Quivers with potentials for Grassmannian cluster algebras
aa r X i v : . [ m a t h . R T ] A ug QUIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTERALGEBRAS
WEN CHANG AND JIE ZHANG
Abstract.
We consider (iced) quiver with potential ( Q ( D ) , F ( D ) , W ( D )) associ-ated to a Postnilov Diagram D and prove the mutation of the quiver with potential( Q ( D ) , F ( D ) , W ( D )) is compatible with the geometric exchange of the Postnikov dia-gram D . This ensures we may define a quiver with potential for a Grassmannian clusteralgebra. We show such quiver with potential is always rigid (thus non-degenerate) andJacobian-finite. And in fact, it is the unique non-degenerate (thus unique rigid) quiverwith potential associated to the Grassmannian cluster algebra up to right-equivalence,by using a general result of Geiß-Labardini-Schr¨oer [16]. As an application, we verifythat the auto-equivalence group of the generalized cluster category C ( Q,W ) is isomor-phic to the cluster automorphism group of the associated Grassmannian cluster algebra A ( Q,W ) with trivial coefficients. Contents
Introduction 1Conventions 31. Preliminaries 31.1. Cluster algebras 31.2. Quivers with potentials 41.3. Grassmannian cluster algebras 52. Quivers with potentials of Grassmannian cluster algebras 82.1. The definition 82.2. Rigidity and finite dimension 142.3. The uniqueness 213. Application 233.1. Categorification 233.2. Auto-equivalence groups and cluster automorphism groups 23Acknowledgments 25References 25
Introduction
Since have been introduced by Fomin and Zelevinsky in the year 2000 [13], clusteralgebras have been seeing a tremendous development. It is believed that the coordinaterings of several algebraic varieties related to semisimple groups have cluster structures.This has been verified for various varieties. An important and early example is the
Wen Chang is supported by the NSF of China (Grant No. 11601295), Shaanxi Province and ShaanxiNormal University.Jie Zhang is supported by the NSF of China (Grant No. 11401022).
Grassmannians [24]. In this paper, we study the quivers with potentials associated toGrassmannians cluster algebras.Recall that as a subalgebra of a rational function field, a (skew-symmetric) clusteralgebra is generated by cluster variables in various seeds , where a seed is a pair consistingof a quiver and a set of indeterminates in the rational function field. Different seeds arerelated by an operate so called mutation . In some sense, the rich combinatorial structureson the cluster algebras are given by these mutations. There is a representation-theoreticinterpretation of quiver mutations given by Derksen, Weyman and Zelevinsky [11]. Theyintroduced a notion quiver with potential and its decorated representation, where apotential is a sum of cycles in the quiver. The mutations of such quiver with potential andthe representation can be viewed as a generalization of Bernstein-Gelfand-Ponomarevreflection functors.On the other hand, the Postnikov diagram D , which is a kind of planar graph ona disc, corresponds to the cluster in a Grassmannian cluster algebra, which consists ofPl¨ucker coordinates. The strands of the diagram cut the disc into some oriented regionsand alternating oriented regions. Then the quiver Q ( D ) of the cluster can be viewedas a kind of dual of the Postnikov diagram, with the alternating oriented regions as thevertices and the crossings of the strands as the arrows. It is proved by Scott in [24] thatthe mutation µ a ( Q ( D )) of the quiver at some special vertex a is compatible with a locallytransformation µ R ( D ), called geometric exchange, on the Postnikov diagram associatedto an alternating oriented quadrilateral cell R. By considering the boundary of the diskas an oriented cycle, we define an iced quiver ( Q ( D ) , F ). Note that the oriented regionsgives us some minimal cycles in the quiver. Then we define the potential W ( D ) for thequiver as an alternating sum of these minimal cycles. We then get the following theoremwhich is a kind of generalization result of Scott in [24], see Theorem 2.6 for more details. Theorem 0.1.
The geometric exchanges µ R ( D ) of the Postnikov diagram D is compat-ible with the mutation of ( Q ( D ) , F, W ( D )) up to right equivalence. This allows us to define the quivers with potentials for a Grassmannian cluster algebrawhich is independent of the choice of the Postnikov diagram.Note that unlike the quiver mutation, the mutation of a quiver with potential canonly be operated at a vertex which is not involved in 2-cycles, and even the initial quiverin a quiver with potential has no 2-cycles, there may appear 2-cycles after mutations. Aquiver with potential is called non-degenerate if there exists no 2-cycles after any iteratedmutations. A more “generic” condition so called rigidity implies the non-degeneration.So the rigid quiver with potential can be viewed as a kind of “good” quiver with potential,respecting to the mutations. We also study the rigidity of the quiver with potential of aGrassmannian cluster algebra: see Theorem 2.19
Theorem 0.2.
The quiver with potentials associated to a Grassmannian cluster algebrais rigid, and in fact it is the unique rigid quiver with potential up to right equivalence.
We can easily get the following corollary:
Corollary 0.3.
Consider the quiver with potential of a Grassmannnian cluster algebras,we have the quiver determines the potential, which means if two mutation-equivalent quiv-ers are isomorphic, then the associated two quivers with potentials are right equivalent.
Moreover, we have
Theorem 0.4.
The Jacobian algebra of a quiver with potential for a Grassmanniancluster algebra is finite dimensional.
UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 3
As an application of above results, we compare two groups associated to the Grass-mannian cluster algebras. One is the cluster automorphism group, which is introducedin [5] to describe the symmetries of a cluster algebra. It is proved in [5, 6] that if thecluster algebra is of acyclic type, then the cluster automorphism group is isomorphic tothe auto-equivalence group of the corresponding cluster category. We provide a similarisomorphism between these two groups for the Grassmannian cluster algebra with trivialcoefficients, see Theorem 3.5
Theorem 0.5.
Let ( Q, W ) be a quiver with potential for a Grassmannian cluster algebra,then the auto-equivalence group of the generalized cluster category C ( Q,W ) is isomorphic tothe cluster automorphism group of the associated Grassmannian cluster algebra A ( Q,W ) with trivial coefficients. Note that excepting some special Grassmannian cluster algebras, most of them arenon-acyclic. In fact, we prove a more general result, this isomorphism is valid for ageneralized cluster category whose potentials are determined by the quivers. We alsoconjecture that this isomorphism is valid for all the generalized cluster categories.The paper is organized as follows: In section 1, we recall some preliminaries on clus-ter algebras, quiver with potentials and Grassmannian cluster algebras. In section 2,we define the quivers with potentials for Grassmannian cluster algebras and prove itsrigidity and uniqueness. Section 3 is devoted to an application of our main results to thegeneralized cluster categories, we prove the isomorphism between the auto-equivalencegroup of the category and the cluster automorphism group in subsection 3.2.
Conventions
Throughout the paper, we use Z as the set of integers, use N as the set of positiveintegers, and use C as the set of complex numbers. Arrows in a quiver are composedfrom right to left, that is, we write a path j β → i α → k as αβ .1. Preliminaries
In this section, we briefly recall some backgrounds on cluster algebras, quivers withpotentials and Grassmannian cluster algebras.1.1.
Cluster algebras.
We refer to [13, 15] for the notions appearing below.
