Cluster monomials in C[G L n /N] , a simplicial fan in the cone of semi-standard Young tableaux, and the Lusztig basis
aa r X i v : . [ m a t h . R A ] N ov Cluster monomials in C [ GL n /N ], a simplicial fan in the cone ofsemi-standard Young tableaux, and the Lusztig basis G.A.Koshevoy ∗ Abstract
We study the cluster monomials and cluster complex in C [ GL n /N ]. For we consider the tableau basis in C [ GL n /N ]. Namely, an element ∆ T of the tableau basis labeled by a semi-standard Young tableau T is the product of the flag minors corresponding to columns of T .Our main results state: (i) cluster monomials in C [ GL n /N ] can be labeled by semistandardYoung tableaux such that any cluster monomial has the form ∆ T + lexicographically smallerterms; (ii) such labeling distinguish the cluster monomials; (iii) for any seed of the clusteralgebra on C [ GL n /N ], we define a cone in D ( n ) ( D ( n ) is the cone of semi-standard Youngtableaux, D ( n ) is linear isomorphic to the Gelfand-Tseitlin cone) generated by tableauxwhich label the cluster variables of the seed, and these cones form a simplicial fan in D ( n ). One of the main motivation of S.Fomin and A.Zelevinsky for introducing cluster algebras [4] wasthe desire to provide a combinatorial framework to understand the structure of ’dual canonicalbases’ in (homogeneous) coordinate rings of various algebraic varieties related to semisimplegroups. Several such varieties (Grassmann varieties and double Bruhat cells) carry a clusteralgebra structure and certain special functions on that spaces (such as Pl¨ucker coordinates,generalized minors) correspond to distinguished elements called cluster variables .For the general linear group G = GL n ( C ), and the subgroup N ⊂ G of unipotent upper-triangular matrices, the coordinate ring of the base affine space for GL n ( C ), C [ GL n /N ], is theset of regular functions on GL n ( C ) which are invariant under the action of N by the rightmultiplication. According to classical invariant theory, C [ GL n /N ] is generated by the flagminors . (A flag minor ∆ I of a matrix x = ( x ij ) ∈ GL n ( C ) is a minor occupying in rows in I and the first | I | columns.) The ring C [ GL n /N ] is one of prototypical examples of a clusteralgebra ([10])). For the cases n = 3 , ,
5, the cluster monomials form a basis in C [ GL n /N ] asin the vector space, but for n ≥ C [ GL n /N ]. For we consider the tableau basis in C [ GL n /N ] (see, for example [9, 11]). The elements of the tableau basis arelabeled by the semistandard Young tableaux. Namely, an element ∆ T of the tableau basislabeled a semistandard Young tableau T is the product of the flag minors corresponding tocolumns of T (for details see Section 3). Because of the D´esam´enien-Kung-Rota algorithm([9]), the set of such monomials in flag minors is a basis in C [ GL n /N ].Consider the lexicographical order ≻ on the variables x ij , i, j ∈ [ n ], ( i indicates the rownumber and j the column number) x ij ≻ x kl if i < k or i = k and j < l . This order defines ∗ CEMI and Poncelet laboratoty (IMU and CNRS (UMI 2615)), email: [email protected] C [ x ij ]. In particular, we get a total order on the Youngtableaux through the total ordering of ∆ T ’s.The semi-standard Young tableaux filled in the alphabet [ n ] can be viewed as integer pointof the cone of D -tight arrays D ( n ) ⊂ R n ( n +1) / (see [6]). The cone D ( n ) is linear isomorphicto well-known cone of the Gelfand-Tseitlin patterns. For us it is convenient to work with thecone of semi-standard Young tableaux.Our main results (Theorem M1, M2 and M3) state: (i) cluster monomials in C [ GL n /N ] canbe labeled by semistandard Young tableaux such that any cluster monomial has the form ∆ T +lexicographically smaller terms; (ii) such labeling distinguish the cluster monomials; (iii) forany seed of the cluster algebra on C [ GL n /N ], we define a cone in D ( n ) generated by tableauxwhich label the cluster variables of the seed; these cones form a simplicial fan in D ( n ).In the case the cluster algebra C [ GL n /N ] is of finite type, that is the case for n = 3 , , D ( n ), and thus the cluster monomialsform a basis in C [ GL n /N ].For n ≥
