Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers
aa r X i v : . [ m a t h . R T ] O c t NAKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS ANDCOMBINATORICS OF AUSLANDER-REITEN QUIVERS
DENIZ KUS AND BEA SCHUMANN
Abstract.
We obtain an explicit crystal isomorphism between two realizations of crystalbases of finite dimensional irreducible representations of simple Lie algebras of type A and D .The first realization we consider is a geometric construction in terms of irreducible componentsof certain Nakajima quiver varieties established by Saito and the second is a realization interms of isomorphism classes of quiver representations obtained by Reineke. We give a homo-logical description of the irreducible components of Lusztig’s quiver varieties which correspondto the crystal of a finite dimensional representation and describe the promotion operator intype A to obtain a geometric realization of Kirillov-Reshetikhin crystals. Introduction
Let g be a Kac-Moody algebra with symmetric Cartan matrix. A groundbreaking result byLusztig was the construction of the canonical basis of the negative part U − q of the quantizedenveloping algebra of g (see [10, 11]). The canonical basis has remarkable properties and yieldsa basis for any irreducible highest weight g -module V ( λ ) of highest weight λ . The main toolof Lusztig’s work is given by a certain class of quiver varieties (for a precise definition seeSection 3.1) known in the literture as Lusztig’s quiver varieties.Motivated by Lusztig’s work Nakajima introduced another class of quiver varieties associatedto irreducible highest weight g -modules (see [12] for details). Here a geometric descriptionof the action on V ( λ ) is obtained and certain Lagrangian subvarieties of Nakajima’s quivervarieties are defined whose irreducible components yield a geometric basis of V ( λ ) .Using ideas from [10] an alternative construction of Lusztig’s canonical bases of U − q and V ( λ ) was given by Kashiwara in [5]. These bases have well-behaved combinatorial analogues, calledthe crystal basis B ( ∞ ) of U − q and B ( λ ) of V ( λ ) , respectively. The crystal B ( ∞ ) has a geo-metric realization in terms of irreducible components of Lusztig’s quiver varieties establishedin [7]; we denote this realization by B g ( ∞ ) . Moreover, Saito has shown in [14] that B ( λ ) alsoadmits a geometric realization in terms of irreducible components of Lagrangian subvarietiesof Nakajima’s quiver varieties. We denote the geometric realization of Saito in the rest of thepaper by B g ( λ ) and view the crystal graph of B g ( λ ) as a full subgraph of B g ( ∞ ) by identify-ing the irreducible components in B g ( λ ) with the subset of irreducible components of Lusztig’squiver varieties satisfying a certain stability condition (see Section 3).The motivation of this paper is to give a homological interpretation of the actions of the crystaloperators in B g ( λ ) using the combinatorics of Auslander-Reiten quivers of a fixed Dynkin quiver Q of the same type as g . This is achieved by constructing an explicit crystal isomorphism to arealization of B ( λ ) in terms of isomorphism classes of Q -modules introduced by Reineke in [13]; we denote this realization by B h ( λ ) . We first give in Theorem 3 a homological description ofthe stable irreducible components and then make use of the homological description of B g ( ∞ ) developed by the second author in [16] and show in Theorem 4 that the embedding of B g ( λ ) into B g ( ∞ ) is compatible with this description when g is of finite type A or D .In the last part of the paper we consider the standard orientation of the type A quiver andextend the classical crystal structure on the set of irreducible components to an affine crystalstructure isomorphic to the Kirillov-Reshetikhin crystal (KR crystal for short). The main ideais to define the promotion operator pr (see Definition 5.2) which is the analogue of the cyclicDynkin diagram automorphism i i + 1 mod ( n + 1) on the level of crystals. This gives,together with the homological description in Theorem 4, a geometric realization of KR crystals(see Corollary 5.4); for combinatorial descriptions of KR crystals in type