Quivers:
Recall that a quiver is a quadruple Q = ( Q , Q , s, t ) consisting of a finite set of vertices Q , of a finite set of arrows Q , and of two maps s, t from Q to Q which mapeach arrow α to its source s ( α ) and its target t ( α ), respectively. An iced quiver is a pair( Q, F ) where Q is a quiver and F is a subset (the set of frozen vertices ) of Q . We willalways write the vertex set Q = { , , . . . , n + m } with F = { n + 1 , n + 2 , . . . , n + m } .The full subquiver of Q with vertex set Q \ F (the set of exchangeable vertices ) is calledthe principal part of Q , denoted by Q pr . We call an arrow α ∈ Q • an internal arrow if s ( α ) , t ( α ) / ∈ F ; • a boundary arrow if either s ( α ) ∈ F or t ( α ) ∈ F ; • an external arrow if s ( α ) , t ( α ) ∈ F .Denote by Q ′ , Q ′′ and Q ′′′ the set of internal arrows, of boundary arrows, and of externalarrows of ( Q, F ) respectively.Note that a quiver Q = ( Q, ∅ ) is a special iced quiver with the frozen vertices empty.For an iced quiver ( Q, F ), we call ( Q, ∅ ) the (non-iced) quiver associated to ( Q, F ), andwrite it as Q for brevity. WEN CHANG AND JIE ZHANG
Let (
Q, F ) be an iced quiver without loops nor 2-cycles. A mutation of (
Q, F ) atexchangeable vertex i is an iced quiver ( µ i ( Q ) , F ), where µ i ( Q ) is obtained from Q by: • inserting a new arrow γ : j → k for each path j β → i α → k ; • inverting all arrows passing through i ; • removing the arrows in a maximal set of pairwise disjoint 2-cycles (2 -cyclesmoves ). Seeds:
By associating each vertex i ∈ Q an indeterminate element x i , one gets a set ˜ x = { x , x , . . . , x n + m } = { x , x , . . . , x n } ⊔ { x n +1 , x n +2 , . . . , x n + m } = x ⊔ p . We call thetriple Σ = ( Q, F, ˜ x ) a seed . An element in x (resp. in p ) is called a cluster variable (resp. coefficient variable ), and x is called a cluster .Let x i be a cluster variable, the mutation of the seed Σ at x i is a new seed µ i (Σ) =( µ i ( Q ) , F, µ i ( x )), where µ i ( x ) = ( x \ { x i } ) ⊔ { x ′ i } with(1.1) x i x ′ i = Y α ∈ Q ; s ( α )= i x t ( α ) + Y α ∈ Q ; t ( α )= i x s ( α ) . Cluster algebras:
Denote by X the union of all possible clusters obtained from an initial seed Σ =( Q, F, ˜ x ) by iterated mutations. Let P be the free abelian group (written multiplicatively)generated by the elements of p . Let F = QP ( x , x , . . . , x n ) be the field of rationalfunctions in n independent variables with coefficients in QP . The cluster algebra A ( Q,F ) is a ZP -subalgebra of F generated by cluster variables in X , that is A ( Q,F ) = ZP [ X ] . Quivers with potentials.
The references of this subsection are [7, 11, 16, 22],especially [22] for the case of iced quiver with potentials.
Quivers with potentials and Jacobian algebras:
Let (
Q, F ) be an iced quiver without loops. We denote by C h Q i the path algebra of Q over C . By length ( p ) we denote the length of a path p in C h Q i . The complete path algebra C hh Q ii is the completion of C h Q i with respect to the ideal m generated by the arrows of Q . A potential W on Q is an element in the closure P ot ( Q ) of the space generated by allcycles in Q . We say two potentials W and W ′ are cyclically equivalent if W − W ′ belongsto the closure C of the space generated by all differences α s · · · α α − α α s · · · α for thecycle α s · · · α α . Denote by [ l ] the set of cycles which cyclically equivalent to a cycle l . We call a pair ( Q, W ) a quiver with potential or a QP for brevity, if no two termsin W ∈ P ot ( Q ) are cyclically equivalent. We call a triple ( Q, F, W ) an iced quiver withpotential or an IQP for brevity, if no two terms in W ∈ P ot ( Q ) are cyclically equivalent,and each term in W includes at least one unexternal arrow. As for the quiver, we alsoview a QP as a special IQP.Let ( Q, F , W ) and ( e Q, e F , f W ) be two IQPs with Q = e Q and F = e F . Their directsum , denoted by ( Q, F , W ) ⊕ ( e Q, e F , f W ), is a new IQP ( Q, F, W ), where Q is a quiverwith Q = Q (= e Q ), F = F (= e F ), Q = Q ⊔ e Q , and W = W + f W .For an arrow α of Q , we define ∂ α : P ot ( Q ) → C hh Q ii the cyclic derivative withrespect to α , which is the unique continuous linear map that sends a cycle l to the sum P l = pαq pq taken over all decompositions of the cycle l . Let J ( Q, F, W ) be the closure
UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 5 of the ideal of C h Q i generated by cyclic derivatives in { ∂ α W, α ∈ Q ′ ⊔ Q ′′ } . We call J ( Q, F, W ) the
Jacobian ideal of (
Q, F, W ) and call the quotient P ( Q, F, W ) = C hh Q ii /J ( Q, F, W )the
Jacobian algebra of (
Q, F, W ).For an IQP (
Q, F, W ), we call it trivial if each term in W is a 2-cycle and P ( Q, F, W )is the product of copies of C , we say it is reduced if each term of W includes at least oneunexternal arrow and ∂ β W ∈ m for all β ∈ Q ′ ⊔ Q ′′ (see [22, Definition 3.3]). Right-equivalences:
Two IQPs (
Q, F, W ) and ( e Q, e F , f W ) are right-equivalent if Q and e Q have the same setof vertices and frozen vertices, and there exists an algebra isomorphism ϕ : C hh Q ii → C hh e Q ii whose restriction on vertices is the identity map and ϕ ( W ) and f W are cyclicallyequivalent. Such an isomorphism ϕ is called a right-equivalence .It is proved in [22, Theorem 3.6] (in [11, Theorem 4.6] that for QP) that for any IQP( Q, F, W ), there exist a trivial IQP ( Q tri , F tri , W tri ) and a reduced IQP ( Q red , F red , W red )such that ( Q, F, W ) is right-equivalent to ( Q tri , F tri , W tri ) ⊕ ( Q red , F red , W red ). Further-more, the right-equivalence class of each of ( Q tri , F tri , W tri ) and ( Q red , F red , W red ) is de-termined by the right-equivalence class of ( Q, F, W ). Mutations of iced quivers with potentials:
Let (
Q, F, W ) be an IQP, and let i be an exchangeable vertex of Q such that thereis no 2-cycles at i and no cycle occurring in the decomposition of W starts and ends at i . The pre-mutation e µ i ( Q, F, W ) of (
Q, F, W ) is a new potential ( e Q, e F , f W ) defined asfollows, where e F = F . The new quiver e Q is obtained from Q by • adding a new arrow [ αβ ] : j → k for each path j β → i α → k ; • replacing each arrow α incident to i with an arrow α ∗ in the opposite direction.The new potential f W is the sum of two potentials f W and f W . The potential f W is obtained from W by replacing each composition αβ by [ αβ ] for any α and β with s ( α ) = t ( β ) = i . The potential f W is given by f W = X α,β [ αβ ] β ∗ α ∗ , where the sum ranges over all pairs of arrows α and β with s ( α ) = t ( β ) = i . We denoteby µ i ( Q, F, W ) the reduced part of e µ i ( Q, F, W ), and call µ i the mutation of ( Q, F, W ) atthe vertex i . We call two IQPs mutation equivalent if one can be obtained from anotherby iterated mutations. Note that the mutation equivalence is an equivalent relation onthe set of right-equivalence classes of IQPs.1.3. Grassmannian cluster algebras.
We recall in the subsection some definitionson Grassmannian cluster algebras, and we refer to [21] and [24] for more details onPostnikov diagrams and Grassmannian cluster algebras, respectively.Let Gr( k, n ) be the Grassmannian of k -planes in C n and C [Gr( k, n )] be its homoge-neous coordinate ring. When k = 2, Fomin and Zelevinsky proved that C [Gr( k, n )] hasa cluster algebra structure [14]. Scott generalised this result to the case of any Grass-mannian Gr( k, n ), where the proof relies on a correspondence between a special kind ofclusters in the cluster algebra and a kind of planar diagram - the Postnikov diagram.For k, n ∈ N with k < n , a ( k, n ) -Postnikov diagram D is a collection of n ori-ented paths, called strands , in a disk with n marked points on its boundary, labeled by WEN CHANG AND JIE ZHANG , , . . . , n in clockwise orientation. The strands, which are labeled by 1 i n , startat point i and end at point i + k . These strands obey the following conditions: • Any two strands cross transversely, and there are no triple crossings betweenstrands. • No strand intersects itself. • There are finite many crossing points. • Following any given strand, the other strands alternately cross it between fromleft to right and from right to left. • For any two strands i and j the configuration shown in Figure 1 is forbidden. · · ·· · · ij Figure 1.
