maximal backwards
+Hom-orthogonal sequences of bricks, we have that G′ is
+B1, B2, . . . , Br
+whilst G is
+B1, . . . , Bi−1, Bi+1, B′
+1, . . . , B′
+s, Bi, Bi+2 . . . , Br
+for s ⩾ 1.
+Proof. The forwards direction is Lemma 4.2. For the backwards direction, let
+T be the basic two-term silting complex corresponding to the torsion class
+Tors(Bi, Bi+1, . . . , Br) and T ′ be the basic two-term silting complex correspond-
+ing to the torsion class Tors(Bi+2, . . . , Br). Then T ′ is obtained from T by two
+green mutations since the corresponding torsion classes differ by two minimal
+inclusions. Hence T ∼= E ⊕ X ⊕ X′ and T ′ ∼= E ⊕ Y ⊕ Y ′ where X, X′, Y , and
+Y ′ are all indecomposable. Then, by Lemma 3.9, the only other path from T
+to T ′ in 2-silt Λ is the other path around the polygon determined by E. This
+must be the path taken in G. Since s ⩾ 1, the polygon determined by E is ori-
+ented. Hence G and G′ only differ in that G′ contains the length two path around
+the oriented polygon whilst G contains the longer path, so G′ is an increasing
+elementary polygonal deformation of G.
+□
+4.2. Partial orders. We can now define the three partial orders on equivalence
+classes of maximal green sequences.
+Definition 4.5.
+(1) The partial order ⩽� on equivalence classes of maximal
+green sequences is defined via its covering relations, which are that [G]⋖�
+[G′] if and only if [G′] is an increasing elementary polygonal deformation
+of [G]. We refer to this as the deformation partial order.
+(2) The partial order ⩽S is defined via [G] ⩽S [G′] if and only if Ss(G) ⊇
+Ss(G′). This is evidently also equivalent to having Sτ(G) ⊇ Sτ(G′). We
+refer to this as the summand partial order.
+
+36
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+(3) The partial order ⩽HN is defined by [G] ⩽HN [G′] if and only if, for any
+module M, we have
+SFG′(M) =
+�
+B∈SFG(M)
+SFG′(B).
+If [G] ⩽HN [G′], then we say that the G′-HN filtrations refine the G-HN
+filtrations.
+Informally, [G] ⩽HN [G′] if the G′-stable factors of any Λ-
+module M can be obtained by breaking up the G-stable factors of M
+into their G′-stable factors. We refer to this as the Harder–Narasimhan
+or HN partial order.
+Remark 4.6. As shown in [Wil22a, Theorem 4.3.1, Theorem 4.4.4] [Wil22b, The-
+orem 3.4, Theorem 5.6], the deformation order here should be seen as a higher-
+dimensional incarnation of the order on silting complexes given by green muta-
+tion, whilst the summand partial order should be seen as a higher-dimensional
+incarnation of the order on silting complexes given by inclusion of aisles [AI12,
+Definition 2.10, Theorem 2.11]. These orders are known to have the same Hasse
+diagram [AI12, Theorem 2.35]. Analogous orders exist on tilting modules [RS91,
+2.2] and support τ-tilting pairs [AIR14, Section 2.4], which are likewise known
+to have the same Hasse diagram [HU05, Theorem 2.1] [AIR14, Theorem 2.33].
+The Harder–Narasimhan order is new and does not have an analogue on silting
+complexes.
+The other new feature here is of course that one must introduce the equivalence
+relation on maximal green sequences in order to see the partial orders. Note that
+a partial order on equivalence classes is exactly the same thing as a preorder, so
+one could instead consider preorders on maximal green sequences. We prefer to
+keep the equivalence relation and the partial orders conceptually separate.
+Remark 4.7. The reason why we must use the stable factors rather than the
+semistable factors to define the Harder–Narasimhan order is as follows.
+The
+plausible alternative definition using the semistable factors would be that [G] ⩽HN
+[G′] if and only if for all Λ-modules M, we have that
+SSFG′(M) =
+�
+F ∈SSFG(M)
+SSFG′(F).
+Consider the path algebra of the A2 quiver 1 ← 2. This has two maximal green
+sequences, namely G given by the sequence of bricks
+2, 2
+1, 1,
+and G′ given by the sequence of bricks
+1, 2.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+37
+Then we have that [G] ⩽HN [G′], but we have
+SSFG′
+�
+2 ⊕ 2
+1�
+= {1, 2 ⊕ 2}
+̸= {1, 2, 2}
+= SSFG′
+�
+2
+1�
+⊔ SSFG′(2)
+=
+�
+F ∈SSFG
+�
+2⊕2
+1� SSFG′(F).
+In order for this to work correctly, we need to break up 2 ⊕ 2 into 2, 2 by con-
+sidering stable factors rather than semistable factors.
+Remark 4.8. Given two maximal green sequences G and G′ such that [G] ⩽HN
+[G′], one might wonder whether, up to equivalence, the G′-HN filtration of a
+Λ-module M can be obtained from the G-HN filtration of M by breaking up
+the G-semistable factors according to their G′-HN filtrations, without doing any
+rearranging of the orders of the factors. To be more precise, suppose that, up to
+equivalence, the G-HN filtration of M is
+M = M0 ⊃ M1 ⊃ · · · ⊃ Ml−1 ⊃ Ml = 0
+with Fi := Mi−1/Mi, and suppose that the G′-HN filtration of each Fi is
+Fi = Li0 ⊃ Li1 ⊃ · · · ⊃ Lili = 0
+with Hij := Li(j−1)/Lij. One might hope that, again up to equivalence, the
+G′-HN filtration of M is
+M = M10 ⊃ M11 ⊃ . . . M1(l1−1) ⊃ M20 ⊃ . . .
+⊃ Mi0 ⊃ · · · ⊃ Mi(li−1) ⊃ · · · ⊃ Ml(ll−1) = 0
+where Mi(j−1)/Mij = Hij for 1 ⩽ i ⩽ l and 0 ⩽ j ⩽ li −2 and Mi(li−1)/M(i+1)0 =
+Hili for 1 ⩽ i ⩽ l − 1.
+Unfortunately, this is not generally true. In general, one has to reorganise
+the semistable factors Hij to obtain the G′-HN filtration, even if one replaces G
+and G′ by equivalent maximal green sequences. This is shown in the following
+example. Consider the path algebra of the following algebra of type �A4.
+2
+4
+1
+3
+This algebra has a maximal green sequence G given by the sequence of bricks
+1, 1
+2, 1
+4,
+1
+2 4, 3, 3
+2, 2, 3
+4, 4
+
+38
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+and a maximal green sequence G′ given by the sequence of bricks
+4, 2, 3, 1.
+Since the bricks of G′ are precisely the simple modules, it is clear that we have
+[G] ⩽HN [G′]. Now consider the module
+M = 1 3
+2 4.
+The G-HN filtration of M is
+1 3
+2 4 ⊃ 3
+4 ⊃ 0,
+with SSFG(M) =
+�
+1
+2, 3
+4�
+, whilst the G′-HN filtration of M is
+1 3
+2 4 ⊃ 1 3
+2 ⊃ 1 ⊕ 3 ⊃ 1 ⊃ 0,
+with SSFG′(M) = {1, 2, 3, 4}.
+Now, if we were to try to construct the G′-HN filtration of M as above, by
+sticking together the G′-HN filtration of
+1
+2
+with the G′-HN filtration of
+3
+4 ,
+then the G′-semistable factors would appear in the order 2, 1, 4, 3, whereas in
+actuality they appear in the order 4, 2, 3, 1. Moreover, no amount of deformation
+across squares for either G or G′ can change this. Indeed, the G-HN filtration of
+M is unique in the equivalence class, whilst deformation across squares cannot
+change the order in which 1 and 4 occur in G′, since Ext1
+Λ(4, 1) ̸= 0.
+The HN order on maximal green sequences implies inclusion of bricks.
+Lemma 4.9. If [G] ⩽HN [G′], then B(G) ⊇ B(G′). Furthermore, if [G] <HN [G′],
+then B(G) ⊃ B(G′).
+Proof. Let B′ ∈ B(G′), so that SFG′(B′) = {B′}. Since [G] ⩽HN [G′], we have
+{B′} = SFG′(B′)
+=
+�
+B∈SFG(B′)
+SFG′(B).
+This means that we must have #SFG(B′) = 1, so SFG(B′) = {B′}. This gives
+that B′ ∈ B(G), as desired.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+39
+For the second statement, suppose that we have [G] <HN [G′], so that, in
+particular, there exists a module M such that SFG(M) ̸= SFG′(M). However, we
+still have refinement, so that
+SFG′(M) =
+�
+B∈SFG(M)
+SFG′(B).
+Therefore, we must have some B ∈ SFG(M) such that SFG′(B) ̸= {B}, otherwise
+the refinement property would give us that SFG′(M) = SFG(M). But this implies
+that we cannot have B ∈ B(G′). We conclude that B(G) ⊃ B(G′).
+□
+The converse to the statement of Lemma 4.9 is not true in general: we might
+have [G] and [G′] incomparable in the Harder–Narasimhan partial order, but such
+that B(G) ⊃ B(G′). See Remark 4.17 for an example in type �A4.
+Proposition 4.10.
+(1) The relation ⩽� is a well-defined partial order.
+(2) The relation ⩽S is a well-defined partial order.
+(3) The relation ⩽HN is a well-defined partial order.
+Proof. In each case, what needs to be shown is that the relation respects equiv-
+alence classes, and that it is reflexive, anti-symmetric, and transitive.