A we refer the readerto [18, 9]). It will be interesting to discuss the promotion operator for various orientations ofthe quiver. The connection to Young tableaux (see Section 5.3) is not known in that case butthe description is still possible with several technical difficulties. This construction will be partof forthcoming work.This paper is organized as follows. In Section 2 we present background material. We recallfacts on representations of quivers and the construction of Auslander-Reiten quivers of Dynkinquivers as well as facts on quantum groups and crystal bases. In Section 3 we recall thegeometric construction of crystal bases in terms of irreducible components of quiver varietiesdue to Kashiwara-Saito [7] and Saito [14]. In Section 4.1 we develop combinatorics on thegeometric constructions using Auslander-Reiten quivers. For this we recall results by Reineke[13], prove a criterion for an irreducible component of Lusztig’s quiver variety to contain astable point (Theorem 3) and construct an explicit crystal isomorphism from B g ( λ ) to B h ( λ ) (Theorem 4). In the final Section 5 we apply the results of Section 4.1 to give a geometricrealization of KR-crystals in type A .2. Background and Notation
Let Q be a Dynkin quiver of type A or D with path algebra C Q . Let I be the vertex setof Q and Q the arrow set. For each arrow h of Q pointing from the vertex i to the vertex j wewrite out( h ) = i and in( h ) = j . Let g be the simple Lie algebra associated to the underlyingdiagram of Q over C with simple system { α i : i ∈ I } , simple coroots { h i : i ∈ I } , Cartan matrix C = ( c i,j ) i,j ∈ I , weight lattice P and dominant integral weights P + . By Gabriel’s theorem theisomorphism classes of finite dimensional indecomposable representations of Q over C are inbijection to the set Φ + of positive roots of g . For a positive root α ∈ Φ + , we denote by M ( α ) arepresentative of the isomorphism class of indecomposable C Q -modules associated to this root.We denote by C Q − mod the abelian category of finite dimensional left C Q -modules. On C Q − mod we have a non-degenerate bilinear form called the Euler form given by: h M, N i R := dim Hom C Q ( M, N ) − dim Ext C Q ( M, N ) . This form is known to depend only on the dimension vectors dim M and dim N and is equal to X j ∈ I dim M j dim N j − X h ∈ Q dim M out( h ) dim N in( h ) . AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 3
The symmetrization of the Euler form ( M, N ) R := h M, N i R + h N, M i R is determined by the Cartan matrix of g ; we have ( M, N ) R = t (dim M ) C (dim N ) (see [4, Lemma3.6.11] for details). Here we recall briefly the construction of the Auslander-Reiten quiver Γ Q of Q from [3,Section 6.5]. The vertices of this quiver are given by the isomorphism classes M of indecompos-able representations of Q while there is an arrow M → N if and only if there is an irreduciblemorphisms M → N in C Q − mod (non-isomorphisms that cannot be written as a compositionof two non-isomorphisms). Let Q ∗ be the quiver with the same set of vertices and reversedarrows. As an intermediate step we construct the infinite quiver Z Q ∗ which has Z × I as set ofvertices and for each arrow h : i → j in Q we draw an arrow from ( r, i ) to ( r + 1 , j ) and ( r, j ) to ( r, i ) for all r ∈ Z . Example.
Consider the Dynkin quiver of type A with orientation → ← . Then Z Q ∗ is given as follows: ( − , ❍❍❍❍❍❍❍❍❍ (0 , ❋❋❋❋❋❋❋❋ (1 , · · · ( − , : : ✉✉✉✉✉✉✉✉✉ $ $ ■■■■■■■■■ (0 , ❋❋❋❋❋❋❋❋ ; ; ①①①①①①①① (1 , ; ; ①①①①①①①① ❋❋❋❋❋❋❋❋ · · · ( − , ; ; ✈✈✈✈✈✈✈✈✈ (0 , ; ; ①①①①①①①① (1 , A slice of Z Q ∗ is a connected full subquiver which contains for each i ∈ I a unique vertexof the form ( r, i ) , r ∈ Z . There is a unique slice S Q which contains (0 , and is isomorphicto Q (in the above example highlighted in blue). The Nakayama permutation ν : Z × I → Z × I gives a bijective correspondence between the indecomposable projectives of C Q and theindecomposable injectives of C Q . For example, in type A n we have ν ( r, i ) = ( r + i − , n + 1 − i ) . Now Γ Q can be identified with the full subquiver of Z Q ∗ formed by the vertices lying between S Q and the image of this slice under ν (see [3, Proposition 6.5]). Furthermore Z Q ∗ has atranslation structure given by the Auslander–Reiten translation τ given by τ ( p, q ) = ( p − , q ) .This function gives rise to a bijection between the isomorphism classes of indecomposable non–projectives and the isomorphism classes of indecomposable non–injectives when restricted to Γ Q . We can describe τ as a function on the dimension vectors. For i ∈ I define r i : Z | I |≥ → Z | I |≥ via r i ( v ) = v − ( v, e i ) R e i . Here we denote the dimension vector of the simple C Q -module S ( i ) by e i . We fix a labeling i , i , . . . , i n of I adapted to Q , that is i is a sink of Q and i is a sink of σ Q and so on DENIZ KUS AND BEA SCHUMANN ( σ revereses the direction of all arrows at vertex ). For an indecomposable non-projective C Q -module M , the indecomposable non-injective C Q -module τ M has dimension vector r i n r i n − · · · r i (dim M ) . If we consider the quiver ← → and denote a vertex of Γ Q by the dimension vector of itsisomorphism class we get " " ❊❊❊❊❊❊❊❊ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ " " ❊❊❊❊❊❊❊❊ < < ②②②②②②②② < < ②②②②②②②② τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ " " ❊❊❊❊❊❊❊❊ < < ②②②②②②②② τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ Let ( C Q) op be the opposite algebra of C Q . We have a functor D : C Q − mod → ( C Q) op − mod , called standard duality functor , between the category of C Q -modules and thecategory of ( C Q) op -modules. For M ∈ C Q − mod we have D ( M ) := Hom C ( M, C ) with ( C Q) op -module structure defined by ( aϕ )( m ) = ϕ ( am ) , m ∈ M, a ∈ ( C Q) op , ϕ ∈ D ( M ) . Furthermore, for
M, N ∈ C Q − mod , f ∈ Hom C Q ( M, N ) , we have D ( f ) : D ( N ) → D ( M ) , φ φ ◦ f . From the definitions it is straightforward to see that D ( τ M ) = τ − D ( M ) . Also wecan identify representations of ( C Q) op − mod with representations of C Q ∗ − mod . Thus theAuslander-Reiten quiver of Q ∗ can be obtained by reversing each arrow in the Auslander-Reitenquiver of Q and interchanging the roles of τ and τ − . Let U q ( g ) be the Q ( q ) -algebra with generators E i , F i , K ± i , i ∈ I and the followingrelations for j ∈ I \ { i } K i K − i = K − i K i = 1 , K i K j = K j K i , K i E i K − i = q E i K i F i K − i = q − F i , E i F j − F j E i = 0 , E i F i − F i E i = K i − K − i q − q − If c i,j = − E i E j + E j E i = ( q + q − ) E i E j E i , F i F j + F j F i = ( q + q − ) F i F j F i ,K i E j K − i = q − E j , K i F j K − i = qF j . If c i,j = 0 : E i E j = E j E i , F i F j = F j F i , K i E j K − i = E j , K i F j K − i = F j . For m ∈ N , let [ m ] q := q m − + q m − + · · · + q − m +1 and define for x ∈ U q ( g ) the divided power x ( m ) := x m [ m ] q ! . (2.1)For λ ∈ P + we denote by V ( λ ) the irreducible U q ( g ) -module of highest weight λ and let U − q ⊆ U q ( g ) be the subalgebra generated by F i , i ∈ I . AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 5
Definition.
An abstract g -crystal B is a set endowed with maps wt : B → P, ε i : B → Z ⊔ {−∞} , ϕ i : B → Z ⊔ {−∞} , ˜ e i : B → B ⊔ { } , ˜ f i : B → B ⊔ { } for i ∈ I. satisfying the following axioms for i ∈ I and b, b ′ ∈ B • ϕ i ( b ) = ε i ( b ) + wt( b )( h i ) , • if b ∈ B satisfies ˜ e i b = 0 then wt(˜ e i b ) = wt( b ) + α i , ϕ i (˜ e i b ) = ϕ i ( b ) + 1 , ε i (˜ e i b ) = ε i ( b ) − , • if b ∈ B satisfies ˜ f i b = 0 then wt( ˜ f i b ) = wt( b ) − α i , ϕ i ( ˜ f a b ) = ϕ i ( b ) − , ε i ( ˜ f i b ) = ε i ( b ) + 1 , • ˜ e i b = b ′ if and only if ˜ f i b ′ = b , • if ε i ( b ) = −∞ , then ˜ e i b = ˜ f i b = 0 .Let B and B be abstract g -crystals. A map ψ : B ⊔ { } → B ⊔ { } satisfying ψ (0) = 0 iscalled a morphism of crystals if for b ∈ B , ψ ( b ) ∈ B and i ∈ I we have wt( ψ ( b )) = wt( b ) , ε i ( ψ ( b )) = ε i ( b ) , ϕ i ( ψ ( b )) = ϕ i ( b ) ,ψ (˜ e i b ) = ˜ e i ψ ( b ) , if ˜ e i b = 0 ψ ( ˜ f i b ) = ˜ f i ψ ( b ) , if ˜ f i b = 0 . A morphism of crystals which commutes with all ˜ e i , ˜ f i is called strict morphism of crystals .An injective (strict) morphism is called a (strict) embedding of crystals and a bijective strictmorphism is called an isomorphism of crystals .Let B and B be abstract g -crystals. The set B ⊗ B := { b ⊗ b | b ∈ B , b ∈ B } is called the tensor product of B and B and admits a g -crystal structure where wt( b ⊗ b ) =wt( b ) + wt( b ) and ε i ( b ⊗ b ) = max { ε i ( b ) , ε i ( b ) − wt( b )( h i ) } ,ϕ i ( b ⊗ b ) = max { ϕ i ( b ) , ϕ i ( b ) + wt( b )( h i ) } , ˜ e i ( b ⊗ b ) = ( ˜ e i b ⊗ b if ϕ i ( b ) ≥ ε i ( b ) b ⊗ ˜ e i b else, ˜ f i ( b ⊗ b ) = ( ˜ f i b ⊗ b if ϕ i ( b ) > ε i ( b ) b ⊗ ˜ f i b else. We recall the definition of the crystal bases B ( ∞ ) and B ( λ ) of U − q and V ( λ ) , respectively,following [5, Sections 2 and 3]. We fix i ∈ I in the rest of the discussion. For P ∈ U − q thereexists unique Q, R ∈ U − q such that E i P − P E i = K i Q − K − i Rq − q − . DENIZ KUS AND BEA SCHUMANN
The endomorphism E ′ i : U − q → U − q given by E ′ i ( P ) = R induces a vector spaces decomposition U − q = M m ≥ F ( m ) i Ker( E ′ i ) . We define the
Kashiwara operators ˜ e i , ˜ f i on U − q by the following rule ˜ f i ( F ( m ) i u ) = F ( m +1) i u, ˜ e i ( F ( m ) i u ) = F ( m − i u, u ∈ Ker( E ′ i ) . Let A be the subring of Q ( q ) consisting of rational functions f ( q ) without a pole at q = 0 . Let L ( ∞ ) be the A -submodule generated by all elements of the form ˜ f i ˜ f i · · · ˜ f i ℓ (1) (2.2)and let B ( ∞ ) ⊆ L ( ∞ ) /q L ( ∞ ) be the subsets of all residues of (2.2). For b ∈ B ( ∞ ) we let wt( b ) be the weight of the element and ε i ( b ) = max { ˜ e ki ( b ) = 0 | k ∈ N } . This endows B ( ∞ ) with the structure of an abstract crystal (see Definition 2.4). The Q ( q ) -antiautomorphism ∗ : U − q → U − q given by E ∗ i = E i , F ∗ i = F i , K ∗ i = K − i has the properties L ( ∞ ) ∗ = L ( ∞ ) , B ( ∞ ) ∗ = B ( ∞ ) and ∗ preserves the weight function (see [6,Theorem 8.3] for details). We can endow B ( ∞ ) with a second structure of an abstract crystaldenoted by B ( ∞ ) ∗ with Kashiwara operators (also called the ∗ -twisted maps) ˜ f ∗ i ( x ) = ( ˜ f i x ∗ ) ∗ , ˜ e ∗ i ( x ) = (˜ e i x ∗ ) ∗ , ε ∗ i ( x ) = ε i ( x ∗ ) By construction ∗ induces a crystal isomorphism between B ( ∞ ) and B ( ∞ ) ∗ . For λ ∈ P + let π λ : U − q → V ( λ ) be the natural U − q -homomorphism sending to v λ and denote by ( L ( λ ) , B ( λ )) the crystal base of V ( λ ) . Then we have (see [5, Theorem 5]) π λ ( L ( ∞ )) = L ( λ ) and we obtainan induced surjective homomorphism π λ : L ( ∞ ) /q L ( ∞ ) → L ( λ ) /q L ( λ ) with the following properties • B ( λ ) is isomorphic to { π λ ( b ) : b ∈ B ( ∞ ) }\{ } by the map π λ . • ˜ f i ◦ π λ = π λ ◦ ˜ f i for all i ∈ I , • If b ∈ B ( ∞ ) is such that π λ ( b ) = 0 , then we have ˜ e i π λ ( b ) = π λ (˜ e i b ) for all i ∈ I .We denote by T λ = { r λ } the abstract g -crystal consisting of one element and wt( t λ ) = λ, ε i ( t λ ) = ϕ i ( t λ ) = −∞ , ˜ e i t λ = ˜ f i t λ = 0 , ∀ i ∈ I. By the above properties we have an embedding of crystals B ( λ ) → B ( ∞ ) ⊗ T λ , π λ ( b ) b ⊗ t λ which commutes with the ˜ e i ’s (but not necessarily with the ˜ f i ’s) whose image is given by (see[6, Proposition 8.2]) { b ⊗ t λ ∈ B ( ∞ ) ⊗ T λ : ε ∗ i ( b ) ≤ λ ( h i ) ∀ i ∈ I } . (2.3)We summarize the above discussion in the following theorem. AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 7
Theorem 1 ([6, Proposition 8.2]) . Let λ ∈ P + . Then the crystal graph B ( λ ) of the irreduciblehighest weight module V ( λ ) can be realized as the full subgraph of B ( ∞ ) consisting of all vertices b ∈ B ( ∞ ) such that ε ∗ i ( b ) ≤ λ ( h i ) for all i ∈ I . For b ∈ B ( λ ) the Kashiwara operators ˜ f λi , ˜ e λi on B ( λ ) are given by ˜ e λi b = ˜ e i b, ˜ f λi b = ( , if ˜ f i b / ∈ B ( λ )˜ f i b, if ˜ f i b ∈ B ( λ ) . (cid:3) Geometric construction of crystal bases
In this section we review a geometric construction of the crystals B ( ∞ ) and B ( λ ) for λ ∈ P + in terms of irreducible components of quiver varieties. We denote by
Q = (
I, H ) the associated double quiver of Q , where for each h ∈ Q , H contains two arrows with the same endpoints, one in each direction. For an arrow h ∈ H , wedenote by h the arrow with out( h ) = in( h ) and in( h ) = out( h ) . In this notation, the doublequiver Q = (
I, H ) has as set of arrows H = Q ⊔ Q . Example.