Forbidden crossingPostnikov diagrams are identified up to isotopy. We say that a Postnikov diagram isof reduced type if no untwisting move shown in Figure 2 can be applied to it. The forth ij ij
Figure 2.
Untwisting movecondition ensures that the strands divide the disc into two types of regions: orientedregions , where all the strands on their boundary circles clockwise or counterclockwise,and alternating oriented regions , where the adjacent strands alternate directions. Aregion is said to be internal if it is not adjacent to the boundary of the disk, and theother regions are referred to as boundary regions . See Figure 3 for an example of (3 , D and an alternating oriented quadrilateral cell R inside D , a new Postnikov diagram e µ R ( D ) is constructed by the local rearrangement shownin Figure 4. We call e µ R a pre-geometric exchange at R . Note that there may appearnew configurations in e µ R ( D ) as shown in the left side of Figure 2. Let µ R ( D ) be thePostnikov diagram obtained from e µ R ( D ) after untwisting these new configurations. Wecall µ R a geometric exchange at R . Note that if D is of reduced type then so does µ R ( D ). Grassmannian cluster algebras:
For a Postnikov diagram D , one may associate it an iced quiver ( Q ( D ) , F ( D )), whosevertices are labeled by the alternating oriented regions of D and arrows correspondto intersection points of two alternating regions, with orientation as in Figure 5. Theinternal alternating oriented regions correspond to the exchangeable vertices of Q ( D ),while the boundary alternating oriented regions correspond to the frozen vertices. Notethat the set F ( D ) is independent of the choice of the ( k, n )-Postnikov diagram, so weuse F , ( Q ( D ) , F ) instead of F ( D ), ( Q ( D ) , F ( D )) for brevity when ( k, n ) is fixed.Note that there are no external arrows in Q ( D ). It has been proved in [24] thatthe geometric exchange of the Postnikov diagram is compatible with the mutation of the UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 7
Figure 3.
A (3 , ji st RD ji st e µ R ( D ) Figure 4.
Pre-geometric exchange
Figure 5.
Arrow orientation for the quiver Q ( D )quiver, after forgetting the external arrows arising from the quiver mutation, and the co-ordinate ring C [Gr( k, n )] has a cluster algebra structure, more precisely, the localizationof C [Gr( k, n )] at consecutive Pl¨ucker coordinates is isomorphic to the complexificationof A ( Q ( D ) ,F ) as cluster algebras. WEN CHANG AND JIE ZHANG ba m (1) ba m (2)
Figure 6.
The local configuration of boundary regions2.
Quivers with potentials of Grassmannian cluster algebras
We introduce in this section a quiver with potential ( Q ( D ) , F, W ( D )) for each Post-nikov diagram D , and study the compatibility of geometric exchange of D and the mu-tation of the quiver with potential. We also consider the dimension of the correspond-ing Jacobian algebra P ( Q ( D ) , F, W ( D )) and we establish the unique non-degenerate(thus unique rigid) quiver with potential associated to the Grassmannian cluster alge-bra A ( Q ( D ) ,F,W ( D )) up to right-equivalence.Note that the iced quiver ( Q ( D ) , F ), associated to a Postnikov diagram D , does notcontain external arrows, which have no influence on the “cluster structure” of the clusteralgebra A ( Q ( D ) ,F ) . However, by adding some external arrows in ( Q ( D ) , F ), we show theyplay an important role in the (Frobenius) categorification of the cluster algebras (seeSection 3). It is also important for the properties about the IQP itself (see Remarks 2.7,2.13 and 2.15). Some special choice of such arrows ensures that we are able to define theQP associated to the Grassmannian cluster algebras (see Remarks 2.7 and Definition2.8).2.1. The definition.
In order to define a new quiver ( Q ( D ) , F ) by adding externalarrows in ( Q ( D ) , F ), we consider the boundary of D as an oriented cycle, with clockwiseorientation. We write C.W. (resp.
A.C.W. ) for clockwise (resp. anticlockwise) forbrevity. The local configurations of the boundary areas are showed in Figure 6, where a and b are alternating oriented boundary regions which correspond frozen vertices in Q ,and m is a marked point on the boundary. By adding an external arrow from b to a in Q ( D ) for each point m depicted in Figure 6 (1), where the strands are clockwise, we getthe new iced quiver ( Q ( D ) , F ). Remark 2.1.
Furthermore, by adding an external arrow from a to b in ( Q ( D ) , F ) foreach point m in the Figure 6 (2), where the strands incident to m are anticlockwise, onegets another quiver ( Q ( D ) , F ) , which was used by Buar-King-Marsh in [8] as the quiverof the dimer algebra associated to a Postnikov diagram. See Figure 7 for the quivers associated to the Postnikov diagram in Figure 3. In thefollowing, we will mainly consider the three kinds of quivers associated to a Postnikovdiagram D , the quiver ( Q ( D ) , F ), its principal part Q pr ( D ), and the quiver ( Q ( D ) , F )defined above. We call them the quiver of D of type I, of type II and of type IIIrespectively.Each oriented region r in D gives a set of cyclically equivalent cycles in the quiverwith arrows bound the region r , see Figure 8. Depending on the (internal or boundary)location of r , there are two kinds of such cycles,. We call them the fundamental cyclesof type I and type II respectively. For a region r , denote by [ ω r ] the set of the associatedcycles with a representative ω r . Note that the quiver ( Q ( D ) , F ) and its principal part Q pr ( D ) have only the fundamental cycles of type I. The quiver ( Q ( D ) , F ) may have bothfundamental cycles of type I and type II. UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 9
Figure 7.
The quiver of the Postnikov diagram in Figure 3. Startingfrom the quiver (
Q, F ), we add the clockwise red external arrows to obtainthe quiver (
Q, F ), and further add dashed external arrows to obtain thequiver (
Q, F ). ...r ...r
Figure 8.
The fundamental cycles, where the horizonal dashed line isthe boundary of the diagram, and other dashed lines are the strands.The omitted part is at the interior of the diagram. The first cycle canalso occur in the opposite sense, which means inverting the orientationssimultaneously.
Definition 2.2.
For quivers ( Q ( D ) , F ) , Q pr ( D ) , and ( Q ( D ) , F ) , set W ( D ) , W pr ( D ) ,and W ( D ) be potentials in the corresponding quivers, which are signed sums of repre-sentatives of fundamental cycles in the quivers, that is W = X r C.W.ω r ∈ Q ( D ) ω r − X r A.C.W.ω r ∈ Q ( D ) ω r ; W pr = X r C.W.ω r ∈ Q pr ( D ) ω r − X r A.C.W.ω r ∈ Q pr ( D ) ω r ; W = X r C.W.ω r ∈ Q ( D ) ω r − X r A.C.W.ω r ∈ Q ( D ) ω r . Note that for each oriented region, we choose one representative in the potential, sothese potentials are dependent on such choice. However, different choices are cyclicallyequivalent to each other. Clearly, there are no two cyclically equivalent cycles appearingin the potential simultaneously. So ( Q pr ( D ) , W pr ( D )) is a QP, ( Q ( D ) , F, W ( D )) and( Q ( D ) , F, W ( D )) are IQPs. We call them the (iced) QPs associated to D of type I, oftype II and of type III respectively. Note that these (iced) QPs are also reduced bydefinition. Remark 2.3.
Similarly, one may define an iced quiver with potential ( Q ( D ) , F, W ( D )) ,which is also defined in [22] . It will be shown in Remarks 2.7, 2.13 and 2.15 that theproperties of ( Q ( D ) , F, W ( D )) and ( Q ( D ) , F, W ( D )) are very different. We are going to prove that the mutation of ( Q ( D ) , F, W ( D )) is compatible with thegeometric exchanges of the Postnikov diagrams D . For convenience we introduce thefollowing Definition 2.4.
A fundamental cycle ω of ( Q ( D ) , F ) , Q pr ( D ) , or ( Q ( D ) , F ) is called • an internal fundamental cycle if the arrows of ω are all internal arrows; • a boundary fundamental cycle if there exist both boundary arrows and internalarrows in ω ; • an external fundamental cycle if there exist no external arrows in ω . In order to prove the compatibility of geometric exchange of the Postnikov diagramand the mutation of the iced quiver with potential, we need the following lemma.