6, the cluster algebra of C [ GL n /N ] is of infinite type, and the union of the conesis only a part of the cone D ( n ).We conjecture that the cluster monomials correspond to the real elements of the Lusztigbasis. Namely, the specialization of the dual canonical basis at q = 1 is the Lusztig basis .Recall, that the dual canonical basis is a basis of the quantum deformation of C [ GL n /N ]. Thisis a distinguished basis which nicely behaves with respect to implementation of C [ GL n /N ] as arepresentation of GL n (namely, C [ GL n /N ] is the direct sum of the irreducible representationsof GL n each taken with multiplicity one).The Lusztig basis is a basis in C [ GL n /N ] labeled by the integer point of the cone D ( n ). Anelement of Lusztig basis labeled by an integer point of D ( n ) has the form ∆ T + lexicographicallysmaller terms, where T is a semi-standard Young tableau corresponding to the point in D ( n ).The Lusztig basis and the tableau basis are different. For example, for GL , the Lusztigbasis (discovered in 1985 by I.Gelfand and A.Zelevinsky ([12])) is the collection of monomialsin ∆ , ∆ , ∆ , ∆ , ∆ , ∆ , and ∆ which do not contain the product ∆ ∆ , while thetableaux basis is the collection of monomials in ∆ , ∆ , ∆ , ∆ , ∆ , ∆ , and ∆ whichdo not contain the monomial ∆ ∆ .An element of the Lusztig basis is real if its square belongs to the Lusztig basis. We callthe semi-standard Young tableau real if the corresponding element of the Lusztig basis is real.Our conjecture is that cluster monomials are labeled by real semi-standard Young tableaux. Acknowledgments . I thank Vladimir Danilov, Sergey Fomin, Alexander Karzanov, and JanSchr¨oer for useful discussions. A part of this work was made in the Max-Planck Institute f¨urMathematics, Hausdorff Institute for Mathematics (Bonn) and IHES (Bures-sur-Yvette) and Ithanks these institutes for hospitality and financial support.
Since we are interested in the cluster algebra structure on C [ GL n /N ], we remind necessarydefinitions for so-called skew-symmetric cluster algebras.Let G = ( V ( G ) , E ( G )) be a quiver (a directed multigraph) in which the vertex set V ( G )is partitioned into two subsets: a set V of frozen vertices, and a set V of mutable vertices.The (integer) edge multiplicity function is regarded as being skew-symmetric: if vertices u, v α edges going from u to v (which are members of E ( G )), we simultaneouslythink of these vertices as being connected by − α edges going from v to u . To each vertex v of G one associates a cluster variable x v so that { x v : v ∈ V ( G ) } is a transcendence basis ofa field of rational functions F . Such a pair consisting of a quiver and a transcendence basisindexed by its vertices is said to be a cluster seed . Monomials in x v , v ∈ V ( G ) are called cluster monomials . The cluster seed can be mutated at any mutable vertex to produce a newcluster variable and a new seed. Applying mutations in all possible situations produce the setof cluster variables and these variables form a skew-symmetric cluster algebra [4].The quiver and variables are modified by applying the following operations called clustermutations. A cluster mutation µ v applied at a mutable vertex v ∈ V changes one variable,namely, x v , and modifies the quiver G by the following rules. For a vertex v , denote In ( v ) := { v ′ ∈ V ( G ) : ( v ′ , v ) ∈ E ( G ) } and Out ( v ) := { v ′′ ∈ V ( G ) : ( v, v ′′ ) ∈ E ( G ) } .The quiver µ v ( G ) has the same vertex set as G , V ( µ v ( G )) = V ( G ), partitioned into frozenand mutable vertices in the same way as before. The edges E ( µ v ( G )) are obtained from edges E ( G ) by the following rule:(i) the edges in E ( µ v ( G )) incident to the vertex v are exactly the edges in E ( G ) incidentto v but taken with the reverse direction;(ii) for each pair v ′ ∈ In ( v ) and v ′′ ∈ Out ( v ), form the edge ( v ′ , v ′′ ) in E ( µ v ( G )) whosemultiplicity is defined to be γ − α · β , where α = w ( v ′ , v ) ≥ v ′ , v )in E ( G ), β = w ( v, v ′′ ) ≥ v, v ′′ ), and γ ∈ Z is that for ( v ′ , v ′′ );(iii) the other edges of µ v ( G ) are those of G that neither are