+(1) By construction, ⩽� respects equivalence classes and is reflexive and tran-
+sitive.
+To show that ⩽� is anti-symmetric, we must show that there can be
+no sequence of covering relations which gives a cycle. But this is clear, since
+an increasing elementary polygonal deformation reduces the length of the maxi-
+mal green sequence by at least one, and length is invariant under equivalence of
+maximal green sequences.
+(2) The relation ⩽S is clearly reflexive, anti-symmetric, and transitive. To see
+that ⩽S respects equivalence classes, it suffices to note that the set of summands
+of a maximal green sequence is invariant under equivalence by Theorem 3.20.
+(3) It follows from Theorem 3.20 that ⩽HN respects equivalence classes, since
+SFG(M) is invariant on the equivalence class of G. Reflexivity then follows from
+the fact that
+SFG(M) =
+�
+B∈SFG(M)
+SFG(B),
+since SFG(B) = {B} for B ∈ B(G).
+We now show that ⩽HN is transitive.
+Suppose that there are equivalence
+classes of maximal green sequences [G], [G′], and [G′′] such that [G] ⩽HN [G′] and
+
+40
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+[G′] ⩽HN [G′′], and again let M be a Λ-module. Then
+SFG′′(M) =
+�
+B′∈SFG′(M)
+SFG′′(B′)
+∵ [G′] ⩽HN [G′′]
+=
+�
+B′∈�
+B∈SFG(M) SFG′(B)
+SFG′′(B′)
+∵ [G] ⩽HN [G′]
+=
+�
+B∈SFG(M)
+�
+B′∈SFG′(B)
+SFG′′(B′)
+=
+�
+B∈SFG(M)
+SFG′′(B).
+∵ [G′] ⩽HN [G′′] for B
+Hence [G] ⩽HN [G′′], so ⩽HN is transitive.
+We now show that ⩽HN is anti-symmetric. Suppose that there are equiva-
+lence classes of maximal green sequences [G] and [G′] such that [G] ⩽HN [G′] and
+[G′] ⩽HN [G], and let M be a Λ-module. By Lemma 4.9, we have B(G) ⊇ B(G′) ⊇
+B(G), so B(G) = B(G′). Then
+SFG′(M) =
+�
+B∈SFG(M)
+SFG′(B)
+∵ [G] ⩽HN [G′]
+=
+�
+B∈SFG(M)
+{B}
+∵ B(G) = B(G′)
+= SFG(M).
+Thus [G] = [G′], by Theorem 3.20, so ⩽HN is anti-symmetric.
+□
+4.2.1. Comparing the orders. The three partial orders each have their own ad-
+vantages. The HN partial order has the most interesting defining property, but
+it is in principle difficult to compute because it requires checking many filtrations
+(in general, infinitely many). It is likewise difficult to verify whether there is a
+sequence of increasing elementary polygonal deformations relating two maximal
+green sequences. In contrast, the partial order ⩽S is the easiest to compute, since
+one only has to compare two finite sets. The advantage of the order ⩽� is that
+its local structure is clear, since we know its covering relations. This also makes
+it the easiest order to prove things about. Knowing that these orders were the
+same would give a single partial order on equivalence classes of maximal green
+sequences with all of these virtues.
+Conjecture 4.11. For a finite-dimensional K-algebra Λ and two maximal green
+sequences G and G′ of Λ, the following are equivalent.
+(1) [G] ⩽� [G′].
+(2) [G] ⩽S [G′].
+(3) [G] ⩽HN [G′].
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+41
+Note that the orders on silting complexes analogous to these orders on equiv-
+alence classes of maximal green sequences discussed in Remark 4.6 in general
+only have the same Hasse diagram, rather than being equal. This is because
+the order on silting complexes given by green mutations only applies when the
+sequence of mutations is finite, whereas there may be inclusion of aisles between
+two silting complexes not related by such a finite sequence of mutations. In the
+present case, we believe that the orders should actually be equal to each other,
+since two maximal green sequences related by the orders should only be related
+by a finite sequence of covering relations.
+Remark 4.12. Conjecture 4.11 can also be seen as a refined version of the “no-
+gap” conjecture [BDP14, Conjecture 1.22], cases of which were proven in [GM19;
+HI19]. Said conjecture says that the set of lengths of maximal green sequences
+of a quiver should have no gaps. In these cases, the only oriented polygons are
+pentagons, and so increasing elementary polygonal deformations only change
+the length of the maximal green sequence by one. For an acyclic quiver Q, there
+exists a maximal green sequence G′ of KQ given by only mutating at sources.
+Furthermore, G′ is of minimal length, and for any other maximal green sequence
+G, we have that Ss(G) ⊇ Ss(G′), since Ss(G′) only consists of projectives and
+shifted projectives.
+Conjecture 4.11 would then give that [G] ⩽� [G′], which
+gives a set of maximal green sequences of every length between that of G and
+G′, thereby establishing the “no-gap” conjecture for acyclic quivers. However,
+Conjecture 4.11 applies to a much larger class of algebras, including those of
+non-simply-laced type and those which are not Jacobian algebras of quivers with
+potential.
+Example 4.13. We consider the algebra from Example 3.26.
+The poset of
+its equivalence classes of maximal green sequences under reverse-inclusion of
+summands is shown in Figure 5.
+It can be verified that equivalence classes
+related by a covering relation in this order are also related by an increasing
+elementary polygonal deformation, so that ⩽S coincides with ⩽� in this instance.
+By computing the Harder–Narasimhan filtrations for the indecomposables, one
+can also check that ⩽HN coincides with the other two orders here.
+We give some particular examples for the deformation order and Harder–
+Narsimhan order. The maximal green sequences G and G′ with
+Ss(G) =
+�
+1, 2
+1, 2, 2
+3, 2
+1[1], 2[1], 2
+3[1]
+�
+and
+Ss(G′) =
+�
+2
+1, 2, 2
+3, 2
+1[1], 2[1], 2
+3[1]
+�
+
+42
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+are such that [G]⋖� [G′] due to the increasing elementary polygonal deformation
+across the oriented polygon
+2
+1 ⊕ 2 ⊕ 2
+3[1]
+2
+1[1] ⊕ 2 ⊕ 2
+3[1]
+2
+1[1] ⊕ 2[1] ⊕ 2
+3[1]
+2
+3[1] ⊕ 2
+1 ⊕ 1
+2
+3[1] ⊕ 2[1] ⊕ 1
+as can be seen from Figure 2.
+As sequences of bricks, we have that G is
+3, 2, 2
+1, 1
+and G′ is
+3, 1, 2.
+The fact that [G] <HN [G′] can be seen from the fact that
+SFG
+�
+2
+1�
+=
+�
+2
+1�
+and
+SFG′
+�
+2
+1�
+= {1, 2}.
+Hence, we have
+SFG′
+�
+2
+1�
+= {1, 2}
+=
+�
+B∈
+�
+2
+1� SFG′(B)
+=
+�
+B∈SFG
+�
+2
+1� SFG′(B).
+An important non-example for the Harder–Narasimhan order is given by the
+maximal green sequences G1
+2, 2
+1, 1, 2
+3, 3
+and G2
+2, 2
+3, 3, 2
+1, 1
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+43
+Figure 5. The poset of reverse-inclusion of indecomposable di-
+rect summands of two-term silting complexes from Example 4.13
+�
+2
+1, 2, 2
+3, 2
+1[1], 2[1], 2
+3[1]
+�
+�
+1, 2
+1, 2, 2
+3, 2
+1[1], 2[1], 2
+3[1]
+�
+�
+1, 2
+1,
+2
+1 3, 2, 2
+3, 2
+1[1], 2[1], 2
+3[1]
+�
+�
+2
+1, 2, 2
+3, 3, 2
+1[1], 2[1], 2
+3[1]
+�
+�
+2
+1,
+2
+1 3, 2, 2
+3, 3, 2
+1[1], 2[1], 2
+3[1]
+�
+�
+1, 2
+1,
+2
+1 3, 2, 2
+3, 3, 2
+1[1], 2[1], 2
+3[1]
+�
+from Example 3.26. In this case, we have that
+SFG1
+�
+2
+1 3�
+=
+�
+1, 2
+3�
+and
+SFG2
+�
+2
+1 3�
+=
+�
+3, 2
+1�
+.
+Hence, one can show that [G1] ̸⩽HN [G2], since
+SFG2
+�
+2
+1 3�
+=
+�
+3, 2
+1�
+̸=
+�
+1, 2
+3�
+=
+�
+B∈SFG1
+�
+2
+1 3� SFG2(B).
+The argument that [G2] ̸⩽HN [G1] is similar.
+We show that the summand order holds whenever the deformation order holds.
+Theorem 4.14. Let Λ be a finite-dimensional algebra over a field K with G and
+G′ two maximal green sequences of Λ. Then [G] ⩽� [G′] implies that [G] ⩽S [G′].
+Proof. It suffices to show that [G] ⋖� [G′] implies [G] ⩽S [G′]. Hence, we shall
+show that if G′ is an increasing elementary polygonal deformation of G, then
+Ss(G) ⊇ Ss(G′). Locally, an increasing elementary polygonal deformation looks as
+
+44
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+in Figure 4. The maximal green sequence passing along the top of this polygon is
+G and the maximal green sequence passing along the bottom is G′. By inspection,
+every summand from the bottom path also occurs in the top path.
+Hence,
+Ss(G) ⊇ Ss(G′), as desired.
+□
+We now show that the deformation order also always implies the Harder–
+Narasimhan order.