For
Q = 1 h ←− h ←− , the double quiver Q looks as follows. Q = 1 h / / h o o h / / . h o o We define the function ǫ : H → {± } h (cid:26) , if h ∈ Q − , if h / ∈ Q . The preprojective algebra
Π(Q) is the quotient of the path algebra of the double quiver Q bythe ideal generated by X h ∈ H, in( h )= i ǫ ( h ) h ¯ h, i ∈ I. For a fixed finite–dimensional I –graded vector spaces V = L i ∈ I V i over C , we define Lusztig’squiver variety Λ V to be the variety of representations of Π(Q) with underlying vector space V ,i.e. Λ V := ( x h ) h ∈ H ∈ M h ∈ H Hom C ( V out( h ) , V in( h ) ) : X h ∈ H, in( h )= i ǫ ( h ) x h x ¯ h = 0 for all i ∈ I . Let
Rep V (Q) be the variety of representations of Q with underlying vector space V , that is Rep V (Q) = M h ∈ Q Hom( V out( h ) , V in( h ) ) , which is clearly a closed subvariety of the affine variety Λ V . From now on we constantly identifythe points of Λ V (resp. Rep V (Q) ) with the corresponding modules over Π(Q) (resp. C Q ) and DENIZ KUS AND BEA SCHUMANN write expressions like M ∈ Λ V for M = ( V, x ) ∈ Π(Q) − mod . We have an action of the group G v = Q i GL ( V i ) on Λ V and Rep V (Q) by base change, that is for M = ( V, x ) ∈ Λ V , g.M = f M ,where f M = ( V, e x ) ∈ Π(Q) − mod with e x h := g in( h ) x h g − h ) , h ∈ H and analogously for M ∈ Rep V (Q) . The orbits of this action on Λ V (resp. Rep V (Q) ) areexactly the isomorphism classes of representations of Π(Q) − mod (resp. C Q − mod ) with fixeddimension vector v := dim( V ) . For M ∈ Rep V (Q) , we denote the corresponding orbit by O M and let gl v = L i ∈ I gl v i the Lie algebra of G v . Remark.
The definition of preprojective algebras is motivated by symplectic geometry, namelyLusztig’s quiver variety can be viewed as the zero fibre of the moment map for the action of G v on Rep V (Q) . The additional nilpotency condition on the elements of Λ V is omitted, since werestrict ourselves to preprojective algebras of Dynkin quivers and this condition is automaticallysatisfied (see [11, Proposition 14.2(a)]).Note that, up to isomorphism, Λ V depends only on the graded dimension of V . Therefore wealso denote Λ V by Λ( v ) , regarding the graded dimension of the vector spaces as part of thedatum of the representations of Π(Q) . The next lemma describes the irreducible componentsof the variety Λ( v ) . Lemma.
An element ( x = ( x h ) , x = ( x h )) h ∈ Q ∈ Rep V (Q) ⊕ Rep V (Q ∗ ) lies in the quivervariety Λ( v ) if and only if tr([ a, x ] x ) := X h ∈ Q tr(( a in ( h ) x h − x h a out ( h ) ) x h ) = 0 , for all a ∈ gl v . The irreducible components are exactly the closures of conormal bundles to G v -orbits in Rep V (Q) : Λ( v ) = [ O M ∈ Rep V (Q) /G v [ x ∈O M ( { x } × X x ) . where X x = { x ∈ Rep V (Q ∗ ) : tr([ a, x ] x ) = 0 ∀ a ∈ gl v } . Proof.