incident to v nor connect pairs v ′ , v ′′ as in (ii).For u = v , we put µ v ( x u ) := x u and define µ v ( x v ) = x newv by the following rule: x newv · x v = Y v ′ ∈ In ( v ) x w ( v ′ ,v ) v ′ + Y v ′′ ∈ Out ( v ) x w ( v,v ′′ ) v ′′ . This gives the new seed: the quiver µ v ( G ) and variables µ v ( x u ), u ∈ V ( µ v ( G )) = V ( G ).Obviously, there holds µ v = id .The variables x v , v ∈ V do not change and are called coefficients . Let X denotes the setof cluster variables. Then the cluster algebra is k [ x v | v ∈ V ]-subalgebra of F generated by X .We need the following important results on cluster algebras. Namely, the Laurent phe-nomenon ([4]) and recently proven Positivity conjecture for skew-symmetric cluster algebras[16]. Theorem CL . ([4]) For any initial seed ( G ; x v , v ∈ V ( G )), any cluster variable x ∈ X is aLaurent polynomial in variables x v , v ∈ V . Theorem CP . ([16]) For a skew-symmetric cluster algebra, the coefficients in such Laurentpolynomials are positive. C [ GL n /N ] and a fan in the cone of semi-standardYoung tableaux For given n , let us consider cluster algebras A ( n ) with the initial seed specified by the quiver Q ( n ) being a triangular grid of size n with cyclically oriented triangles (we depicted Q (5)below). 3e are interested in such a cluster algebra, because the quiver Q ( n ) corresponds to apseudoline arrangement for the reduced decomposition of the longest permutation w = s s . . . s n − s . . . s n − . . . s s s (see [10] and [2]). (cid:0)(cid:0)✠ (cid:0)(cid:0)✠ (cid:0)(cid:0)✠ (cid:0)(cid:0)✠ ❅❅■ ❅❅■ ❅❅■ ❅❅■❅❅■ ❅❅■ ❅❅■❅❅■ ❅❅■❅❅■ (cid:0)(cid:0)✠(cid:0)(cid:0)✠(cid:0)(cid:0)✠ (cid:0)(cid:0)✠(cid:0)(cid:0)✠ (cid:0)(cid:0)✠✛ ✲✲✲✲ ✲✲✲ ✲✲ ✲ (1 ,
1) (5 , , ,
1) (3 , , Q ( n ) by ( i, j ), i + j ≤ n + 1, i , j ≥
1, suchthat the vertex labeled by (1 ,
1) is at the left corner of the triangular grid, and ( i, j ) labels theend point of any path in Q which has i − j − Q ( n )). (We depicted the vertices (1,1), (3,1), (1,3), (1,5), (5,1)and (3,3) in the above picture.) The vertices at the left and right sides of the triangle arefrozen, that is the set V of the frozen vertices consists of vertices labeled by ( i, i = 1 , . . . , n ,and ( i, j ) with i + j = n + 1, and the set V of mutable vertices is the vertices labeled by { ( i, j ),either i = 1 or i + j < n + 1 } .The cluster structure on C [ GL n /N ] is a specialization of the cluster algebra A ( n ). Namely,we view n ( n + 1) / { i,i +1 ,...,i + j − } labeling the vertices ( i, j ), i + j ≤ n + 1, i, j ≥
1, as formal indeterminates. There are ( n − n − / Q ( n ), we use the quiver to write the corresponding exchange relations:∆ { i,...,i + j − } Ω ij = ∆ { i,...,i + j − } ∆ { i +1 ,...,i + j } ∆ { i − ,...,i + j − } +∆ { i,...,i + j } ∆ { i +1 ,...,i + j − } ∆ { i − ,...,i + j − } , j > , and, for j = 1, we have ∆ i Ω i = ∆ i − ∆ i,i +1 + ∆ i − ,i ∆ i +1 , i ≥ . Using the Pl¨ucker relations, we get the new cluster variables Ω i = ∆ i − ,i +1 , i = 2 , . . . , n − ij = ∆ { i − ,...,i + j − ,i + j } ∆ { i +1 ,...,i + j − } − ∆ { i,...,i + j − } ∆ { i +1 ,...,i + j } . For each new quiver and the corresponding cluster variables we proceed new mutations andso on (we use Pl¨ucker relations for calculating cluster variables as we demonstrated above).The resulting set of cluster variables constitute the algebra P ( n ) being such a specialization of A ( n ). (For n = 4 see [10], Section 2.)Recall that C [ GL n /N ] is the set of regular functions on GL n ( C ) which are invariant underthe action of N by the right multiplication, that is C [ GL n /N ] = C [ x ij ] N .From the next theorem follows that P ( n ) is a cluster algebra on C [ GL n /N ]. (Recently, in[13, 14] have shown that the standard quantum deformation of the coordinate ring of a doubleBruhat cell is a quantum cluster algebra confirming the Berenstein-Zelevinsky conjecture [3].Passing to the classical limit, this shows that the coordinate ring of a double Bruhat cell is infact a genuine cluster algebra.) Theorem M0 . 