Lemma 2.5.
Let D be a Postnikov diagram. let ε : { r } → {± } be a function on theset of oriented regions in D . Define potentials W ε = X r C.W.ω r ∈ Q ( D ) ε ( r ) ω r − X r A.C.W.ω r ∈ Q ( D ) ε ( r ) ω r ; W prε ( D ) = X r C.W.ω r ∈ Q pr ( D ) ε ( r ) ω r − X r A.C.W.ω r ∈ Q pr ( D ) ε ( r ) ω r ; W ε ( D ) = X r C.W.ω r ∈ Q ( D ) ε ( r ) ω r − X r A.C.W.ω r ∈ Q ( D ) ε ( r ) ω r . Then ( Q ( D ) , F, W ε ( D )) (resp. ( Q pr ( D ) , W prε ( D )) , ( Q ( D ) , F, W ε ( D )) ) is right-equivalentto ( Q ( D ) , F, W ( D )) (resp. ( Q pr ( D ) , W pr ( D )) , ( Q ( D ) , F, W ( D )) ).Proof. We only deal with the case of ( Q ( D ) , F ), other cases are similar. Because theunderlying graph of ( Q ( D ) , F ) is a planar graph with non-trivial boundary, there existsa function ξ : Q ( D ) → {± } UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 11 on the arrows of Q ( D ) such that for any r with ω r = α m · · · α α , we have m Y i =1 ξ ( α i ) = ε ( r )for any ε . So the map φ : Q ( D ) → Q ( D ) , α ξ ( α ) α induces an algebra isomorphism Φ from C hh Q ( D ) ii to C hh Q ( D ) ii which maps W ( D ) to W ε ( D ). Then Φ is a right-equivalence which completes the proof. (cid:3) Now we are ready to give the main result in this subsection.
Theorem 2.6.
The mutation of ( Q ( D ) , F, W ( D )) is compatible with the geometric ex-changes of the Postnikov diagram. More precisely, let D be a reduced Postnikov diagramwith an alternating oriented quadrilateral cell R , which associates to an exchangeablevertex a of Q ( D ) . Then we have µ a ( Q ( D )) = Q ( µ R ( D )) and µ a ( W ( D )) = W ( µ R ( D )) .Proof. Since R is quadrilateral in D , a is an end point of four arrows α, β, γ, δ in( Q ( D ) , F ). On the other hand, since a is exchangeable, we have α, β, γ, δ ∈ Q ′ ( D ) ⊔ Q ′′ ( D ) . Without loss of generality, we assume that a = s ( α ) = t ( β ). Let ω r = αβp be afundamental cycle in W ( D ) corresponding to an oriented region r of D that containsboth α and β , where p is a path from t ( α ) to s ( β ).Essentially, there are four possibilities of ω r , which are showed in Figure 9 (1 . . .
0) and (4 . length ( p ) > length ( p ) = 1, then the local configuration of D is as showed in Figure 9 (1 . . . .
1) or (2 . . r is the oriented region encircled by the dashed strands.In Figure 9 (1 . . .
1) and (4 . r is an internal region, and α, β ∈ Q ′ ( D ). InFigure 9 (2 .
3) and (4 . r is a boundary region with α, β ∈ Q ′′ ( D ), where the orientationof r coincides with the orientation of the boundary.If length ( p ) >
1, then the local configuration is as showed in Figure 9 (1 . . .
2) or (4 . length ( p ) = 0 if there is no cycle in W ( D ) containing both α and β . In this case, the orientation of r is opposite to theorientation of the boundary, see Figure 9 (1 . . α and β . For example, we may choose the configurations (1 . . .
2) and (4 .
1) as the representatives. Now we consider the local confugration of a depicted in the first picture of Figure 10, which contains all the four representatives.So we only prove the result for this situation. Other situations can be proved similarly.We write the potential W ( D ) = W ′ ( D ) + sαδ − rγδ + qγβ − qt, where length ( r ) > length ( t ) > W ′ ( D ) does not contain any oneof α, β, γ, δ, q , and s . Then by a pre-geometric exchange e µ R on D and a pre-mutation e µ a on ( Q, F ), we obtain the second picture in Figure 10. Meanwhile, by applying thepre-mutation e µ a on W ( D ), we get a new potential e µ a ( W ( D )) = W ′ ( D )+ s [ αδ ] − r [ γδ ]+ q [ γβ ] − qt +[ αβ ] β ∗ α ∗ +[ αδ ] δ ∗ α ∗ +[ γδ ] δ ∗ γ ∗ +[ γβ ] β ∗ γ ∗ . (1.0) β αap β αp a (1.1) β αp a (1.2) β αp a (1.3)(2.0) βαa p βα pa (2.1) βα pa (2.2) βα pa (2.3)(3.0) βαa p βα pa (3.1) βα pa (3.2) βα pa (3.3)(4.0) β αap β αp a (4.1) β αp a (4.2) β αp a (4.3) Figure 9.
Local configuration of a fundamental circle. The rotationsaround a are also allowedNote that W ′ ( D ) is not changed because any arrow appearing in any cycle of W ′ ( D ) isnot involved in a . To reduce the IQP ( e µ a ( Q ( D )) , F, e µ a ( W ( D ))), we consider a unitrian-gular automorphism f on C hh e µ a ( Q ( D )) ii , where f ([ γβ ]) = [ γβ ] + t, f ( q ) = q − β ∗ γ ∗ , f ( s ) = s − δ ∗ α ∗ , f ( u ) = u for other arrows u in e µ a (( Q ( D )). Then f ( e µ a ( W ( D ))) = W ′ ( D ) − r [ γδ ] + β ∗ γ ∗ t + β ∗ α ∗ [ αβ ] + δ ∗ γ ∗ [ γδ ] + s [ αδ ] + q [ γβ ]with the reduced part f ( e µ a ( W ( D ))) red = W ′ ( D ) − r [ γδ ] + β ∗ γ ∗ t + β ∗ α ∗ [ αβ ] + δ ∗ γ ∗ [ γδ ] . Then the third picture of Figure 10 shows the final Postnikov diagram µ R ( D ) andthe final quiver µ a ( Q ( D )) after applying the untwisting moves on the diagram e µ R ( D ) UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 13 β α δγa sq rt pre-mutationspre-geometricexchanges(
D, Q ) α ∗ β ∗ γ ∗ δ ∗ [ αδ ][ γδ ][ αβ ][ γβ ] sq rt a e µ R ( D ) , e µ a ( Q )) α ∗ β ∗ γ ∗ δ ∗ [ δγ ][ βα ] rt a ( µ R ( D ) , µ a ( Q )) Figure 10.
Mutations and geometric exchangesand the 2-cycles moves on the quiver e µ a ( Q ( D )) respectively. From the picture, we have Q ( µ R ( D )) = µ a ( Q ( D )). By the definition, W ( µ R ( D )) = W ′ ( D ) − r [ γδ ] − β ∗ γ ∗ t + β ∗ α ∗ [ αβ ] + δ ∗ γ ∗ [ γδ ] . By Lemma 2.5, there is a sign changing of arrows φ on C hh µ a ( Q ( D )) ii such that φ ( f ( e µ a ( W ( D ))) red ) = W ( µ R ( D ))up to the equality µ a ( Q ( D )) = Q ( µ R ( D )). So by the right-equivalent φf , we obtain theIQP µ a ( Q ( D ) , F, W ( D )), as well as the wanted equalities µ a ( Q ( D )) = Q ( µ R ( D )) and µ a ( W ( D )) = W ( µ R ( D )) . (cid:3) Remark 2.7.
Since ( Q pr ( D ) , W pr ( D )) is the principal part of ( Q ( D ) , F, W ( D )) , theabove theorem yields that the mutation of ( Q pr ( D ) , W pr ( D )) is also compatible withthe geometric exchanges of the Postnikov diagram. However, this is not true for thecase of ( Q ( D ) , F, W ( D )) and ( Q ( D ) , F, W ( D )) . The compatibility ensures the followingdefinition. Definition 2.8.