For the first claim see [8, Lemma 5.6] and hence Λ( v ) is in fact the union of theseconormal bundles. Since G v is irreducible, we also have that the closures of conormal bundlesare irreducible subvarieties of Λ( v ) . The fact that they are as well the irreducible componentsfollows from Gabriel’s theorem, since the orbit space Rep V (Q) /G v is finite. (cid:3) We denote the irreducible components of Λ( v ) by Irr Λ( v ) and the closure of the conormalbundle corresponding to the orbit O M by X M . Now, we recall the geometric construction of Kashiwara operators on the set of irre-ducible components
Irr Λ( v ) from [7]. For i ∈ I and M ∈ Λ( v ) , define ε i ( M ) to be thedimension of the S ( i ) -isotypic component of the head of M . For M = ( V, x ) ∈ Π(Q) − mod ,that is ε i ( M ) = dim Coker M h :in( h )= i V out( h ) x h −→ V i . (3.1) AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 9
For c ∈ Z ≥ , we further introduce the subsets Λ( v ) i,c := { M ∈ Λ( v ) | ε i ( M ) = c } . Let e i ∈ Z | I |≥ be as usual the i -th unit vecor and fix c ∈ Z ≥ such that v i − c ≥ . We define Λ( v, c, i ) := { ( M, N, ϕ ) : M ∈ Λ( v ) i,c , N ∈ Λ( v − ce i ) i, , ϕ ∈ Hom
Π(Q) ( N, M ) injective } . Considering the diagram Λ( v − ce i ) i, p ←− Λ( v, c, i ) p −→ Λ( v ) i,c , (3.2)where p ( M, N, ϕ ) = N and p ( M, N, ϕ ) = M . It is shown in [7, Lemma 5.2.3] that themap p is a principal G v -bundle and the map p is a smooth map whose fibres are connectedvarieties. Standard algebraic geometry arguments then show (for Λ( v ) i,c = ∅ ) that there is aone–to–one correspondence between the set of irreducible components of Λ( v − ce i ) i, and theset of irreducible components of Λ( v ) i,c , i.e. Irr Λ( v − ce i ) i, ∼ = Irr Λ( v ) i,c . (3.3)Let X ∈ Irr Λ( v ) and define for i ∈ I the integer ε i ( X ) := min M ∈ X ε i ( M ) . The function ε i given in (3.1) is upper semi-continuous, i.e. { M ∈ X : ε i ( M ) ≥ ε i ( X ) + 1 } is a closed subset. So there is an open dense subset of X such that ε i is constant (namely thevalue of ε i on this subset is ε i ( X ) ). Let IΛ( v ) i,c := { X ∈ Irr Λ( v ) | ε i ( X ) = c } . So if X ∈ IΛ( v ) i,c , we obtain from the prior considerations that there is an open dense subset U X of X such that U X ⊆ Λ( v ) i,c . Since Λ( v ) and Λ( v ) i,c have pure dimension dim Rep V (Q) (see [11, Theorem 12.3]), we get a bijection IΛ( v ) i,c → IΛ( v − ce i ) i, , X p p − ( U X ) . (3.4)Suppose that ¯ X ∈ IΛ( v − ce i ) i, corresponds to the component X ∈ IΛ( v ) i,c und the bijection3.4. We define maps ˜ f ci : IΛ( v − ce i ) i, → IΛ( v ) i,c , ¯ X X ˜ e ci : IΛ( v ) i,c → IΛ( v − ce i ) i, , X ¯ X The data of these maps yields a crystal structure on B g ( ∞ ) := F v Irr Λ( v ) together with thefollowing maps for X ∈ IΛ( v ) i,c ⊆ B g ( ∞ ) : ˜ f i ( X ) := ˜ f c +1 i ˜ e ci ( X ) , ˜ e i ( X ) := ( ˜ f c − i ˜ e ci ( X ) , if c > , otherwise wt( X ) := − X i ∈ I v i α i for X ∈ Irr Λ( v ) , ϕ i ( X ) := ε i ( X ) + wt( X )( h i ) . It is shown in [7, Theorem 5.3.2] that B g ( ∞ ) is isomorphic to the crystal B ( ∞ ) of U − q . In [14], Saito gave a realization of the crystal B ( λ ) via Nakajima’s quiver varieties. Inthis section we recall the definition of those spaces. To define them we consider a framing onthe double quiver Q by adding an extra vertex i ′ and an extra arrow t i : i → i ′ for all i ∈ I . Example.