4. Any cluster variable of P ( n ) belongs to C [ x ij ] N ;2. For every I ⊂ [ n ], the flag minor ∆ I is a cluster variable of P ( n ).For the proof of the item 2 see [7]. For the proof of the item 1, we consider a special seedof P ( n ). Namely, consider the quiver T ( n ) corresponding to the pseudo-line arrangement forthe following reduced decomposition (see [2] and [7] for algorithms how to associate a quiverto a pseudo-line arrangement) w = [( n − / Y i =0 s i +1 s i +2 . . . s n − i − s n − i s n − i − . . . s i +2 s i +1 . An important property of the quiver T ( n ) is that (i) it can be reached by quiver mutationsfrom Q ( n ), (ii) the cluster variables (in P ( n )) which label the vertices of T ( n ) are flag minors,and (iii) any seed which is obtained by a single mutation from T ( n ) has properties (i) and (ii).For n = 8, we draw below a tiling (dual object to pseudo-line arrangement) and the corre-sponding quiver T (8). ❍❍❍❍❨❅❅■❆❆❆❆❑❇❇❇❇❇❇▼✻✁✁✁✁✕(cid:0)(cid:0)✒✟✟✟✟✯ ✟✟✟✟✯(cid:0)(cid:0)✒✁✁✁✁✕✻❇❇❇❇❇❇▼❆❆❆❆❑❅❅■❍❍❍❍❨❅❅■❆❆❆❆❑❇❇❇❇❇❇▼✻✁✁✁✁✕(cid:0)(cid:0)✒✟✟✟✟✯❍❍❍❍❨❍❍❍❍❨❍❍❍❍❨ ❍❍❍❍❨ ❍❍❍❍❨ ❍❍❍❍❨ ❅❅■❆❆❆❆❑❇❇❇❇❇❇▼✻✁✁✁✁✕(cid:0)(cid:0)✒✟✟✟✟✯✟✟✟✟✯✟✟✟✟✯✟✟✟✟✯✟✟✟✟✯ ❆❆❆❆❑❇❇❇❇❇❇▼✻✁✁✁✁✕(cid:0)(cid:0)✒ ❆❆❆❆❑❇❇❇❇❇❇▼✻✁✁✁✁✕(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■ (cid:0)(cid:0)✒❅❅■ ❅❅■ ❇❇❇❇❇❇▼✻✁✁✁✁✕ ❇❇❇❇❇❇▼✻✁✁✁✁✕❆❆❆❆❑ ❆❆❆❆❑ (cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■ (cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘✻(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❄✻❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘(cid:0)(cid:0)✠❅❅❘✻(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❅❅■(cid:0)(cid:0)✒❄❅❅❘(cid:0)(cid:0)✠❅❅❘✻(cid:0)(cid:0)✒ On the left hand side we depicted, for n = 8, the tiling corresponding to the above definedreduced decomposition, and on the right hand side is the quiver T (8).For T ( n ), the corresponding flag minors are labeled by the following sets { } , { , } , . . . , { , . . . n } ,and pairs { } , { , } , . . . , { , . . . , n − , n } , { , n } , { , , n } , . . . , { , . . . , n − , n } , . . . , { k, n − k +3 , . . . , n − , n } , { k, k +1 , n − k +3 , . . . , n − , n } . . . , { k, k +1 , . . . , n − k +2 , n − k +3 , . . . , n − , n } , { k, n − k + 2 , , n − k + 3 , . . . , n − , n } , { k, k + 1 , n − k + 2 , n − k + 3 , . . . , n − , n } . . . , { k, k +1 , . . . , n − k, n − k + 2 , n − k + 3 , . . . , n − , n } , k = 3 , . . . , [ n/ T ( n ).Let us recall the following facts on C [ x ij ] N (see, for example [11]).5 The coordinate ring C [ x ij ] N is a unique factorization domain. • For any I ⊂ [ n ], the flag minor ∆ I is irreducible polynomial in C [ x ij ] (and hence in C [ x ij ] N ). • Irreducible flag minors ∆ I are non-equivalent among themselves. Proof Theorem M0 . We prove item 1. We use the seed for the quiver T ( n ). Suppose there isa cluster variable P which is not a polynomial in flag minors, that is P C [ x ij ] N . According toTheorem CL, P is a Laurent polynomial in the variables being the flag minors for the seed withthe quiver T ( n ). Then by assumption, this Laurent polynomial does not belong to C [ x ij ] N .Firstly, we claim that the denominator of this Laurent polynomial can not have multipliesin non-frozen variable. In fact, suppose this not the case and the denominator has a multiplies,say ∆ I , corresponding to a non-frozen vertex v of T ( n ). Then let us apply the mutation at thisvertex. The new seed T v ( n ) has cluster variables being also flag minors. The latter set doesnot contain ∆ I . Since P is a Laurent polynomial in the flag minors of the seed T v ( n ), we getthat the denominator of this Laurent polynomial does not contain ∆ I . This contradicts to theabove facts on C [ x ij ] N .Secondly, we claim that cluster variables can not have denominator with multiplies in thefrozen flag minors. We proceed by induction on the number of mutations from the seed T ( n ).The base of induction is valid by the construction of the seed T ( n ). Consider the first case whena seed has a cluster variable with the denominator being a monomial in frozen flag minors.This means that for some polynomials S , T , U and V ∈ C [ x ij ] N , we have S + TU = VM , where M is a monomial in the frozen flag minors. Hence M ( S + T ) = V U . Because of the