+Theorem 4.15. Let Λ be a finite-dimensional algebra over a field K with G and
+G′ two maximal green sequences of Λ. Then [G] ⩽� [G′] implies that [G] ⩽HN [G′].
+Proof. It suffices to show that [G] ⋖� [G′] implies [G] ⩽HN [G′], so we suppose
+that G′ is an increasing elementary polygonal deformation of G. Let B and B′
+be the brick labels for the short path around the polygon, so that B < B′ in
+G′, but B′ < B in G.
+Further, let M be a Λ-module.
+If B /∈ SFG′(M) or
+B′ /∈ SFG′(M), then the G′-HN filtration of M is also a valid G-HN filtration,
+and so SFG(M) = SFG′(M).
+We thus assume that B, B′ ∈ SFG′(M).
+Let N be the subquotient of M
+spanned by the two G′-semistable factors in Filt(B) and Filt(B′). Then N ∈
+Filt(B, B′). By Lemma 4.3, the long path around the polygon therefore induces
+an HN filtration of N in the abelian category Filt(B, B′), which gives us the G-
+HN filtration of N. The G-HN filtration of M is then obtained from the G′-HN
+filtration of M by replacing the filtration of N with this filtration. We then have
+that
+SFG′(N) =
+�
+B′′∈SFG(N)
+SFG′(B′′),
+since the elements of SFG(N) lie in Filt(B, B′) and B and B′ are the relatively
+simple objects in Filt(B, B′). Indeed, Filt(B, B′) is an abelian category, so ap-
+plying SFG′(−) just gives the multiset of composition factors in this category.
+Using the fact that N is a subquotient of both the G-HN filtration and the
+G′-HN filtration, we deduce that
+SFG′(M) = (SFG′(M) \ SFG′(N)) ⊔ SFG′(N)
+= (SFG(M) \ SFG(N)) ⊔ SFG′(N)
+= (SFG(M) \ SFG(N)) ⊔
+�
+B′′∈SFG(N)
+SFG′(B′′)
+=
+�
+B′′∈SFG(M)
+SFG′(B′′),
+since for B′′ ∈ SFG(M) \ SFG(N), we have SFG′(B′′) = {B′′}. This gives us that
+[G] ⩽HN [G′], as desired.
+□
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+45
+Corollary 4.16. Let Λ be a finite-dimensional algebra over a field K with G
+and G′ two maximal green sequences of Λ. Then [G] ⩽� [G′] implies that B(G) ⊇
+B(G′).
+Proof. This follows immediately either from Lemma 4.4, or from combining The-
+orem 4.15 and Proposition 4.9.
+□
+Remark 4.17. Given maximal green sequences G and G′ such that [G] ⩽HN [G′], it
+is natural to attempt to prove the converse of Theorem 4.15 by using Lemma 3.19
+to find a decreasing elementary polygonal deformation G′′ of G′ such that [G] ⩽HN
+[G′′] ⋖� [G′].
+However, simply applying Lemma 3.19 does not always give a
+deformation which works. Indeed, suppose that we applied this lemma and found
+[G′′] which differs from [G′] by a sequence of deformations across squares and a
+single decreasing deformation across a polygon with the short side B < B′ inside
+G′, such that B > B′ in G and in G′′. The issue is that we do not automatically
+have [G] ⩽HN [G′′], as can be shown by applying the example from Remark 4.8.
+Recall here that we have the path algebra Λ of
+2
+4
+1
+3
+of type �A4 with maximal green sequences G
+1, 1
+2, 1
+4,
+1
+2 4, 3, 3
+2, 2, 3
+4, 4
+and G′
+4, 2, 3, 1.
+All of the adjacent pairs of bricks in G′ are ordered differently in G, so we could
+obtain any of them from Lemma 3.19. But, in particular, we could obtain the
+pair 2 and 3.
+Deforming these across a pentagon yields the maximal green
+sequence G′′
+4, 3, 3
+2, 2, 1.
+But now we do not have [G] ⩽HN [G′′], since
+SFG
+�
+3
+2 4�
+=
+�
+4, 3
+2
+�
+,
+whereas
+SFG′′
+�
+3
+2 4�
+=
+�
+2, 3
+4
+�
+.
+It is an exercise for the reader to verify that in this case we still have [G] ⩽� [G′],
+so that this does not provide a counter-example to Conjecture 4.11. This all
+contrasts with the later situation in Theorem 5.11 for the case of Nakayama
+
+46
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+algebras, where any deformation obtained from Lemma 3.19 will work.
+The
+difference is in the converse of Lemma 4.9: it holds for Nakayama algebras by
+Corollary 5.17, but does not hold in general. The failure is illustrated by the
+present example: we have B(G′′) ⊆ B(G), but do not have [G] ⩽HN [G′′]. One can
+also see that we do not have [G] ⩽� [G′′]. In order to deform [G′′] into [G], one
+would need to move 1 past 3
+2, but when one does this one is forced to insert the
+brick
+1 3
+2 ,
+which is not a brick of [G].
+We prove Conjecture 4.11 in the simple case where the algebra Λ only has two
+simple modules up to isomorphism, for which we need the following lemma.
+Lemma 4.18. If G is a maximal green sequence given as a maximal backwards
+Hom-orthogonal sequence of bricks B1, B2, . . . , Br, then both B1 and Br are sim-
+ple Λ-modules.
+Proof. That B1 must be a simple Λ-module follows from the fact that it must
+be a relatively simple object in the first torsion class of G, which is mod Λ. That
+Br must also be a simple Λ-module follows from the duality between torsion
+and torsion-free, but can also be seen by the following direct argument.
+As
+explained in Section 2.4.2, we must have that the final non-zero torsion class of
+G is Filt(B). But for this to be a torsion class, we must have that B has no
+proper factor modules, which implies that B is a simple Λ-module.
+□
+Theorem 4.19. Let Λ be a finite-dimensional algebra over a field K with two
+isomorphism classes of simple modules. Let G and G′ be maximal green sequences
+of Λ. Then the following are equivalent.
+(1) [G] ⩽� [G′].
+(2) [G] ⩽S [G′].
+(3) [G] ⩽HN [G′].
+(4) B(G) ⊇ B(G′).
+Proof. We already know that [G] ⩽� [G′] implies [G] ⩽S [G′], [G] ⩽HN [G′], and
+B(G) ⊇ B(G′) by Theorem 4.14, Theorem 4.15, and Corollary 4.16. We wish to
+show the two converse implications. Note first that the Hasse diagram of the
+poset of two-term silting complexes of Λ is 2-regular by [AIR14, Corollary 3.8(a)].
+Hence Λ has at most two maximal green sequences. If Λ has fewer than two
+maximal green sequences, then the result is trivial, so suppose that Λ has two
+maximal green sequences G and G′.
+Suppose that [G] ̸⩽� [G′]. Since we are assuming that G and G′ are the two
+distinct maximal green sequences of Λ, the only option is that G′ is a maximal
+green sequence of length greater than two, by Definition 4.1.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+47
+To show that [G] ̸⩽S [G′], we note that there exists X ∈ Ss(G′) which is neither
+a projective nor a shifted projective, since G′ has length greater than two. We
+have that X completes to precisely two different basic two-term silting complexes
+X ⊕ Y and X ⊕ Y ′ [AIR14, Corollary 3.8(a)]. Both X ⊕ Y and X ⊕ Y ′ therefore
+occur in G′ and neither is Λ or Λ[1].
+Hence X /∈ Ss(G), since neither of the
+two-term silting complexes it is an indecomposable summand of occurs in G. We
+conclude that [G] ̸⩽S [G′], as desired.
+To show that [G] ̸⩽HN [G′], we note that there exists B ∈ B(G′) which is
+not a simple. Let S1 and S2 be the simple Λ-modules. By Lemma 4.18, these
+must be the first and last bricks in G and G′. Suppose that these are ordered
+S1 < B < S2 by G′, in which case the simples must be ordered S2 < S1 in G.
+By backwards Hom-orthogonality, the top of B cannot contain S1, so it must
+contain S2.
+Similarly, the socle of B cannot contain S2, so it must contain
+S1.
+This implies that B /∈ B(G), since B would have to occur after S2 and
+before S1 by Lemma 4.18, which would contradict backwards Hom-orthogonality.
+Hence, we have B(G) ̸⊇ B(G′). By Lemma 4.9, we then have [G] ̸⩽HN [G′]. Note
+also that bricks therefore determine the equivalence class for algebras with two
+simples.
+□
+One implication of Theorem 4.19 is that unoriented polygons are “unoriented”
+in all of the partial orders. That is, maximal green sequences which differ by a
+deformation across an unoriented polygon are not related in any of the orders.
+4.2.2. Maxima and minima. We show that in certain cases the partial orders
+have unique maxima or unique minima. We first consider cases where the partial
+orders have unique maxima.
+Proposition 4.20. Let Λ = KQ/I be a path algebra with relations, where Q is
+acyclic. Then there is a maximal green sequence Gmax whose equivalence class is
+the unique maximum for ⩽S and ⩽HN, and which is maximal in ⩽�.
+Proof. If Q is an acyclic quiver, then it is clear that the vertices of Q may be
+labelled 1, 2, . . . , n such that HomΛ(Pi, Pj) = 0 for i < j, where Pi is the indecom-
+posable projective at vertex i. Then there is a maximal green sequence Gmax of
+Λ given by the sequence of exchange pairs (P1, P1[1]), (P2, P2[1]), . . . , (Pn, Pn[1]).