The framed double quiver for the Dynkin graph of type A is given as follows. h / / t (cid:15) (cid:15) h o o t (cid:15) (cid:15) h / / h o o t (cid:15) (cid:15) ′ ′ ′ For v, λ ∈ Z | I |≥ , we choose I -graded vector spaces V and W of graded dimension v, λ , respec-tively and define Λ( v, λ ) := Λ( v ) × M i ∈ I Hom( V i , W i ) . The action of the group G v can be extended on Λ( v, λ ) via g ( x, t ) := ( g i ) i ∈ I (( x h ) h ∈ H , ( t i ) i ∈ I ) = (cid:16) ( g in( h ) x h g − h ) ) h ∈ H , ( t i g − i ) i ∈ I (cid:17) . We consider the subset Λ( v, λ ) st of stable points in Λ defined as Λ( v, λ ) st := { ( x, t ) ∈ Λ( v, λ ) : \ out( h )= i (ker x h ∩ ker t i ) = 0 } . Remark.
This definition is equivalent to the one given in [12] stating that there is no non-trivial x -stable subspace of V contained in the kernel of t , see [2, Lemma 3.4]. Moreover, thisis a stability condition in the sense of Mumford with respect to the character θ = ( θ i ) ∈ Z | I | of G v with θ i = − for all i ∈ I (see [12, Section 3.2]).The subset Λ( v, λ ) st is open in Λ( v, λ ) and we clearly have an induced action of the group G v on Λ( v, λ ) st . We further have the following. Lemma ([12, Lemma 3.1]) . The action of G v on Λ( v, λ ) st is free and Λ( v, λ ) st is a non-singularsubvariety of Λ( v, λ ) . (cid:3) The
Nakajima quiver variety is defined to be the geometric quotient of Λ( v, λ ) st by G v L ( v, λ ) := Λ( v, λ ) st /G v . We denote by
Irr L ( v, λ ) the set of irreducible components of L ( v, λ ) . Now using Lemma 3.3we can make the following observations. We have Irr L ( v, λ ) = n Y λM : M ∈ C Q − mod and Y λM = ∅ o (3.5)where Y λM := X M × M i ∈ I Hom( V i , W i ) ! ∩ Λ( v, λ ) st ! /G v . Moreover, we have the following identification
Irr L ( v, λ ) ∼ = { Y ∈ Irr Λ( v, λ ) : Y ∩ Λ( v, λ ) st = ∅} . (3.6) AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 11
We conclude that the irreducible components of L ( v, λ ) are in one-to-one correspondence to theirreducible components of Λ( v, λ ) that contain a stable point. In [14] Saito describes a crystalstructure on Irr L ( v, λ ) using similar arguments as in [7]. The key point for our approach isthe following theorem. Theorem 2 ([14, Theorem 4.6.4, Lemma 4.6.3]) . The map i : Irr L ( v, λ ) → B g ( ∞ ) , Y λM X M is an embedding of crystals which commutes with the operators ˜ e i , i ∈ I . Moreover, Irr L ( v, λ ) is isomorphic to the crystal B g ( λ ) and hence B g ( λ ) is the full subgraph of B g ( ∞ ) with vertices { X M ∈ B g ( ∞ ) : Y λM = ∅} . (cid:3) A description of the irreducible components (3.5) will be given purely combinatorial in theAuslander-Reiten quiver Γ Q in Theorem 3.4. Crystals via the combinatorics of Auslander-Reiten quivers
In this subsection we recall some fundamental definitions from [13] which arise in thecontext of an explicit crystal structure on certain Lusztig PBW basis.We define the posets P i (Q) := { M ∈ C Q − mod : M is indecomposable and dim Hom C Q ( M, S ( i )) = 0 }P ∨ i (Q) := { M ∈ C Q − mod : M is indecomposable and dim Hom C Q ( S ( i ) , M ) = 0 } together with the relation (cid:22) given by N (cid:22) M ⇐⇒ Hom C Q ( N, M ) = 0 . Furthermore we define the poset S i (Q) := V = k M j =1 M j : { M , . . . , M k } forms an antichain in P i (Q) together with the relation E given by V E V ′ ⇐⇒ dim Hom k Q ( B, V ′ ) = 0 for each indecomposable direct summand B of V . In the same way we can define S ∨ i (Q) withthe relation E ∨ given by V E ∨ V ′ ⇐⇒ dim Hom k Q ( V, B ) = 0 for each indecomposable direct summand B of V ′ . Example.