uniquefactorization property, we have that U divides M , and U is obtained by the smaller number ofmutations than V . Now, we consider the first case in the subsequence of mutations between T ( n ) and a seed containing U , such that it appears a cluster variable which divides a monomialin frozen flag minor. That is, for some polynomials S ′ , T ′ , U ′ , V ′ ∈ C [ x ij ] N , it holds that S ′ + T ′ U ′ = V ′ M ′ , where M ′ is a monomial in frozen variables, and S ′ , T ′ , U ′ do not divide frozenvariables. This contradicts that C [ x ij ] N is the unique factorization domain. (cid:3) We need more facts and notions to formulate our main results.Recall, that a partition λ = ( λ ≥ . . . ≥ λ k >
0) can be identified with a Young diagram: aleft-justified shape of k rows of boxes of lengths λ , . . . , λ k (numbered from bottom to top forthe French style). A semi-standard Young tableau in the alphabet [ n ] is a filling (assignment anumber in [ n ] to each box) of λ non-decreasing along rows and increasing along columns. Hereis an example of a semi-standard Young tableau | || | | | | | || | | | | | | | | | | | of the shape (11 , , D -tightarrays ([6]). Namely, let us identify the space R n ( n +1)2 and the space of of upper triangular n × n -matrices ( a ij ), 1 ≤ j ≤ i ≤ n . Since we consider the French style to draw Youngdiagrams, we write matrices in Descardes coordinates. Because of this, such a written matrix6s called an array . A typical upper-triangular array A is . . . a nn . . . a n − n − a n n − ... ... . . . ... ...0 a . . . a n − a n a a . . . a n − a n .For a semi-standard Young tableau T , let us define an array A ( T ) by the following rule: a ij ( T ) is equal to the number of boxes in the j -th row filled with i . The array correspondingto the above tableau is .The set of arrays corresponding the semi-standard Young tableaux is the set of integerpoints in the cone of D -tight arrays. Namely, we have ([6]) Proposition . An (integer) n × n array A is an array corresponding to a semistandardYoung tableau if and only if A belongs to the cone D ( n ) of D - tight arrays , that is, for any j < n and i ∈ [ n ], it holds a (1 , j + 1) + . . . + a ( i, j + 1) ≤ a (1 , j ) + . . . + a ( i − , j ) . A Gelfand-Tsetlin pattern is a triangular array x nn x n − n x n − n . . . x n x n − n − x n − n − . . . x n − x n − n − . . . x n − . . . x such that 0 ≤ x ij ≤ x i − j − ≤ x i − j . The set of all G-T patterns is the cone GT ( n ).There is a linear isomorphism sending the cone D ( n ) to the Gelfand-Tseitlin cone GT ( n ).Namely, this linear isomorphism is: x = a , x = a , x = a + a , . . . , x ij = a ii + . . . + a ji , . . . , x n = a + a + . . . a n .It is convenient to us to work with the cone D ( n ). We endow this cone with a tropicalsemi-ring structure. Namely, the multiplication ⊙ = + is the sum of arrays, A ⊙ B := A + B .To define the sum, consider the lexicographical order and the corresponding total order onmonomials as in Introduction. Then A ≻ B iff X A is bigger than X B wrt the total order onmonomials, where X A = x a x a · · · x a n n · x a x a · · · x a n n · · · x a n n x a n n · · · x a nn nn . Then, for a pair of arrays A , B we define max( A, B ) to be equal A if A ≻ B and be equal to B , otherwise. Then, we set A ⊕ B = max( A, B ).There is a distinguished basis in C [ x ij ] N , the tableau basis. Elements of the tableau basis are labeled by semi-standard Young tableaux. Namely, for a semi-standard Young tableau T of shape λ = ( λ ≥ λ ≥ . . . ≥ λ n ), we denote T , . . . , T λ the collections of subsets of [ n ] beingfillings of the columns of T .For each T i , we consider the flag minor ∆ T i . Thus, for a semi-standard Young tableau T ,we define a monomial in flag minors as follows∆ T = ∆ T · · · ∆ T λ . T is X A ( T ) .For the above example of semi-standard Young tableau, the corresponding monomial is∆ (∆ ) ∆ (∆ ) ∆ (∆ ) .The importance of such monomials is that they form a linear basis in C [ GL n /N ]. Theorem A . ([9, 11]). The set of polynomials ∆ T , while T runs the set of semi-standardYoung tableaux filled from the alphabet [ n ], form a linear basis of C [ x ij ] N . Remark.