+Then Ss(Gmax) consists of the indecomposable projectives and indecomposable
+shifted projectives. Hence, for any other maximal green sequence G′ of Λ, we
+have that [G′] ⩽S [Gmax], which proves that [Gmax] is the unique maximum for ⩽S.
+To show that [Gmax] is the unique maximum for ⩽HN, we first note that B(Gmax)
+consists of only the simple Λ-modules {S1, . . . , Sn}, since B(Gmax) must contain
+all simple modules by Theorem 2.7 and #B(Gmax) = #Ss(Gmax) = n. By the
+ordering of Q0 we choose earlier, we in fact have that S1, S2, . . . , Sn is a backwards
+maximal Hom-orthogonal sequence of bricks corresponding to Gmax. Now let G
+
+48
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+be a maximal green sequence of Λ and M be a Λ-module. Then SFGmax(M) just
+consists of the composition factors of M, so it is clear that
+SFGmax(M) =
+�
+B∈SFG(M)
+SFGmax(B).
+Finally, it is clear that Gmax is maximal in ⩽�, because it does not admit any
+increasing elementary polygonal deformations.
+□
+It is not obvious if Gmax is the unique maximal green sequence which does not
+admit any increasing elementary polygonal deformations. A priori, there may
+be other maximal green sequences which do not admit any increasing elementary
+polygonal deformations — maximal green sequences where one has got stuck, so
+to speak. However, due to Conjecture 4.11, we expect this not to happen.
+For the cases where the partial orders have unique minima, we need the
+following concepts, which can be found in [Rin84]. A path in mod Λ is a tuple
+(M0, . . . , Ms) of Λ-modules with s ⩾ 1 such that for each 1 ⩽ i ⩽ s, there
+exists a map Mi−1 → Mi which is neither zero nor an isomorphism. An algebra
+Λ is representation-directed if and only if there exists no path (M0, . . . , Ms)
+with M0 ∼= Ms.
+It is known that every representation-directed algebra is
+representation-finite [Rin84].
+Proposition 4.21. If Λ is representation-directed, then there is a maximal green
+sequence Gmin whose equivalence class is the unique minimum for all orders.
+Proof. Since Λ is representation-directed, it is representation-finite, and so
+has finitely many bricks. Moreover, the fact that Λ is representation-directed
+means that these bricks can be ordered B1, B2, . . . , Br in such a way that
+HomΛ(Bj, Bi) = 0 for i < j. We therefore have a maximal backwards Hom-
+orthogonal sequence of bricks, which gives us a maximal green sequence Gmin.
+Since #B(Gmin) = #Sτ(Gmin), as both are equal to the length of Gmin, we have
+that Sτ(Gmin) must consist of all indecomposable τ-rigid modules of Λ, because
+these are in bijection with the bricks of Λ by [DIJ19, Theorem 4.1]. Hence, by
+Theorem 3.20, having B(Gmin) consist of all of the bricks of Λ determines Gmin
+up to equivalence.
+Since Sτ(Gmin) consists of all indecomposable τ-rigid modules, [Gmin] is clearly
+minimal in ⩽S. To show that [Gmin] is the unique minimum in ⩽�, let G be a
+maximal green sequence of Λ. If B(Gmin) = B(G), the above argument shows
+that G ∼ Gmin, so it is sufficient to check that [Gmin] ⩽� [G] for G such that
+B(Gmin) ⊃ B(G). For such G, we can use Lemma 3.19 to find bricks B and B′
+which are adjacent with B < B′ in G but B′ > B in Gmin. If swapping B and B′ in
+G results in a deformation across a square, then we can repeat this process, so we
+can assume that swapping B and B′ results in an increasing elementary polygonal
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+49
+deformation G′ of G. Since B(Gmin) consists of all bricks, we have that B(Gmin) ⊇
+B(G′) ⊃ B(G). By induction, we conclude that [Gmin] ⩽� [G]. (The induction
+process stops since Λ is representation-finite, and so #(B(Gmin)\B(G)) < ∞.)
+By Theorem 4.15, we have that [Gmin] ⩽HN [G], so that [Gmin] is also the unique
+minimum in ⩽HN.
+□
+Remark 4.22. Note that for algebras which have infinite global dimension, the
+posets of equivalence classes of maximal green sequences do not always have
+unique maxima and minima. For example, one can compute the posets for the
+path algebra of the three-cycle with relations given by paths of length two.
+Furthermore, for hereditary algebras which are representation-infinite, the
+posets may not have unique minima. For instance, [AI20, Theorem M3] shows
+that for path algebras of type �A4, there are maximal green sequences which are
+of maximal length but have different sets of bricks. More generally, maximal
+green sequences of maximal length for tame hereditary algebras are studied in
+[AI20] and [KN21], whilst minimal length maximal green sequences are studied
+in [GMS18].
+4.3. Exchange pairs. In this subsection, we consider how the partial orders
+on maximal green sequences interact with exchange pairs. Note first that one
+cannot define a partial order using inclusion of exchange pairs.
+Proposition 4.23. If maximal green sequences G and G′ of Λ are such that
+E(G) ⊇ E(G′), then we have G ∼ G′.
+Proof. Suppose that we have E(G) ⊇ E(G′). Suppose for contradiction that there
+is an exchange pair (X, Y ) of G which is not an exchange pair of G′. We may
+choose (X, Y ) to be the first exchange pair of G for which this is the case.
+We claim that X ∈ Ss(G′). This is certainly true if X is projective. If X is not
+projective, then there exists an exchange pair (W, X) ∈ E(G). This must occur
+before (X, Y ) in G by Lemma 3.4. Hence, by the choice of (X, Y ), we have that
+(W, X) ∈ E(G′), and so X ∈ Ss(G′) in this case too.
+However, if X ∈ Ss(G′), we must have, for some Y ′ ̸∼= Y , that (X, Y ′) is an
+exchange pair of G′ and therefore of G, since X cannot be a shifted projective
+due to the exchange pair (X, Y ). But this contradicts Lemma 3.4. Hence E(G) =
+E(G′), and so G ∼ G′ by Theorem 3.20.
+□
+Moreover, the data of the exchange pairs gives the data of the indecomposable
+presilting summands, so it would seem impossible to define a partial order using
+exchange pairs by other means, without its collapsing into the summand partial
+order.
+However, the exchange pairs of a maximal green sequence do exhibit
+interesting behaviour with respect to the deformation order.
+
+50
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Proposition 4.24. Let Λ be a finite-dimensional algebra over a field K. Let
+E ⊕ X ⊕ A
+E ⊕ Z ⊕ A
+E ⊕ Z ⊕ C
+E ⊕ X ⊕ C′
+E ⊕ Z′ ⊕ C′
+. . .
+E ⊕ X′ ⊕ C
+be an oriented polygon in 2-silt Λ. There are then commutative diagrams of ex-
+change triangles
+X
+Y ′
+Z′
+X[1]
+X
+Y
+Z
+X[1]
+and
+A
+B′
+C′
+A[1]
+A
+B
+C
+A[1],
+where Y , Y ′, B, and B′ are the two-term complexes which appear in the middle
+of the relevant exchange triangles.
+Proof. By definition of silting mutation, we have that X → Y ′ is a minimal left
+add(E ⊕ C′)-approximation and that f : X → Y is a minimal left add(E ⊕ A)-
+approximation. We claim that f is in fact a minimal left add E-approximation.
+We prove this claim using silting reduction [IY18]. Let Z = (⊥0E[> 0]) ∩ (E[<
+0]⊥0). We have that Z/[E] is a triangulated category by [IY08, Theorem 4.2],
+with shift functor denoted by ⟨1⟩. Let �
+(−): Z → Z/[E] be the quotient map
+and Γ = EndZ/[E]( �
+X ⊕ �A). Then, as in Remark 3.11, two-term silting complexes
+in 2-silt Γ correspond to the different completions of E to a two-term silting
+complex.
+Then, since �E⊕ �Z⊕ �C is the minimum of 2-silt Γ, we must have that the images
+of the exchange pairs on the short path are ( �
+X, �
+X⟨1⟩) and ( �A, �A⟨1⟩). Thus, we
+have �Z ∼= �
+X⟨1⟩ and �C ∼= �A⟨1⟩. Since �A ⊕ �Z ∼= �A ⊕ �
+X⟨1⟩ is silting in Z/[E], we
+must have that 0 = HomZ/[E]( �
+X⟨1⟩, �
+A⟨1⟩) ∼= HomZ/[E]( �
+X, �A). This means that
+every morphism X → A must factor through E. Letting Y ∼= YE ⊕ A⊕m where
+YE ∈ add E, m ≥ 0, we then have that there is a factorisation
+X
+YE ⊕ E′
+YE ⊕ A⊕m,
+g
+f
+h
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+51
+where E′ ∈ add E. Then, since f is a left add(E ⊕ A)-approximation and YE ⊕
+E′ ∈ add E ⊆ add(E ⊕ A), we have that there is a map l: YE ⊕ A⊕m → YE ⊕ E′
+such that g = lf. But then we have that f = hg = hlf. Since, f is left minimal,
+we must have that hl is an isomorphism.
+This implies that YE ⊕ A⊕m is a
+direct summand of YE ⊕ E′, which in turn implies that A⊕m = 0 = E′. Thus,
+f : X → Y is in fact a minimal left add E-approximation as claimed.
+Therefore, since X → Y ′ is a left add(E ⊕ C)-approximation, we must get a
+factorisation
+X
+Y ′
+Y.