Let D be any ( k, n ) -Postnikov diagram and let C [ Gr ( k, n )] be the Grass-mannian cluster algebra. We call • a QP which is mutation equivalent to ( Q pr ( D ) , W pr ( D )) the QP of C [ Gr ( k, n )] ,and denote it by ( Q pr , W pr ) ; • a IQP which is mutation equivalent to ( Q ( D ) , F, W ( D )) the IQP of C [ Gr ( k, n )] ,and denote it by ( Q, F, W ) . Rigidity and finite dimension.
We prove in this subsection that any iced quiverwith potential of a Grassmannian cluster algebra C [ Gr ( k, n )] is rigid and Jacobi-finite.Recall that an IQP ( Q, F, W ) 2-acyclic if there is no 2-cycles in the quiver. Note thatthere may appear 2-cycles in the quiver of µ i ( Q, F, W ) after IQP mutations, even if(
Q, F, W ) is 2-acyclic. If all possible iterations of mutations are 2-acyclic, then we say(
Q, F, W ) is non-degenerate . We call (
Q, F, W ) rigid if every cycle in Q is cyclicallyequivalent to an element of the Jacobian ideal J ( Q, F, W ). It is known that a rigidIQP is always non-degenerate. We call (
Q, F, W ) Jacobi-finite if the Jacobian algebra P ( Q, F, W ) is finite dimensional.For the further study, we need some special Postnikov diagram, see Figures 11, wherethe diagrams depend on the parities of k and n , and any pair ( k, n ) matchs a uniquediagram shown in these figures. These diagrams are of special importance, they areused by Scott as the initial diagrams which give the initial quivers of the Grassmanniancluster algebras [24].Denote by ( Q ini , F ), Q prini and ( Q ini , F ) the quivers of type I, of type II and of typeIII associated to these initial Postnikov diagrams. The quivers ( Q ini , F ) are given inFigures 12 and 13, where the boundary arrows are blue and the external arrows are red.As depicted in Figure 12, we endow the points of the quiver with coordinates . Denoteby a ( i, j ) the vertex with coordinate ( i, j ). We also separate the fundamental cycles intodifferent levels , see Figure 12 for example.Let l be a path which forms a cycle in Q ini . Let a ( i , j ) be a vertex on l , we sayit is a leftmost vertex of l if i i for any vertex a ( i, j ) on l . Similarly, we define the rightmost vertex , lowest vertex and highest vertex of l as a ( i , j ), a ( i , j ) and a ( i , j )respectively. We call width ( l ) = i − i the width of l , and call height ( l ) = j − j the height of l .To prove the rigidity and the Jacobian finite property of ( Q, F, W ), we need thefollowing lemma.
Lemma 2.9.
For any cycle l of ( Q ini , F ) with end point ( i , j ) , let ω be a fundamentalcycle with end point ( i , j ) , then there exists a positive integer m such that l − ω m ∈ J ( Q ini , F, W ini ) .Proof. We prove this for the quivers in Figure 12, that is the case when k and n areboth odd. For the rest results in the paper, we also only consider these quivers, othercases are similar. Let ω be a fundamental cycle with end point ( i , j ). Without loss ofgenerality, we may assume that ( i , j ) is at the top left corner of ω . Then ω is located UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 15 · · ·· · ···· ··· n − k +32 n − k +22 n n +12 (1) k odd , n odd 1 n − k +32 n n n − k +32 · · ·· · ···· ··· (2) k odd , n even12 n − k +32 2 n − k n n +12 · · ·· · ···· ··· (3) k even , n odd 1 n +22 n − k n − k +22 n · · ·· · ···· ··· (4) k even , n even Figure 11.
Initial Postnikov diagramat the j -th level of ( Q ini , F ), where j = j − if j / ∈ N ; j if j ∈ N and i = 0; j − i , j ) = (0 , n − k ) . The proof is proceeded in two steps.
Step 1:
We claim that there exists a cycle ξ satisfying the following conditions:(1) the end point of ξ is ( i , j );(2) l − ξ ∈ J ( Q ini , F, W ini );(3) any highest vertex of ξ is located at the top of the j -th level of ( Q ini , F ).Let a be a highest vertex of l . Up to the left-right symmetries, there are nine kindsof possible positions of a which we show in Figure 14, where the bold arrows form asubpath of some cycle in [ l ]. Now for each case, we construct a new cycle l ′ from l withend point ( i , j ) such that l − l ′ ∈ J ( Q ini , F, W ini ). · · ·· · ·· · ·· · ···· ······ ··· level 1level 2level 3level n − k − n − k (0 ,
0) (1 ,
1) (2 , ,
2) ( k − , ,
0) ( k, , )(0 , n − k − )(0 , n − k ) ( , n − k ) ( k, n − k ) ••••• ••••• ••••• ••••• ••••• •••••••••• •••••• • •• • • Figure 12.
Initial quiver ( Q ini , F ) ( k odd, n odd) · · ·· · ·· · ·· · ···· ······ ··· •••••• •••••• •••••• •••••• •••••• ••••••••••• •••••• • •• • • (2) k odd , n even · · ·· · ·· · ·· · ···· ······ ··· •••••• •••••• •••••• •••••• •••••• •••••••••• • •• • (3) k even , n odd · · ·· · ·· · ·· · ···· ······ ··· ••••• ••••• ••••• ••••• ••••• ••• •••• • •• • • • • (4) k even , n even Figure 13.
Initial quiver ( Q ini , F ) UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 17
Case I. a = a (0 , s + ),1 s n − k − or a = a (0 , n − k ), and the local configurationof l neighbouring a is depicted in Figure 14 (1). We assume that none of the end pointsof γ is ( i , j ). Otherwise, a is already located at the top of the j -th level of ( Q ini , F ),so it is unnecessary to consider such a . Therefore δγβ is a subpath of l , and we maywrite l = qδγβp with p and q the subpaths of l , where p and q maybe trivial paths. Let l ′′ = qνµp . Then the end point of l ′′ is still the ( i , j ), and l − l ′′ = q ( δγβ − νµ ) p = p ( ∂ α W ini ) q ∈ J ( Q ini , F, W ini ) . If a is still a vertex on l ′′ , we repeat above constructionuntill a is never a vertex on a cycle l ′ , which makes sense since the length of l is finite.The final cycle l ′ is what we want.Case II. a = a (0 , n − k ) is depicted in Figure 14 (2). Similarly, we also assume thatnone of the end points of β is the ( i , j ), and write l = qγβp with subpaths p and q .Let l ′′ = qνµδp . Repeating this construction, we obtain a cycle l ′′′ that satisfies theconditions of l in the Case I. So we reduce this case to the first case.Case III. a = a (1 , s ), 1 s n − k − , is depicted in Figure 14 (3). If ( i , j ) = a thenthere is nothing to prove. Otherwise βγ is a subpath of l , and we write l = qβγp anduse the partial ∂ α to construct cycle l ′ similar to Case I.Case IV. a = a (2 s + , n − k ),0 s k − , is depicted in Figure 14 (4). We may write l = qγβp and use the partial ∂ α to construct cycle l ′ similar to Case I.Case V. a = a (2 s, s k − , is depicted in Figure 14 (5). We may write l = qδµνp and use the partial ∂ α to construct cycle l ′ similar to Case I.Case VI. a = a (1 + 2 s, t ), 0 s k − , t n − k − or a = a (2 + 2 s, t ),0 s k − , t n − k − and the local configuration of l neighbouring a is depictedin Figure 14 (6). We may write l = qβγδp and use the partial ∂ α to construct cycle l ′ similar to Case I.Case VII. a = a (1 + 2 s, s k − , is depicted in Figure 14 (7). Note that in thiscase, ω is at the second level of Q ini and l already satisfies the conditions in the claim.Case VIII. a = a (2 s, s k − , is depicted in Figure 14 (8). Note that in thiscase, ω is at the first level of Q ini and l already satisfies the conditions in the claim.Case IX. a = a (0 , ) and the local configuration of l neighbouring a is depicted inFigure 14 (9). Note that in this case, l is in fact a power of a fundamental cycle in[ γβα ] = [ ω ], so there is nothing to prove.Now the cycle l ′ obtained in each case has the following properties,(1) l − l ′ ∈ J ( Q ini , F, W ini );(2) a is never a highest vertex of l ′ ;(3) no new highest vertex arises in l ′ with respect to l .Thus by inductively constructing the cycle l ′ , we may find a cycle ξ satisfies the conditionsin the claim. Step 2:
For the cycle ξ produced in Step I, we consider the lowest, the leftmost andthe rightmost vertices, similar to the analysis used in Step I, we obtain a cycle ζ suchthat(1) the end point of ζ is ( i , j );(2) l − ζ ∈ J ( Q ini , F, W ini );(3) ζ is at the j -level and width ( ζ ) = 1.By the items (1) and (3), ζ is a power of a fundamental cycle ω ′ whose end point is( i , j ). By our assumption, ( i , j ) is an end point of ω , so ω ′ ∈ [ ω ]. Moreover, notethat ω ′ can be any fundamental cycle in [ ω ]. Therefore there exists a positive integer m such that l − ω m ∈ J ( Q ini , F, W ini ). (cid:3) aβ δνγαµ (1) a δ αµ νβ γ (2) aγβα (3) aδ αµ νβ γ (4) aδ µα νβ γ (5) a (6) γαµ ρνδ βa (7) γαδ β a (8) βα γ aα βγ (9) Figure 14.