There are many other linear bases indexed by semi-standard Young tableaux.These bases differ in rules of forming (triangulations of D ( n )), for a given tableau T , a monomialin flag minors (see [8]).Now we formulate main results of the paper. Theorem M1 . Any cluster variable P in P ( n ) has the form P = ∆ T + P ′ , for some semi-standard Young tableau T and some P ′ ∈ C [ x ij ] N which is smaller ∆ T wrt thetotal lexicographical order. Theorem M2 . Let P = ∆ T + P ′ and Q = ∆ T + Q ′ be cluster variables in P ( n ) representedaccording to Theorem M1 with the same semi-standard Young tableau T . Then P = Q. Because of these theorems, for each seed S in P ( n ), we have a collection of semi-standardYoung tableaux, T ( S ) , . . . , T ( n − n − / ( S ), corresponding to mutable variables, and the ’frozen’one-column Young tableaux T fi := | i | ... | || || | , i = 1 , . . . , n and T fn − i := | n || n − | ... | i + 1 || i | , i = 2 , . . . , n .Denote C ( S ) the cone in D ( n ) spanned by the arrays corresponding to this set of semi-standard Young tableaux for the mutable and frozen variables. By the construction the cone C ( S ) ⊂ D ( n ) is simplicial. Moreover, the generating set of arrays of this cone is unimodular ,that is a basis in Z n ( n +1) / . The unimodularity of generators of the cone C ( S ) follows from:(i) due to the definition P ( n ), the vertices of the quiver Q ( n ) are labeled by the flag minorsof one-column Young tableaux filled from intervals; (ii) the the set of arrays corresponding theone-column interval tableaux is unimodular in Z n ( n +1) / ; (iii) cluster mutations according tothe quiver rule preserve unimodularity (see Remark after the proof of Theorem M1). Theorem M3 . The cones C ( S ), while S runs the set of seeds in P ( n ), form a simplicialfan in D ( n ). Remark . From Theorem M3 follows that cluster monomials in C [ GL n /N ] are linear in-dependent (for skew-symmetric cluster algebras, linear independence of cluster monomials isproven in [15] using categorification).For n = 3 , , C [ GL n /N ] is the finite cluster algebra of types A , A and D , respectively.The union of the cones in the fan is the whole D ( n ), n = 3 , ,
5. Namely, we have8 orollary . For n = 3 , ,
5, the union of cones of the fan coincides with D ( n ).For n ≥ D ( n ).We end this section with examples for n = 4 , n = 4Here is the picture on the sphere (see also [10]) in R (we consider the quotient space by theset of frozen arrays) illustrating the fan of cones of seed of the cluster algebra with the initialseed Q (4) (such an algebra is finite A -algebra). ✟✟✟✟✟✟✟✟✟✟(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡❇❇❇❇ ✁✁✁✁✁PPPPPPPPPPPPPP❆❆❆❆❆ ✂✂✂✂❍❍❍❍❍❍❍❍❍❍✏✏✏✏ PPPP ❅❅❅❅❅❅❅❅❅ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ ❞ ❞ ❞❞ ❞❞ ❞❞ ❞ ✁✁✘✘✘✘✘✘✘✘✘✘✘✘✘✘ C C
134 1241413 243 223 C C C C C C C C C C C C There are 14 cones (the cone C is the triangle 14 , ,
124 on the 2-sphere), 9 variables, 8of them are of the form A I , where I is a non-frozen subset of [4] (2 − − | || || | | . This Youngtableau is a vertex in cones C , C , C , C .In the quotient space by the frozen arrays, the images of the arrays 2, 3, 24, 14, 23, 13, 134,and 124 are related by the following relations:3 + 14 = 0 (because there holds 3+14=1+34 in D ( n )); 2 + 134 = 0 (2+134=1+234);23 + 124 = 0 (23+124=12+234); 2 + 14 = 24 (2+14=1+24); 23 + 134 = 13 (23+134=13+234).For unique non-extreme ray cluster variable we have 421 3 = 124 + 3.Let the vectors 3, 23 and 2 be a basis of the 3-dim quotient space. Then these clustervectors are of the form3 = (1 , , − , ,
0) , 23 = (0 , , , − , , , , , − Let φ : R n ( n +1) / → R ( n − n − / be the quotient by frozen arrays A ( T fi ) and A ( T fn − i ). Then it is easy toverify that the collection φ ( C ( S )), S runs the cluster seeds, is a simplicial fan.