+f
+Using the axioms of triangulated categories, we can extend this to the desired
+commutative diagram between exchange triangles given in the statement of the
+proposition.
+The proof of the second claim is similar. We have that �
+X ⊕ �A is silting in
+Z/[E], so we must have 0 = HomZ/[E]( �A, �
+X⟨1⟩) = HomZ/[E]( �A, �Z).
+Hence,
+any homomorphism from A to Z must factor through E. This means that the
+minimal left add(E ⊕ Z)-approximation A → B′ of A must in fact be a minimal
+left add E-approximation by the same argument as above. Since A → B is a
+minimal left add(E ⊕ X)-approximation of A, we have a factorisation
+A
+B
+B′.
+We again use the axioms of triangulated categories to extend this to the com-
+mutative diagram shown in the statement of the proposition.
+□
+Corollary 4.25. Let Λ be a finite-dimensional algebra over a field K with G′
+and G two maximal green sequences of Λ. Suppose that [G′] ⩽� [G] and let (X, Z)
+be an exchange pair of G with exchange triangle
+X → Y → Z → X[1].
+Then there is an exchange pair (X, Z′) of G′ with exchange triangle
+X → Y ′ → Z′ → X[1]
+and a commutative diagram
+X
+Y ′
+Z′
+X[1]
+X
+Y
+Z
+X[1]
+.
+
+52
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Proof. We show the result by induction on the number of deformations from [G′]
+to [G]. The base case, where [G′] = [G] is trivial, since we have E(G′) = E(G) by
+Theorem 3.20.
+Hence, we suppose that [G] is the result of at least one increasing elementary
+polygonal deformation of [G′].
+Hence, there is a maximal green sequence G′′
+such that [G′] ⋖� [G′′] ⩽� [G]. Let (X, Z) be an exchange pair of G. Then, by
+Theorem 4.14, we must have exchange pairs (X, Z′′) of G′′ and (X, Z′) of G′.
+By applying Proposition 4.24 to [G′] ⋖� [G′′] and the induction hypothesis to
+[G′′] ⩽� [G], we obtain a commutative diagram
+X
+Y ′
+Z′
+X[1]
+X
+Y ′′
+Z′′
+X[1]
+X
+Y
+Z
+X[1]
+.
+This then gives us the desired diagram
+X
+Y ′
+Z′
+X[1]
+X
+Y
+Z
+X[1]
+completing the proof.
+□
+Note that Proposition 4.24 and Corollary 4.25 are both dualisable via fixing
+the second entry of the exchange pairs, rather than the first. The intuition for
+Corollary 4.25 is that as we go higher up in the deformation order, the exchange
+pairs become closer to (X, X[1]).
+4.4. An example from the twice-punctured torus. In this section, we con-
+sider the maximal green sequences of the Jacobian algebra Λ associated to the
+triangulation of the twice-punctured torus shown in Figure 6. In this figure, A
+and B are the two punctures, and the triangulation has six different arcs, labelled
+1 to 6, dividing the torus into four triangles. The quiver of this triangulation is
+given by a clockwise oriented three-cycle within each triangle and is shown in
+Figure 7. The associated potential is
+W = λδκ + ιαδ + ζηγ + µǫβ + κµζθ + διαηγǫβλ.
+We then have that Λ = KQ/⟨∂W⟩, that is, Λ is the path algebra of Q modulo
+the cyclic derivatives of the potential, in the usual way from [DWZ08].
+We
+do not give background on cluster algebras from triangulated surfaces and the
+associated quivers with potential and cluster categories. Relevant background
+can be found in [FST08; DWZ08; Lab09a; Lab09b; Ami09; Dom17].
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+53
+Figure 6. A triangulation of the twice-punctured torus, with its
+associated quiver
+α
+β
+γ
+δ
+ǫ
+ζ
+η
+θ
+ι
+κ
+λ
+µ
+1
+1
+2
+2
+3
+4
+5
+6
+A
+A
+A
+A
+B
+•
+•
+•
+•
+•
+This quiver Q was used in [KY20, Example 1] to give an example of a quiver
+whose exchange graph had a fundamental group not generated by squares and
+pentagons, following earlier work in [FST08, Remark 9.19]. We discuss the im-
+plications for posets of equivalence classes of maximal green sequences, since this
+fact about the exchange graph makes the algebra Λ a natural place to look for a
+counter-example to Conjecture 4.11. However, as we shall explain, [KY20, Ex-
+ample 1] does not give a counter-example to the conjecture that the summand
+order is equal to the deformation order.
+Using the work of [KY20, Example 1], two maximal green sequences of this
+algebra are
+(1)
+5, 6, 4, 3, 6, 5, 1, 2, 4, 3, 6, 5, 3, 4, 2, 1
+and
+(2)
+1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 5, 6,
+given as sequences of vertices for mutation, as explained in Section 2.3.5. It is
+shown in [KY20], using arguments from [FST08], that these two maximal green
+sequences cannot be deformed into each other across squares and pentagons. In
+
+54
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Figure 7. The quiver Q of the algebra Λ considered in Sec-
+tion 4.4
+1
+2
+3
+5
+4
+6
+α
+β
+γ
+δ
+ǫ
+ζ
+η
+θ
+ι
+κ
+λ
+µ
+fact, these are the only types of polygon in this case. Hence the poset ⩽� of
+maximal green sequences for Λ has at least two connected components.
+The intuitive reason why these two maximal green sequences cannot be de-
+formed into each other across squares and pentagons is as follows. The quiver Q
+is of infinite cluster type, as can be seen from the fact that it contains subquivers
+of affine type �D4. Moreover, the poset of two-term silting complexes of this al-
+gebra is therefore also infinite. One cannot deform one maximal green sequence
+into the other, since doing so would require traversing regions of the exchange
+graph which are infinite, and, naturally, this cannot be done.
+The proof of that these two maximal green sequences cannot be deformed
+into each other across squares and larger polygons uses theory from tagged tri-
+angulations. Generalising [Lab09b], by [Dom17], tagged triangulations of the
+twice-punctured surface are in bijection with cluster-tilting objects in the as-
+sociated generalised cluster category of Λ which are connected to the initial
+cluster-tilting object corresponding to Λ via mutation. See [Ami11, Section 3.4]
+for a summary. It then follows from [AIR14, Theorem 4.7] that cluster-tilting
+objects in the generalised cluster category of Λ are in bijection with two-term
+silting complexes over Λ. These bijections are moreover induced by a bijection
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+55
+Figure 8. The poset of two-term silting objects of Λ divided
+into strata
+Ω++
+Ω+0
+Ω0+
+Ω+−
+Ω−+
+Ω0−
+Ω−0
+Ω−−
+between tagged arcs, indecomposable rigid objects in the generalised cluster cat-
+egory, and indecomposable two-term presilting complexes. In the last two cases,
+the indecomposable objects and complexes must respectively be summands of
+cluster-tilting objects and two-term silting complexes connected to Λ, respec-
+tively to the corresponding cluster-tilting object, by mutation.
+We briefly outline some of the theory of tagged triangulations.
+Arcs in a
+tagged triangulation may be “notched” at either end of the arc, or notched at
+both ends, or simply plain. For a tagged triangulation T of the twice-punctured
+torus, there is an associated signature δT : {A, B} → {−, 0, +} on the set of
+punctures, where
+δT (X) =
+
+
+
+
+
++,
+if all arc-ends incident to X are plain
+−,
+if all arc-ends incident to X are notched
+0.
+if there are both plain and notched arc-ends
+Tagged triangulations of the twice-punctured torus fall into eight disjoint sets,
+known as ‘strata’, according to the signatures at the two punctures. These eight
+strata are denoted
+Ω++, Ω+0, Ω0+, Ω+−, Ω−+, Ω0−, Ω−0, Ω−−,
+where triangulations T ∈ Ωxy have δT (A) = x and δT (B) = y. Dividing up
+the associated two-term silting complexes into these strata yields the depiction
+of the poset 2-silt Λ shown in Figure 8 — see [FST08, Remark 9.19]. Here Ωxy
+represents the the subposet corresponding to the relevant stratum.
+Hence, the two-term silting complex of projectives lies in the stratum Ω++,
+whilst the two-term silting complex of shifted projectives lies in the stratum Ω−−.
+It is clear from Figure 8 that there are only two possible routes for a maximal
+green sequence of Λ through the strata. The maximal green sequence (1) takes
+
+56
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+the left-hand route, whilst the maximal green sequence (2) takes the right-hand
+route [KY20, Example 1]. Bear in mind that the strata Ωxy have cardinality
+larger than 1: each of these maximal green sequences has length 16, while each
+route changes strata 4 times. In particular, one should not think of Figure 8 as
+depicting an unoriented octagon from Subsection 3.2.
+If there were a maximal green sequence taking one route through the strata
+which contained all the summands of a maximal green sequence taking the other
+route through the strata, then we would have an immediate counter-example to
+Conjecture 4.11, since these maximal green sequences could not be connected
+by deformations across squares and oriented polygons, as we have explained.
+However, we now show that two such maximal green sequences cannot exist,
+meaning that there is no such apparent counter-example.
+Proposition 4.26. Let G be a maximal green sequence of Λ passing through the
+stratum Ω+− and G′ a maximal green sequence of Λ passing through the stratum
+Ω−+. Then we have [G] ̸⩽S [G′] and [G′] ̸⩽S [G].