Local configuration neighbouring the highest vertex a of a cycle Remark 2.10.
Let l be any cycle in Q ini , note that for any fundamental cycle ω ap-pearing in the above lemma, the positive integer m is the same. We call m the essentiallength of l , and denote it by elength ( l ) . So a cycle is of essential length one if andonly if it is a fundamental cycle. Since two paths may have different length in a relation ∂ α W , α ∈ Q ′ ⊔ Q ′′ , it implies two cycles with the same essential length may also havedifferent length. However, we can estimate the gap between the length and the essentiallength by inequality elength ( l ) length ( l ) elength ( l ) , since a fundamental cycle in Q ini is a triangle or a quadrangle. Theorem 2.11.
Any IQP ( Q, F, W ) of a Grassmannian cluster algebra is rigid. Because the IQP-mutations preserve the rigidity, it suffices to prove the theorem forthe initial IQP ( Q ini , F, W ini ). So we have to show that any cycle in ( Q ini , F ) is cyclicallyequivalent to a cycle in the Jacobian idea J ( Q ini , F, W ini ). This is easy for the case k = 2or k = n −
2. Now we assume k = 2 and k = n −
2. As before, we only consider thequivers in Figure 12. The following lemma is useful.
Lemma 2.12.
Let ω and ω be two fundamental cycles of ( Q ini , F ) sharing a commonarrow α . For any positive integer m , if ω m is cyclically equivalent to an element in J ( Q ini , F, W ini ) , then ω m is also cyclically equivalent to an element in J ( Q ini , F, W ini ) .Proof. Recall that C is the closure of the span of all elements of the form α s · · · α α − α α s · · · α , UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 19 where α s · · · α α is a cycle. Since ω m is cyclically equivalent to an element in the ideal J ( Q ini , F, W ini ), there is a potential ω ∈ J ( Q ini , F, W ini ) such that ω m − ω ∈ C . Assumethat αp (resp. αp ) is the fundamental cycle which is cyclically equivalent to ω (resp. ω ), where p and p are paths with head t ( α ) and tail h ( α ). Since α belongs to twodifferent fundamental cycles, α ∈ Q ′ ⊔ Q ′′ , we can use the partial derivation ∂ α to obtainthat αp − αp ∈ J ( Q ini , F, W ini ) . Moreover, since αp − αp is a factor of ( αp ) m − ( αp ) m ,( αp ) m − ( αp ) m ∈ J ( Q ini , F, W ini ) . Note that ( αp ) m − ω m ∈ C and ω m − ω ∈ C , thus ( αp ) m − ω ∈ C . Therefore ω m − [( αp ) m − ( αp ) m + ω ] = [ ω m − ( αp ) m ] + [( αp ) m − ω ] ∈ C, where ( αp ) m − ( αp ) m + ω ∈ J ( Q ini , F, W ini ). This completes the proof. (cid:3) Proof of the theorem: we separate the proof into three steps.
Step 1:
For the iced quiver in Figure 12, there exists a boundary arrow α and afundamental cycle αp such that the only fundamental cycles which contain α are thosein [ αp ]. Actually, one may always choose the leftmost fundamental triangle on thebottom level of the quiver. So( αp ) m = ( α∂ α W ) m ∈ J ( Q ini , F, W ini )for any positive integer m . Therefore ω m is cyclically equivalent to an element in J ( Q ini , F, W ini ) for any fundamental cycle ω in [ αp ] and any positive integer m . Step 2:
For any fundamental cycle ω and any positive integer m , by recursively usingLemma 2.12, we find a fundamental cycle ω appearing in Step I, such that ω m is cycli-cally equivalent ω m . Thus ω m is cyclically equivalent to an element in J ( Q ini , F, W ini ). Step 3:
For any cycle l in ( Q ini , F ), by Lemma 2.9, there is a power ω m of funda-mental cycle with l − ω m ∈ J ( Q ini , F, W ini ). By Step II, there is an element ω m in J ( Q ini , F, W ini ) such that ω m − ω m ∈ C . That is l − [ l − ω m + ω m ] ∈ C, where l − ω m + ω m ∈ J ( Q ini , F, W ini ). This means that l is cyclically equivalent to anelement in J ( Q ini , F, W ini ). Thus the IQP ( Q ini , F, W ini ) is rigid. Remark 2.13.
By using a similar method, one may show that the quiver with potential ( Q pr , W pr ) of C [ Gr ( k, n )] is also rigid. However, the IQP ( Q ini , F, W ini ) is not rigid.In particular, any fundamental cycle in ( Q ini , F, W ini ) is not cyclically equivalent to acycle in J ( Q ini , F, W ini ) . Theorem 2.14.
For each IQP ( Q, F, W ) of the Grassmannian cluster algebra, the Ja-cobian algebra P ( Q, F, W ) is finite dimensional.Proof. Since the Jacobi-finiteness of a IQP is invariant under the IQP-mutations, weonly prove this for the initial IQP ( Q ini , F, W ini ). We have to prove that if the lengthof a cycle is large enough, then the cycle belongs to the Jacobian ideal.By the Lemma 2.9, and the inequality in Remark 2.10, we only need to show that forany fundamental cycle ω , there is a positive integer m such that ω m ∈ J ( Q ini , F, W ini ) . ··· ωω ω ab αδ γ β Figure 15.
Jacobi-finiteness of the IQPThis can be done by iteratively using the relations in J ( Q ini , F, W ini ). For example, weconsider ω m with ω shown in Figure 15, where the end points of fundamental cycles ω , ω and ω are a , a , and b respectively. Then we have ω m − ω m ∈ J ( Q ini , F, W ini ) ,ω m − δω m − γβα ∈ J ( Q ini , F, W ini ) , and thus ω m − δω m − γβα ∈ J ( Q ini , F, W ini ) . As long as m is large enough, repeating this process, we can find a fundamental cycle ω ′ locating at the second level (see Figure 14 (7)) or the first level (see Figure 14 (8)) of Q ini , which belongs to J ( Q ini , F, W ini ), such that ω m − q ( ω ′ ) m ′ p ∈ J ( Q ini , F, W ini ) , where m ′ is a positive integer, p and q are paths in Q ini . Therefore ω m ∈ J ( Q ini , F, W ini )which completes the proof. (cid:3) Remark 2.15.
In fact, if the level of ω in the proof is m , then m + 1 is large enoughsuch that ω m +1 ∈ J ( Q ini , F, W ini ) . Similar to the above proof, the QP ( Q pr , W pr ) of C [ Gr ( k, n )] is Jacobi-finite. However, the IQP ( Q ini , F, W ini ) is not Jacobi-finite. Inparticular, any power of a fundamental cycle of Q init is non-zero in the Jacobian algebra P ( Q ini , F, W ini ) Remark 2.16. In [16] , the authors proved that the representation type of the Jacobianalgebra associated to a -acyclic quiver with non-degenerated potential is invariant underQP-mutations, and gave a classification of such Jacobian algebras by the representationtype. Then by [16, Theorem 7.1] , we have the following classification of the Jacobianalgebras associated to the Grassmannian cluster algebras.finite type Gr (2 , n ) Gr (3 , Gr (3 , Gr (3 , tame type Gr (3 , Gr (4 , wild type others UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 21
The uniqueness.