94 = ( − , , , , − , − , ❅❅✟✟✟ ❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✟✟✟ ❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
124 + 3 ✛
23 322413 12413414 (cid:0)(cid:0)✠ ❙❙♦❩❩⑦
For n = 5 the fan has 672 simplicial cones and 36 one-dimensional rays corresponding tomutable variables and 9 rays corresponding to frozen. The quotient of this fan by frozen rays isthe normal fan to the D -associahedron. This is 6-dimensional polytope which has 672 verticesand 36 facets.The mutable variables have the leading term ∆ T (Theorem M1) of the following form:either T is an one column semistandard Young tableaux whose filling is a subset I ⊂ [5], suchthat I is not an interval containing either 1 or n , (in such a case, cluster variables coincide withthe leading terms) or T is one of the following set of two column tableaux:421 3 , 521 3 , 521 4 , 532 4 , 531 4 , 42 51 3 , 53 41 2 , 5321 4 , 532 41 3 , 532 41 2 , 532 41 1 ,5421 3 , 542 41 3 , 542 51 3 . Proof of Theorem M1 . Because of Theorem M0 (item 1), each cluster variable is a polynomialin flag minors. So, we have to prove that each such cluster polynomial P has the coefficient 1with the leading monomial wrt the lexicographic order. Proceed by induction on the numberof mutations from the initial seed with the quiver Q ( n ). For one mutations we gave explicitformulae for Ω ij and Ω i above, and we are done.Suppose we are done for some number of mutations, and consider one more mutation.10onsider the quiver mutation rule x newv · x v = Y v ′ ∈ In ( v ) x w ( v ′ v ) v ′ + Y v ′′ ∈ Out ( v ) x w ( v,v ′′ ) v ′′ . By induction, for each v , v ′ ∈ In ( v ), and v ′′ ∈ Out ( v ), we have x v = ∆ T ( v ) + P ′ v , x v ′ =∆ T ( v ′ ) + P ′ v ′ and x v ′′ = ∆ T ( v ′′ ) + P ′ v ′′ . By Theorem M0, x newv = a ∆ T + P ′ . Then, because theproduct of leading terms on the left hand side equals the product of the leading terms on theright hand side, we have aX A ( T ) X A ( T ( v )) = X max( P v ′∈ I ( v ) w ( v ′ ,v ) A ( T ( v ′ )) , P v ′′∈ Out ( v ) w ( v,v ′′ ) A ( T ( v ′′ ))) , where the operation max for the cone D ( n ) is defined above.Because of this, we have a = 1. (cid:3) Remark . Let us note that from the proof of this theorem, for any seed of P ( n ) and anyvertex v , we havemax( X v ′ ∈ I ( v ) w ( v ′ , v ) A ( T ( v ′ )) , X v ′′ ∈ Out ( v ) w ( v, v ′′ ) A ( T ( v ′′ ))) − A ( T ( v ) ∈ D ( n ) . (4.1)From this relation follows that if, for a seed S , the set of arrays { A ( T v ) } (vectors in Z n ( n +1) / ), v runs the vertices of the quiver for S , is unimodular set of vectors in Z n ( n +1) / .Then, for any seed, which is obtained by a mutation, and hence, for all seeds in P ( n ), theunimodularity is valid. In fact, due to (4.1), the transformation matrix for mutations of clustercones is unimodular. For the initial quiver Q ( n ), the interval tableaux, the labels of the verticesof Q ( n ), form a unimodular set. Therefore every cone C ( S ) is unimodular. Proof of Theorem M2 . Suppose there are two seeds S and S ′ and cluster variables w and w ′ in these seeds such that w = ∆ T + P and w ′ = ∆ T + P ′ , where P and P ′ are polynomialsin the flag minors which are smaller (wrt the lexicographical order) ∆ T .Suppose w = w ′ . By Theorem CL, w is a Laurent polynomial of the cluster variables ofthe seed S ′ . We claim that this Laurent polynomial has the form w = w ′ + p ′ ( S ′ \ w ′ ) , (4.2)where p ′ ( S ′ \ w ′ ) is a Laurent polynomial in variables of the seed S ′ except w ′ . In fact, theleading term of w