+Proof. In every triangulation of the twice-punctured torus, there must be a
+tagged arc connecting A and B. In the stratum Ω+−, this arc must be plain
+at A and notched at B. Such an arc, however, cannot exist in any of the strata
+in the right-hand path through the maximal green sequences in Figure 8, since
+its presence in the triangulation implies that the signature at A is + or 0 and
+the signature at B is 0 or −. Therefore, there exists a tagged arc γ which is
+in the sequence of triangulations corresponding to G but not in the sequence of
+triangulations corresponding to G′. This then corresponds to an indecomposable
+two-term presilting complex X such that X ∈ Ss(G) but X /∈ Ss(G′). Hence, we
+obtain that [G′] ̸⩽S [G], and the converse can be shown symmetrically.
+□
+Corollary 4.27. The poset of equivalence classes of maximal green sequences of
+Λ with respect to ⩽S also has at least two connected components.
+Remark 4.28. Note that it is itself unremarkable for these posets to have several
+connected components, since unoriented polygons may prevent connectedness.
+For instance, one can compute that the algebra from [BST19, Example 3.30] has
+three connected components in its poset of equivalence classes of maximal green
+sequences. What is interesting about the example considered here is firstly that
+the disconnectedness does not result from unoriented polygons, secondly that
+the fundamental group of the poset 2-silt Λ is not generated by polygons (in the
+sense of Definition 3.6), and thirdly how the disconnectedness of the posets of
+equivalence classes of maximal green sequences can be understood in terms of
+the punctured surface.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+57
+5. Nakayama algebras
+Recall that a finite-dimensional K-algebra Λ is a Nakayama algebra if every
+finite-dimensional indecomposable Λ-module is uniserial, meaning that it has a
+unique composition series. In this section, we show that equivalence of maximal
+green sequences for Nakayama algebras is given by having the same sets of bricks.
+This makes it possible to define a partial order on equivalence classes of maximal
+green sequences in terms of reverse-inclusion of bricks. This in turn allows us to
+prove Conjecture 4.11 in this special case.
+5.1. Equivalence using bricks. For Nakayama algebras, using the following
+lemma, we can show that the bricks of a maximal green sequence do determine its
+equivalence class, for which we need the following series of lemmas. We first show
+that filtrations of indecomposables by any bricks are unique, not just filtrations
+by simples.
+Lemma 5.1. Given an indecomposable module M over a Nakayama algebra
+Λ such that M ∈ Filt(B1, B2, . . . , Bs) where B1, B2, . . . , Bs are pairwise Hom-
+orthogonal bricks, we have that the filtration of M in terms of B1, B2, . . . , Bs is
+unique.
+Proof. We show that there is a unique Bi such that Bi is a submodule of M,
+whence the claim follows by induction on the length of the filtration. Note that
+M/Bi must also be indecomposable, otherwise M cannot be uniserial. Suppose
+that Bi and Bj are both submodules of M. Then, since M is uniserial, we either
+have Bi ֒→ Bj or Bj ֒→ Bi. But this contradicts the pairwise Hom-orthogonality
+unless i = j, which is what we wanted to show.
+□
+In the proof of Theorem 5.11 we will need to consider not only bricks, but
+indecomposables which are self-extensions of a single brick. In the following few
+lemmas, we show how such indecomposables behave somewhat like bricks.
+Lemma 5.2. Suppose that L ∈ Filt(B) and N ∈ Filt(B′) are indecompos-
+able modules, where B and B′ are bricks over a Nakayama algebra Λ such that
+HomΛ(B′, B) = 0. Then HomΛ(N, L) = 0.
+Proof. Assume that there exists a non-zero map f : N → L. Choose a maxi-
+mal submodule X of L such that X ⊂ im f and X ∈ Filt(B). If we consider
+the quotient p: L → L/X, then L/X ∈ Filt(B). Moreover, L/X must be in-
+decomposable and so uniserial, otherwise L cannot be uniserial. Since im(pf)
+and B are both submodules of L/X, we have im(pf) ⊆ B. Indeed, since L/X
+is uniserial, the only alternative would be that B ⊂ im(pf), but in this case X
+would not be maximal with respect to its defining properties. Since X ⊃ im f,
+im(pf) ∼= im f/X is non-zero.
+
+58
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Now choose a minimal submodule i: Y ֒→ N such that Y ∈ Filt(B′) but
+ker(pf) ⊂ Y . Then coim(pfi) ∼= Y/ ker(pfi) ̸∼= 0, since ker(pf) ⊂ Y as sub-
+modules of N. Moreover, coim(pfi) and B′ are both factor modules of Y , and
+since Y must be indecomposable and hence uniserial, we must either have that
+coim(pfi) ։ B′ as a proper factor module or B′ ։ coim(pfi). However, the
+former possibility cannot hold, since then we could replace Y by its submodule
+Y ′ such that Y/Y ′ ∼= B′, and we would still have that ker(pf) ⊂ Y ′. Thus, we
+have that B′ ։ coim(pfi).
+The composition B′ ։ coim(pfi) ∼= im(pfi) ֒→ im(pf) ֒→ B is then non-zero.
+This contradicts the assumption that HomΛ(B′, B) = 0. Thus, such non-zero f
+cannot exist and HomΛ(N, L) = 0.
+□
+Corollary 5.3. If B and B′ are bricks over a Nakayama algebra Λ with
+HomΛ(B′, B) = 0, then Filt(B) ∩ Filt(B′) = {0}.
+Proof. This follows from letting L ∈ Filt(B) ∩ Filt(B′) and applying Lemma 5.2
+with N = L.
+□
+A similar argument to Lemma 5.2 also shows the following lemma.
+Lemma 5.4. Suppose that M ∈ Filt(B) is an indecomposable module over a
+Nakayama algebra Λ, where B is a brick. Then every endomorphism f : M → M
+is either
+(1) zero,
+(2) an isomorphism, or
+(3) has im f ∈ Filt(B) being a proper submodule of M, equivalently, has
+ker f ∈ Filt(B) being a proper submodule of M.
+The following lemma can be seen as a stronger version of Lemma 3.16 which
+holds for Nakayama algebras.
+Lemma 5.5. Suppose that L ∈ Filt(B) and N ∈ Filt(B′) are indecompos-
+able modules, where B and B′ are bricks over a Nakayama algebra Λ such that
+HomΛ(B′, B) = 0. Then every indecomposable M which is a non-split extension
+0 → L
+f−→ M
+g−→ N → 0
+of L and N is a brick.
+Proof. We suppose that h: M → M is an endomorphism of M and seek to show
+that this must either be zero or an isomorphism. Since Λ is a Nakayama algebra,
+we are in one of the following two cases:
+(1) L ⊆ ker h;
+(2) ker h ⊂ L.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+59
+Case (1) is similar in flavour to Lemma 3.16, but we will refrain from copying
+out the commutative diagrams once more. We have that hf = 0, so that there
+is an induced map a: N → M such that h = ag.
+Since N ∈ Filt(B′), by
+Lemma 5.4, its endomorphim ga is either an isomorphism, zero, or there is a
+proper submodule X = im(ga) ∈ Filt(B′) of N which is also a proper factor
+module of N. If ga is an isomorphism, then a(ga)−1 gives a splitting of g, which
+contradicts the assumption that the sequence is non-split. If ga = 0, then there
+is an induced map b: N → L such that fb = a. But, since HomΛ(N, L) = 0 by
+Lemma 5.2, we have that b = 0, and so h = ag = fbg = 0.
+We are left with the last option, where there is a proper submodule X ∈
+Filt(B′) which is also a proper factor module. First, note that im a = im h,
+since h = ag and g is epic.
+Then, we have that X = im(ga) = g(im a) =
+g(im h) ∼= im h/L. Secondly, note that there exists a submodule m: M′ ֒→ M
+such that M/M′ ∼= X, as X is a factor module of N and so also a factor module
+of M. Then N ′ ∼= M′/L ∈ Filt(B′) is a proper non-zero submodule l: N ′ ֒→ N
+and, moreover, N ′ = ker(ga) as X = im(ga) ∼= coim(ga). We denote the map
+M′ ։ N ′ by g′. We have a map a′ : N ′ → M′ defined by a′ = al. Now we have
+that ga′ = gal = 0, since N ′ = ker(ga), which gives an induced map b′ : N ′ → L
+such that fb′ = a′.
+M′
+N ′
+M
+N
+L
+M
+N
+m
+g′
+b′
+a′
+l
+h
+g
+a
+f
+g
+Then b′ = 0, by Lemma 5.2, which gives that hm = agm = alg′ = a′g′ =
+fb′g′ = 0. This then implies that M′ ⊆ ker h, and so that im h ∼= M/ ker h ∼=
+(M/M′)/(ker h/M′) ∼= X/(ker h/M′) by the third isomorphism theorem. But
+im h cannot be a quotient of X, since X ∼= im h/L and L ̸= 0.
+This is a
+contradiction, and so the last option cannot hold.
+In case (2), we have that L/ ker h is a non-zero factor module of L. If ker h = 0,
+then h is an isomorphism, so we can ignore this case and assume that L/ ker h
+is a proper factor module of L. We have an induced monomorphism L/ ker h ֒→
+M/ ker h ∼= im h. Thus, L/ ker h is a submodule of M. Since M is uniserial,
+we have L/ ker h ⊂ L or L ⊆ L/ ker h.
+By length considerations, only the
+former is possible. Thus, we have an endomorphism L ։ L/ ker h ֒→ L which
+is certainly not an isomorphism. Since we assume that ker h ̸= 0, we must have
+that ker h ∈ Filt(B) by Lemma 5.4.