We study in this subsection the uniqueness of the QPs of aGrassmannian cluster algebea. This is based on a general result of Geiß-Labardini-Schr¨oer [16]. They give a criterion which guarantees the uniqueness of a non-degenerateQP.We first recall some definitions in [16]. If W is a finite potential , i.e. the potential withfinite many items in the decomposition, then we denote by long ( W ) the length of thelongest cycle appearing in W . For a non-zero element u ∈ C h Q i , denote by short ( u ) theunique integer such that u ∈ m short ( u ) but u / ∈ m short ( u )+1 . We also set short (0) = + ∞ .(see [16, Section 2.5]). The following two propositions are important for our main result. Proposition 2.17. [16, Proposition 2.4]
Let ( Q, W ) be a QP over a quiver Q , and let I be a subset of Q such that the following hold: (1) The full subquiver Q | I of Q with vertex set I contains exactly m arrows α , · · · , α m ; (2) l := α · · · α m is a cycle in Q ; (3) The vertices s ( α ) , · · · , s ( α m ) are pairwise different; (4) W is non-degenerate.Then the cycle l appears in W . Proposition 2.18. [16, Theorem 8.20]
Suppose ( Q, W ) is a QP over a quiver Q thatsatisfies the following three properties (1) W is a finite potential; (2) Every cycle l in Q of length greater then long ( W ) is cyclically equivalent to anelement of the form P α ∈ Q η α ∂ α W with short ( η α ) + short ( ∂ α W ) > length ( l ) for all α ∈ Q ; (3) Every non-degenerate potential on Q is right-equivalent to W + W ′ for somepotential W ′ with short ( W ′ ) > long ( W ) .Then W is non-degenerate and every non-degenerate QP on Q is right-equivalent to W . Theorem 2.19.
Let Q pr be a principal part quiver of a Grassmannian cluster algebra,then the QP ( Q pr , W pr ) is the unique non-degenerate QP on Q pr up to right-equivalence,and thus the unique rigid QP on Q pr up to right-equivalence.Proof. By Remark 2.13, ( Q pr , W pr ) is rigid, so if it is unique as a non-degenerate QPthen it must be unique as a rigid QP. Since the mutations of two right-equivalent QPsare still right-equivalent, we only need to prove the theorem for the initial QP.To do this, we check that the conditions (1)-(3) in Proposition 2.18 hold for ( Q prini , W prini ).The condition (1) is clear. Since the cluster algebra of Gr (2 , n ) is of acyclic type, sothere is a unique rigid QP. Otherwise, there exists at least one internal fundamentalcycle on Q prini , and long ( W prini ) = 4. We prove the condition (2) in two steps. As beforewe only consider the case when k and n are odd numbers, see the principal part of thequiver in Figure 12. Step I:
Let ω be a fundamental cycle of Q prini and m be a positive integer number. Weclaim that ω m is cyclically equivalent to P α ∈ Q η α ∂ α W prini , where the length of a pathappearing in non-zero η α is 4 m −
3, and the length of all paths appearing in ∂ α W prini is 3.We prove this by induction on the level of ω . Assume the level of ω is 2 and α be thebottom arrow of ω , then it is cyclically equivalent to α∂ α W prini . Moreover, ω m is cyclically equivalent to (( α∂ α W prini ) m − α ) ∂ α W prini , where η α = ( α∂ α W prini ) m − α and ∂ α W prini satisfy the conditions in the claim.Now let ω be a fundamental cycle located at level t . Assume that the claim holds forthe fundamental cycle αρνµ , which is located at level t −
1, see the Figure 16. Here we ωαδ γ βν µρ
Figure 16.
Uniqueness of the QPonly consider the clockwise cycle αρνµ , another case is similar. So we may assume that( αρνµ ) m is cyclically equivalent to a potential P α ′ ∈ Q η α ′ ∂ α ′ W prini satisfying the claim.Then ω m is cyclically equivalent to ( αδγβ ) m , which equals to ( αρνµ − α∂ α W prini ) m .Note that we may write the expansion of ( αρνµ − α∂ α W prini ) m as the form of ( αρνµ ) m + P k S k , where S k is a multiplication of αρνµ and − α∂ α W prini with the term − α∂ α W prini appearing in it at least once. We write S k = S ′ k α∂ α W prini S ′′ k , where S ′ k and S ′′ k aremultiplications (maybe empty) of αρνµ and − α∂ α W prini . Then S k − S ′′ k S ′ k α∂ α W prini ∈ C .Thus ( αδγβ ) m − [ X α ′ ∈ Q η α ′ ∂ α ′ W prini + ( P k S ′′ k S ′ k α ) ∂ α W prini ]=[( αρνµ ) m + P k S k ] − [ X α ′ ∈ Q η α ′ ∂ α ′ W prini + ( P k S ′′ k S ′ k α ) ∂ α W prini ]=[( αρνµ ) m − X α ′ ∈ Q η α ′ ∂ α ′ W prini ] + P k ( S k − S ′′ k S ′ k α∂ α W prini ) ∈ C. So ( αδγβ ) m , and therefore ω m , is cyclically equivalent to X α ′ ∈ Q η α ′ ∂ α ′ W prini + ( P k S ′′ k S ′ k β ) ∂ β W prini , which satisfies the conditions in the claim.To sum up, for any fundamental cycle ω and any positive integer number m , ω m is cyclically equivalent to P η α ∂ α W prini , where η α = 0 or each path in η α has length4 m −
3, and each path in ∂ α W prini has length 3. So short ( η α ) = + ∞ or 4 m −
3, and short ( ∂ α W prini ) = 3. Therefore short ( η α ) + short ( ∂ α W prini ) > m = length ( ω m ) , and the condition (2) holds for ω m . Step II:
Let l be a cycle of Q pr . We use the notations appearing in Lemma 2.9. Inparticular, l ′ is the new cycle which shares an arrow α with l , p and q are two subpathsof l such that l = l ′ ± q∂ α W prini p . At last we find a fundamental cycle ω with l − ω m ∈ J ( Q pr , W prini ) . Assume that l ′ is cyclically equivalent to P η α ′ ∂ α ′ W prini and condition (2) holds for l ′ ,that is, short ( η α ′ ) + short ( ∂ α ′ W ) > length ( l ′ ) . UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 23
Note that the quiver we consider is the principal part Q pr , so (6) and (7) in Figure 14are the only cases we should deal with. Thus we have length ( l ) = length ( l ′ ) and length ( l ) = length ( pq ) + short ( ∂ α W prini ) . Therefore l is cyclically equivalent to P η α ′ ∂ α ′ W prini ± pq∂ α W prini , which satisfies the con-dition (2). This proves the condition (2) for all cycles over Q pr .Finally, the condition (3) follows immediately from the following two observations.By the Proposition 2.17, all of the fundamental cycles appear in W prini . For any cycle l ,excepting the fundamental cycles, length ( l ) > long ( W prini ). (cid:3) Application
Categorification.
An “additive categorification” of a cluster algebra hac beenwell studied in recent years. Roughly speaking, it lifts a cluster algebra structure on acategory level, that is, one may find a cluster structure (see [6] for precisely definition)on the category. Such category always has some duality property so called 2-Calabi-Yauproperty. In particular, the cluster category is an important example of 2-Calabi-Yautriangulated category with cluster structure, which gives a categorification for the clusteralgebra of acyclic type with trivial coefficients. In [1], Amiot constructed generalizedcluster categories for a quiver with potential (
Q, W ). We denote this generalized clustercategory by C ( Q,W ) .For a cluster algebra with non-trivial coefficients, some stably 2-Calabi-Yau Frobeniuscategory also has cluster structure (see [6, 12]). In our context, such Frobenius categoryis always a kind of subcategory of module categories. For the cluster algebra structureon the coordinate ring(3.1) C [Gr( k, n )] / ( φ { , ,...,k } − φ { , ,...,k } is the consecutive Pl¨uckercoordinate indexed by k -subset { , , . . . , k } , Geiss-Leclerc-Schr¨oer have given in [17] acategorification in terms of a subcategory Sub Q k of the category of finite dimensionalmodules over the preprojective algebra of type A n − . Note that the cluster coefficient φ { , ,...,k } in C [Gr( k, n )] is not realised in the category. More recently, Jensen-King-Su [18] have given a full and direct categorification of the cluster structure on C [Gr( k, n )],using the category CM( B ) of (maximal) Cohen-Macaulay modules over the completionof an algebra B , which is a quotient of the preprojective algebra of type ˜ A n − . Remark 3.1.