is X A ( T ) . Then there exists a Laurent monomial in variables S ′ of the aboveLaurent polynomial, such that its leading term (that is the product of leading term of thismonomial) is equal to X A ( T ) . But the leading term of w ′ is also X A ( T ) . Hence there is a lineardependence between arrays for tableaux corresponding to the variables of S ′ . Due to Remarkafter proof Theorem M1, this is no the case. Thus (4.2) holds true.By the same line of arguments, it holds that w ′ = w + p ( S \ w ) , where p ( S \ w ) is a Laurent polynomial in variables of the seed S \ w .Therefore, we have p ( S \ w ) + p ′ ( S ′ \ w ′ ) = 0 . Because of the positivity of coefficients of Laurent polynomials (Theorem CP) , the latterequality is possible if and only if p = p ′ = 0 .
11n fact, the numerator of p ( S \ w ) + p ′ ( S ′ \ w ′ ) is a Laurent polynomial in variables of the initialseed (with the quiver Q ( n )) with positive coefficients. Since, for a totally positive matrix, allinterval minors are positive we get p ( S \ w ) + p ′ ( S ′ \ w ′ ) >
0. Thus, p = p ′ = 0, and hence w = w ′ . (cid:3) Proof of Theorem M3 .Suppose there are two seeds, S and S ′ , such that the cones C := C ( S ) and C ′ := C ( S ′ )intersect such that it holds int ( C ) ∩ int ( C ′ ) = ∅ ( int ( C ′ ) denotes interior of a cone). Then,because the cones are unimodular, this intersection contains an integer array, say, correspondinga semi-standard Young tableau T . Then there is a cluster monomial y in the variables of theseed S such that y = x A ( T ) + lexicographically smaller terms, and there is a cluster monomial y ′ in the variables of the seed S ′ such that y ′ = x A ( T ) + lexicographically smaller terms.Let us show that y = y ′ . By Theorem CL, the cluster monomial y is the product of Laurentpolynomials in the variables of the seed S ′ . The same reasoning as in the proof Theorem M2shows that y = y ′ + R ( S ′ ). Similarly, we get y ′ = y + R ′ ( S ), where R and R ′ are Laurentpolynomials with positive coefficients (Theorem CP). Thus, R ( C ′ ) + R ( C ) = 0 is possible iff R ( C ′ ) = R ( C ) = 0. Hence y = y ′ .Now, recall that y = Q v s a v v is the monomial in the cluster variables of S . By Theorem CL, y is the product of the Laurent polynomials in cluster variables of S ′ , since each s v is a Laurentpolynomial in S ′ . But y = y ′ and y ′ is a monomial in S ′ . Hence each cluster variable s v is aLaurent monomial in S ′ . And, similarly, each cluster variable in S ′ is a Laurent monomial in S . Since the cones C := C ( S ) and C ′ := C ( S ′ ) have a common interior point, there is a facetof one of this cones which has a common interior point with another cone. Let this facet be afacet of C := C ( S ). Note that the vertices corresponding to the frozen arrays belong to thisfacet. Let S ′′ be a seed obtained through the mutation of S in the vertex, corresponding to theray of C := C ( S ) which does not belong to this facet. Then C ′′ := C ( S ′′ ) and C ′ := C ( S ′ ) havea common interior point (for example, in neighbor of the facet). Hence, by the same reasoningas above, the new cluster variable in S ′′ is a Laurent monomial in cluster variable in S ′ , butthis is not the case, because all cluster variables in S are Laurent monomials in S ′ . (cid:3) Let X ∈ M at q ( n ) be a n × n quantum q -matrix. That is, the following relations hold: x il x ik = q x ik x il ∀ i, ∀ k < l ; x jk x ik = q x ik x jk ∀ i < j, ∀ k ; x jk x il = x il x jk ∀ i
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