+
+60
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+We show that having ker h ∈ Filt(B) leads to a contradiction. By the third
+isomorphism theorem, we have
+N ∼= M
+L
+∼= M/ ker h
+L/ ker h
+∼=
+im h
+L/ ker h ֒→
+M
+L/ ker h,
+so that N is a submodule of M/(L/ ker h).
+We also have that L/(L/ ker h)
+is a submodule of M/(L/ ker h).
+Since M/(L/ ker h) is uniserial as a factor
+module of a uniserial module M, we must either have that N ⊆ L/(L/ ker h)
+or L/(L/ ker h) ⊂ N as submodules of this module. Having N ⊆ L/(L/ ker h)
+contradicts Lemma 5.2, since L/(L/ ker h) ∈ Filt(B).
+Hence, we must have
+L/(L/ ker h) ⊂ N. Having L/(L/ ker h) ∈ Filt(B′) contradicts Corollary 5.3,
+so L/(L/ ker h) /∈ Filt(B′). But then we apply the third isomorphism theorem
+again to obtain
+N
+L/(L/ ker h) ֒→ M/(L/ ker h)
+L/(L/ ker h)
+∼= M
+L
+∼= N.
+We therefore have a composition of maps
+N ։
+N
+L/(L/ ker h) ֒→ N
+which is neither zero nor an isomorphism, with N/(L/(L/ ker h)) /∈ Filt(B′),
+which contradicts Lemma 5.4.
+We conclude that M is a brick.
+□
+We can now show the key lemma which establishes that bricks determine the
+equivalence class of a maximal green sequence over a Nakayama algebra.
+Lemma 5.6. Let G be a maximal green sequence of a Nakayama algebra Λ.
+Suppose that Bi and Bk are bricks of G such that i < k in the sequence of bricks
+and that there is a non-split short exact sequence
+0 → Bi
+f−→ B
+g−→ Bk → 0
+such that B is a brick. Then B ∼= Bj is a brick of G with i < j < k.
+Proof. We use the fact that the brick sequence of G must be maximal with respect
+to the property of being backwards Hom-orthogonal. Hence, if we can show that
+there is a space between Bi and Bk where one can insert B without disrupting
+the backwards Hom-orthogonality, then it follows that B must actually occur in
+the brick sequence of G as B ∼= Bj for i < j < k.
+Let us insert B in the brick sequence of G at some point between Bi and Bk.
+If B cannot be inserted at such a point, then there must exist either a brick L
+in G which occurs after B with a non-zero homomorphism l: L → B, or there
+exists a brick N before B with a non-zero homomorphism n: B → N. In the
+first case, if L occurs after Bk, then the composition L
+l−→ B
+g−→ Bk must be zero.
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+61
+This gives a non-zero map L → Bi, which is a contradiction. One can argue in
+a similar way that N cannot occur before Bi.
+Hence, we have that all such bricks N and L occur between Bi and Bk. If all
+such bricks L occur before all such bricks N, then we can place B between these
+two sets of bricks, thereby preserving backwards Hom-orthogonality. Thus, we
+may assume that N occurs before L.
+We now use the fact that Λ is a Nakayama algebra and so that all the modules
+in question must be uniserial. We have that im l ⊃ Bi, otherwise there is a non-
+zero map L ։ im l ֒→ Bi. Likewise, ker n ⊂ ker g = Bi, otherwise there is a
+non-zero map Bk → N. Thus, we have that im l ⊃ ker n. This means that the
+composition L
+l−→ B
+n−→ N is non-zero, which contradicts the backwards Hom-
+orthogonality. Hence, all of the bricks L occur before all the bricks N between
+Bi and Bk, and so there is a point between Bi and Bk where B can be placed
+without violating backwards Hom-orthogonality, as desired.
+□
+Remark 5.7. Note that Lemma 5.6 does not hold for the non-linearly oriented
+type A algebra from Example 3.26, which is not a Nakayama algebra. Indeed,
+considering the maximal green sequence
+2, 2
+1, 1, 2
+3, 3,
+we have a short exact sequence
+0 → 2
+1 → 2
+1 3 → 3 → 0,
+but the non-uniserial brick
+2
+1 3 does not occur in the maximal green sequence.
+We can then make the following argument showing that maximal green se-
+quences of Nakayama algebras with the same bricks are connected by deforma-
+tions across squares.
+Theorem 5.8. Let G1 and G2 be maximal green sequences of a Nakayama algebra
+Λ such that B(G1) = B(G2). Then we have that G1 ∼ G2.
+Proof. Suppose that we have maximal green sequences G1 and G2 such that
+B(G1) = B(G2). We will prove that G1 and G2 can be deformed into each other
+across squares by induction on the number of bricks after the first brick where
+they differ. In the base case, the maximal green sequences coincide, and so there
+is no point at which they diverge.
+Now we suppose that G1 ̸= G2. We consider the first bricks where G1 and G2
+differ. Let this brick be B1 for G1 and B2 for G2. Since B(G1) = B(G2), we must
+therefore have B1 ∈ B(G2) and B2 ∈ B(G1) as well.
+We claim that, by deforming across squares, we can make B2 the brick immedi-
+ately before B1 in G1. Suppose that, on the contrary, there is a brick B ∈ B(G1)
+
+62
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+such that B2 cannot be moved back past B.
+By Lemma 3.18, this must ei-
+ther be because HomΛ(B, B2) ̸= 0 or because Ext1
+Λ(B, B2) ̸= 0. We must have
+B ∈ B(G2). Furthermore, B must occur in G2 after B2, since it coincides with
+B1 or occurs in G1 after B1, and the respective segments of G1 and G2 before
+B1 and B2 coincide. We then cannot have HomΛ(B, B2) ̸= 0, since B occurs
+after B2 in G2. Suppose, then, that Ext1
+Λ(B, B2) ̸= 0. Since B and B2 appear in
+different orders in G1 and G2, we have HomΛ(B2, B) = HomΛ(B, B2) = 0. Then,
+by Lemma 3.16, there is a brick B′ which is a non-split extension of B and B2,
+0 → B2 → B′ → B → 0.
+Because B2 occurs before B in G2, we must have that B′ ∈ B(G2) by Lemma 5.6.
+Then we also have that B′ ∈ B(G1).
+Due to backwards Hom-orthogonality,
+we must have that B′ occurs after B2 and before B in G1. However, this is a
+contradiction, since B2 occurs after B in G1.
+Thus, we can move B2 back along G1 by deforming across squares to obtain
+a maximal green sequence G′
+1 where B2 occurs before B1. The maximal green
+sequences G′
+1 and G2 then have fewer bricks which occur after they differ, so
+by the induction hypothesis, we have that G′
+1 ∼ G2. This then establishes that
+G1 ∼ G2, as desired, since G1 ∼ G′
+1.
+□
+5.2. Partial order using bricks. For Nakayama algebras we may also consider
+the partial order on maximal green sequences given by reverse inclusion of bricks,
+since maximal green sequences with the same sets of bricks are equivalent by
+Theorem 5.8. For general algebras this will not be a well-defined partial order;
+the relation will not in general be anti-symmetric, since non-equivalent maximal
+green sequences may have the same set of bricks.
+Definition 5.9. Let [G] and [G′] be equivalence classes of maximal green se-
+quences over a Nakayama algebra Λ. We write [G] ⩽B [G′] if B(G) ⊇ B(G′). We
+refer to this as the brick order.
+Lemma 5.10. The partial order ⩽B is well-defined for Nakayama algebras.
+Proof. We first note ⩽B is well-defined on equivalence classes, since B(G) = B( �G)
+for all �G ∈ [G] by Lemma 3.22. The relation ⩽B is clearly transitive and reflexive.
+It is anti-symmetric since B(G) = B(G′) implies [G] = [G′] by Theorem 5.8.
+□
+We can show that the brick order coincides with the deformation order for
+Nakayama algebras, which is the key to showing that in fact all of the orders
+coincide in this case.
+Theorem 5.11. Let Λ be a finite-dimensional Nakayama algebra over a field K,
+with G and G′ two maximal green sequences of Λ. Then [G] ⩽B [G′] if and only
+if [G] ⩽� [G′].
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+63
+Proof. We know that [G] ⩽� [G′] implies [G] ⩽B [G′] by Corollary 4.16.
+We
+now show that [G] ⩽B [G′] implies [G] ⩽� [G′]. Let G and G′ be maximal green
+sequences such that [G] <B [G′]. Hence, every brick of G′ is also a brick of G.
+By Lemma 3.19, there must be an adjacent pair of bricks B and B′ of G′ such
+that B occurs before B′ in G′, but B occurs after B′ in G. Let S be the sequence
+of bricks given by starting with G′ and swapping B and B′ so that B′ now occurs
+before B. This sequence S must in fact be backwards Hom-orthogonal, since B′
+occurs before B in G. If S is maximal backwards Hom-orthogonal, then we have
+deformed across a square and we apply Lemma 3.19 again to find a new pair of
+bricks. Hence, we can assume that S is not maximal backwards Hom-orthogonal.
+Therefore, we can add bricks to S to obtain a maximal backwards Hom-
+orthogonal sequence G′′. By Lemma 3.9, G′′ is unique. Since G′ was maximal
+backwards Hom-orthogonal, the extra bricks in G′′ can only occur between B
+and B′. We have that all the extra bricks lie in Filt(B, B′) by Theorem 2.5.