It has been proved in [8] that for a cluster tilting object T in CM ( B ) corresponding to a Postnikov diagram, the cluster-tilted algebra End ( T ) is isomorphicto the Jacobian algebra J ( Q, F, W ) . Note that CM ( B ) is Hom-infinite and End ( T ) isof infinite dimension, this is compatible with the Hom-infiniteness of J ( Q, F, W ) , seeRemark 2.15.On the other hand, Amiot, Reiten, and Todorov shown in [4] that the generalizedcluster category has some “ubiquity” (see also in [2, 3, 23] ). In our situation, this meansthe stable categories of SubQ k and CM( B ) are both equivalent to a generalized clustercategory. Combining with the result in [8] , this generalized cluster category is exactly thegeneralized cluster category defined by the QP ( Q pr , W pr ) . Auto-equivalence groups and cluster automorphism groups.
Recall thatfor a cluster algebra A , we call an algebra automorphism f a cluster automorphism , ifit maps a cluster x to a cluster x ′ , and is compatible with the mutations of the clusters.Equivalently, an algebra automorphism f is a cluster automorphism if and only if Q ′ ∼ = Q or Q ′ ∼ = Q op , where Q ′ and Q are the associated quivers of x ′ and x respectively. Werefer to [5, 9, 10] for the details of cluster automorphisms.Let C be a 2-Calabi-Yau triangulated category with cluster structure. In particular,there is a cluster tilting object T and a cluster map φ which sends cluster tilting objects,which are reachable by iterated mutations from T in category C , to clusters in algebra A φ ( T ) , where A φ ( T ) is the cluster algebra with initial cluster φ ( T ). In fact A φ ( T ) is thecluster algebra defined by the Gabriel quiver of End C ( T ).Denote by Aut T ( C ) a quotient group consisting of the (covariant and contravariant)triangulated auto-equivalence on C that maps T to a cluster tilting object which isreachable from T itself, where we view two equivalences F and F ′ the same if F ( T ) ∼ = F ( T ′ ).Let F be an auto-equivalence in Aut T ( C ). Denote by Q and Q ′ the Gabriel quivers of End C ( T ) and End C ( F ( T )) respectively. Then Q is naturally isomorphic to Q ′ since F is a triangulated equivalence. Moreover, since F ( T ) is reachable from T , φ ( F ( T )) is acluster in A φ ( T ) , so there is a cluster automorphism f in Aut ( A φ ( T ) ) which maps φ ( T )to φ ( F ( T )). Thus Aut T ( C ) can be viewed as a subgroup of Aut ( A φ ( T ) ). Conversely, wehave the following Conjecture 3.2.
There is an natural isomorphism
Aut T ( C ) ∼ = Aut ( A φ ( T ) ) . If C is algebraic and Q is acyclic, then C is a (classical) cluster category from [19].Then the conjecture has been verified in [5, section 3] and [7, Theorem 2.3]. For the caseof generalized cluster categories, the conjecture is related to the following conjecture,which says that the quivers determine the potentials up to right equivalences. Conjecture 3.3.
Let ( Q, W ) be a non-degenerate QP. Assume that ( Q ′ , W ′ ) is a QPwhich is mutation equivalent to ( Q, W ) . Then(1) ( Q ′ , W ′ ) is right equivalent to ( Q, W ) if Q ′ ∼ = Q ;(2) ( Q ′ , W ′ ) is right equivalent to ( Q op , W op ) if Q ′ ∼ = Q op . Proposition 3.4.
If Conjecture 3.3 is true for a Jacobian-finite QP ( Q, W ) , then Con-jecture 3.2 is true for the generalized cluster category C ( Q,W ) .Proof. Since (
Q, W ) is Jacobian-finite, recall from [1] that, there is a canonical clustertitling object T in C ( Q,W ) whose endomorphism algebra is isomorphic to the Jacobi-algebra J ( Q, W ). Because we already have
Aut T ( C ( Q,W ) ) ⊂ Aut ( A φ ( T ) ) , it suffices to show that any cluster automorphism f can be lifted as an auto-equivalenceon C which maps the canonical cluster tilting object to a reachable one. Assume that f maps φ ( T ) to a cluster µ ( φ ( T )) with quiver Q ′ ∼ = Q , where µ ( φ ( T )) is obtained from φ ( T ) by iterated mutations. Denote by ( Q ′ , W ′ ) = µ ( Q, W ) the QP obtained from(
Q, W ) by the same steps of mutations.On the one hand, by [20, Theorem 3.2], there is an equivalence Φ from C ( Q,W ) to C ( Q ′ ,W ′ ) which maps T to µ ( T ′ ), where T ′ is the canonical cluster tilting object in C ( Q ′ ,W ′ ) whose endomorphism algebra is isomorphic to J ( Q ′ , W ′ ).On the other hand, Conjecture 3.3 ensures that there is a right equivalence between( Q ′ , W ′ ) and ( Q, W ), and then by [20, Lemma 2.9], there is a covariant equivalence Ψfrom C ( Q ′ ,W ′ ) to C ( Q,W ) . Note that Ψ maps T ′ to T , and thus maps µ ( T ′ ) to µ ( T ),since the mutations are obtained by exchanged triangles (see [6] for example) and Ψ istriangulated. Finally, the auto-equivalence ΨΦ is what we wanted, which gives a lift of UIVERS WITH POTENTIALS FOR GRASSMANNIAN CLUSTER ALGEBRAS 25 f . We have a similar proof for the case Q ′ ∼ = Q op . See the following diagram for theequivalences. C ( Q,W ) ΨΦ $ $ ■■■■■ Φ / / C ( Q ′ ,W ′ )Ψ z z ttttttttt C ( Q,W ) (cid:3) Theorem 3.5.
For the non-degenerated QP arising from the Grassmannians clusteralgebra, the Conjecture 3.3 is true. So for the associated generalized cluster category C ,we have an isomorphism Aut T ( C ) ∼ = Aut ( A φ ( T ) ) .Proof. Let (
Q, W ) be a non-degenerate QP of the Grassmannians cluster algebra, andlet ( Q ′ , W ′ ) be a QP which is mutation equivalent to ( Q, W ). Then ( Q ′ , W ′ ) is non-degenerate. On the other hand, by Theorem 2.19, ( Q, W ) is the unique non-degenerateQP on Q , up to right equivalence. So ( Q ′ , W ′ ) is right equivalent to ( Q, W ) if Q ∼ = Q ′ .Note that ( Q op , W op ) also has the non-degenerate uniqueness property since ( Q, W )has. Thus similarly ( Q ′ , W ′ ) is also right equivalent to ( Q op , W op ), if Q ′ ∼ = Q op . So theconjecture 3.3 is true, and Aut T ( C ) ∼ = Aut ( A φ ( T ) ). (cid:3) Remark 3.6.
For the QP arising from a marked Riemann surface with some ”techniqueconditions”, [16, Theorem 1.4] ensures the non-degenerate uniqueness. So we have asimilar isomorphism as in Theorem 3.5 for this case.
Acknowledgments.
The authors would like to thank Claire Amiot, Jeanne Scott, DongYang, Bin Zhu for useful discussions, Bernhard Keller and Osamu Iyama for pointingus to the references [1, 3]. Part of the work was done when the first author visitedUniversity of Connecticut, he thanks the university and especially professor Ralf Schifflerfor hospitality and providing him an excellent working environment.
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