+We claim that, by applying Lemma 5.6 iteratively, we obtain that all the extra
+bricks of G′′ must also be bricks of G lying in between B′ and B, and so there
+can only be finitely many of them. Let B′′ be an extra brick of G′′, which thus
+has a unique filtration with factors B and B′ by Lemma 5.1. Note then that we
+must have Ext1
+Λ(B′, B) = 0 by Lemma 3.17, since B occurs immediately before
+B′ in G′ and Hom(B, B′) = 0 by the the backwards Hom-orthogonality of G.
+Hence any subquotient of B′′ with factors B and B′ where B occurs below B′
+must be isomorphic to B ⊕ B′, with the result that the order of the factors can
+be switched so that B′ occurs below B. This contradicts uniqueness, so in fact
+we must have that in the filtration of B′′ by B and B′, all of the B′ factors occur
+below all of the B factors.
+We now prove the claim that B′′ ∈ B(G) and lies in between B′ and B by
+induction on the length of the filtration of B′′ by B and B′. In the base case,
+where this filtration is of length 1, we either have B′′ ∼= B or B′′ ∼= B′, and in
+either case the claim is immediate. We also use the case where B′′ has only one
+B factor and only one B′ factor as a base case. Since B′ occurs in G before B,
+we then have that B′′ occurs in G between B′ and B by Lemma 5.6. For the
+induction step there are then two cases to consider, namely, either
+• B′′ has at least two B factors, or
+• B′′ has at least two B′ factors.
+The two cases are similar, so we only consider the first one. First note that B′′
+must also have B′ factors in its filtration, otherwise it is clearly not a brick. We
+claim that if L is the unique submodule of B′′ such that B′′/L ∼= B, then we
+have that L is a brick. Indeed, we have that there is a non-split short exact
+sequence 0 → X → L → Y → 0 such that X ∈ Filt(B′) and Y ∈ Filt(B). Also,
+
+64
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+L is indecomposable, since otherwise B′′ would not be uniserial. Hence, L is a
+brick by Lemma 5.5.
+Hence, by the induction hypothesis, we have that L is a brick of G lying
+between B′ and B. Then Lemma 5.6 establishes that B′′ is a brick of G lying
+between L and B, and therefore also between B′ and B. Therefore, by induction,
+we have that all of the extra bricks of G′′ are bricks of G and lie between B′ and B.
+We thus obtain that G′ is an increasing elementary polygonal deformation
+of G′′, and that the bricks of G′′ are contained in the bricks of G, and so [G] ⩽B
+[G′′]⋖�[G′]. Applying this argument inductively gives [G] <� [G′], as desired.
+□
+Corollary 5.12. Let Λ be a finite-dimensional Nakayama algebra over a field
+K, with G and G′ two maximal green sequence of Λ. Then [G] ⩽HN [G′] if and
+only if [G] ⩽� [G′].
+Proof. We already know from Theorem 4.15 that if [G] ⩽� [G′] then [G] ⩽HN [G′].
+For the other direction, we have from Lemma 4.9 that [G] ⩽HN [G′] implies
+[G] ⩽B [G′]. Then, by Theorem 5.11, we have that this implies [G] ⩽� [G′].
+□
+5.3. Bricks versus summands. We now show how one can compute the set
+of τ-rigid summands of a maximal green sequence for a Nakayama algebra from
+the set of bricks, which will allow us to show that the brick order coincides with
+the summand order for Nakayama algebras.
+We begin by letting Λ be a Nakayama algebra, and introducing the set
+B(Λ) := {B ∈ brick Λ | B /∈ simp Λ}
+of bricks over Λ which are not simple. We further introduce
+R(Λ) := {M ∈ ind Λ | M is τ-rigid but M /∈ proj Λ}
+of indecomposable τ-rigid modules which are not projective. Note that τ-tilting
+theory for Nakayama algebras was studied in [Ada16].
+Lemma 5.13. If Λ is a Nakayama algebra, then there is a bijection
+φ: B(Λ) → R(Λ)
+B �→ B/ soc B
+Proof. Note first that, since B is not a simple, we have that B/ soc B is non-zero.
+Moreover, B/ soc B is not a projective due to the non-split short exact sequence
+0 → soc B → B → B/ soc B → 0,
+whilst B/ soc B must be indecomposable since B is a uniserial module.
+We now show that B/ soc B is τ-rigid.
+It follows from [ARS95, Theo-
+rem VI.2.1] that τ(B/ soc B) = rad B (see also [ASS06, Theorem V.4.1]). Thus
+
+A STRUCTURAL VIEW OF MAXIMAL GREEN SEQUENCES
+65
+HomΛ(B/ soc B, τ(B/ soc B)) = HomΛ(B/ soc B, rad B) = 0, since B is a brick.
+Hence B/ soc B is τ-rigid, as desired.
+We now show that φ is injective. Suppose that we have bricks B, B′ ∈ B(Λ)
+such that B/ soc B ∼= B′/ soc B′. We use the fact from [Nak41], [ARS95, Theo-
+rem VI.2.1], [ASS06, Theorem V.3.5] that there exist indecomposable projectives
+P, P ′ ∈ proj Λ such that B ∼= P/ radt P and B′ ∼= P ′ radt′ P ′. Then B/ soc B ∼=
+P/ radt−1 P and B′/ soc B′ ∼= P ′/ radt′−1 P ′. Hence B/ soc B ∼= B′/ soc B′ im-
+plies that P ∼= P ′ since indecomposable projectives are determined by their top,
+and so we also have t = t′ and B ∼= B′, as desired.
+Surjectivity of the map φ then follows from the fact that #B(Λ) = #R(Λ)
+because there is a bijection between indecomposable τ-rigid modules and bricks
+generating functorially finite torsion classes from [DIJ19, Theorem 4.1], as well as
+a well-known bijection between simple modules and indecomposable projectives
+(sending a simple module to its projective cover). Note that, since Nakayama
+algebras are representation-finite [Nak41], [ARS95, Theorem VI.2.1], [ASS06,
+Theorem V.3.5], we have that all bricks generate functorially finite torsion classes
+and that both the sets B(Λ) and R(Λ) are finite.
+□
+Note that this bijection φ is specific to Nakayama algebras; it does not co-
+incide with the bijection between indecomposable τ-rigid modules and bricks
+generating functorially finite torsion classes for general algebras given in [DIJ19,
+Theorem 4.1]. We can then use the map φ to relate the relative simples of a
+torsion class to the relative projectives of the torsion class.
+Lemma 5.14. If B is a relatively simple object in a torsion class T for a
+Nakayama algebra Λ, then φ(B) is relatively projective in T .
+Proof. Let B be a relative simple in T . To show that φ(B) is a relative projective
+in T , it suffices to show that HomΛ(X, τφ(B)) = 0 for any X ∈ T , by the
+Auslander–Reiten formula. As in Lemma 5.13, we have that HomΛ(X, τφ(B)) =
+HomΛ(X, rad B). If there were then a non-zero map f : X → rad B, then im f ∈
+T would be a proper submodule of B, which would contradict the fact that B
+is relatively simple in T .
+□
+Note that it is not true in general that for an arbitrary torsion class T ,
+φ restricts to a bijection between non-simple relatively simple objects and
+non-projective indecomposable relatively projective objects in T . Nonetheless,
+Lemma 5.14 allows us to compute the τ-rigid summands of a maximal green
+sequence of a Nakayama algebra from the bricks.
+Proposition 5.15. Let G be a maximal green sequence of a Nakayama algebra Λ.
+Then
+φ(B(G) \ simp Λ) = Sτ(G) \ proj Λ.
+
+66
+MIKHAIL GORSKY AND NICHOLAS J. WILLIAMS
+Proof. Let B ∈ B(G) \ simp Λ. Then by Theorem 2.7 B is a relative simple in
+some torsion class T in G containing B. By Lemma 5.14, we the obtain that
+φ(B) is a relative projective in T . Recalling that φ(B) /∈ proj Λ, we conclude
+that φ(B) ∈ Sτ(G) \ proj Λ. Thus, φ(B(G) \ simp Λ) ⊆ Sτ(G) \ proj Λ. We then
+obtain the result from the fact that #B(G) = #Sτ(G), since both are equal to
+the length of the maximal green sequence.
+□
+We conclude that the orders on maximal green sequences defined by bricks
+and summands coincide for Nakayama algebras.
+Theorem 5.16. Given a Nakayama algebra Λ and two maximal green sequences
+G and G′ of Λ, we have that
+[G] ⩽B [G′] if and only if [G] ⩽S [G′].
+Proof. Since the simple modules and indecomposable projectives are always con-
+tained in B(G) and Sτ(G) respectively, we can make the following chain of de-
+ductions.
+[G] ⩽B [G′] ⇐⇒ B(G) ⊇ B(G′)
+⇐⇒ B(G) \ simp Λ ⊇ B(G′) \ simp Λ
+⇐⇒ φ(B(G) \ simp Λ) ⊇ φ(B(G′) \ simp Λ)
+⇐⇒ Sτ(G) \ proj Λ ⊇ Sτ(G′) \ proj Λ
+⇐⇒ Sτ(G) ⊇ Sτ(G′)
+⇐⇒ [G] ⩽S [G′].
+□
+Corollary 5.17. For a Nakayama algebra Λ and two maximal green sequences
+G and G′ of Λ, the following are equivalent.
+(1) [G] ⩽� [G′].
+(2) [G] ⩽S [G′].
+(3) [G] ⩽HN [G′].
+(4) [G] ⩽B [G′].
+Proof. Equivalence of (1) and (4) is Theorem 5.11. Equivalence of (1) and (3) is
+Corollary 5.12. Finally, equivalence of (2) and (4) is Theorem 5.16.
+□
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