Let Q be the following quiver
21 3 o o O O . o o The poset P ∨ (Q) is the union of all framed modules:
01 0 0 (cid:24) (cid:24) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷
10 1 0 (cid:24) (cid:24) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
01 1 1 (cid:24) (cid:24) ✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
10 0 0 ! ! ❉❉❉❉❉❉❉❉❉
01 1 0 ! ! ❉❉❉❉❉❉❉❉ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
10 1 1 ! ! ❇❇❇❇❇❇❇❇❇❇ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
11 1 0 (cid:28) (cid:28) ✽✽✽✽✽✽✽✽✽✽✽✽✽✽ = = ③③③③③③③③ F F ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞
11 2 1 (cid:28) (cid:28) ✽✽✽✽✽✽✽✽✽✽✽✽✽✽✽ = = ③③③③③③③③③ F F ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
00 1 1 (cid:26) (cid:26) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
11 1 1 B B ✝✝✝✝✝✝✝✝✝✝✝✝✝✝
00 1 0 C C ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
00 0 1 τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ τ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ We have S ∨ (Q) = (cid:26)
01 0 0 ,
11 1 0 ,
11 1 1 ,
01 1 0 ,
11 1 1 ,
01 1 0 ⊕
11 1 1 ,
11 2 1 ,
01 1 1 (cid:27) . We have two chains of maximal length in S ∨ i (Q) :
01 0 0 E ∨
11 1 0 E ∨
01 1 0 ⊕
11 1 1 E ∨
01 1 0 E ∨
11 2 1 E ∨
01 1 1 and
01 0 0 E ∨
11 1 0 E ∨
01 1 0 ⊕
11 1 1 E ∨
11 1 1 E ∨
11 2 1 E ∨
01 1 1
Fix i ∈ I . For a k Q –module M and an element V ∈ S i (Q) and V ∈ S ∨ i (Q) respectively define F i ( M, V ) := X B ∈P i (Q); B E V µ B ( M ) − µ τB ( M ) . (4.1) F ∨ i ( M, V ) := X B ∈P ∨ i (Q); V E ∨ B µ B ( M ) − µ τ − B ( M ) . (4.2)For a C Q -module M , let V M be a E -maximal element of S i (Q) such that max V ∈S i (Q) F i ( M, V ) = F i ( M, V M ) and let U M be the direct sum of all τ B such that B is an element of P i (Q) with B V M and B is minimal with this property. We define B h ( ∞ ) to be the set of all isomorphism classes of C Q -modules. Definition.
A quiver Q is called cospecial if dim Hom C Q ( S ( i ) , M ) ≤ for all i ∈ I and allindecomposable C Q -modules M . Equivalently, this means that each sink i ∈ I corrspond toa miniscule weight of the Lie algebra of Q (see [16, Corollary 2.22]). We call Q special if Q ∗ is cospecial. This is equivalent to the fact that no thick vertex is a source of Q , where avertex i ∈ I is called thick if there exists an indecomposable C Q -module M = ( V, x ) such that dim V i ≥ . AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 13
For a special quiver Q it has been shown in [13, Proposition 6.1] that the module U M is a directsummand of M and the following defines a crystal structure on B h ( ∞ ) isomorphic to B ( ∞ ) : ˜ f i ( M ) = ( N ⊕ V M ) , where M = N ⊕ U M ε i ( M ) = F i ( M, V M ) , ϕ i ( M ) = ε i ( M ) + wt( M )( h i )wt( M ) = − X j ∈ I (dim M ) j α j Remark. (1) The explicit description of ˜ e i can be found in [16, Proposition 2.19].(2) The isomorphism between B h ( ∞ ) and B ( ∞ ) implies that the module V M is unique.An alternative proof of this fact can be given as in [16, Lemma 3.15].Moreover, if we set ε ∗ i ( M ) = max V ∈S ∨ i (Q) F ∨ i ( M, V ) , and assume that Q is cospecial and special, then the embedding of the crystal graph of B h ( λ ) ∼ = B ( λ ) into B h ( ∞ ) can be described as (see [13, Proposition 7.4]): B h ( λ ) = { M ∈ B h ( ∞ ) : ε ∗ i ( M ) ≤ λ ( h i ) for all i ∈ I } . The remainder of this section develops combinatorics on the geometric construction ofcrystal bases recalled in Section 3 in terms of Auslander-Reiten quivers. Recall the embeddingof irreducible components of Nakajima’s quiver variety into the irreducible components ofLusztig’s quiver variety from (3.6). We decribe the image of this embedding. Namely thefollowing result gives a criterion for the irreducible components of Λ( v, λ ) to contain a stablepoint. Theorem 3.
Let Q be cospecial, M ∈ C Q − mod . The following statements are equivalent. (i) Y λM ∈ Irr L ( v, λ ) , i.e. Y λM = ∅ . (ii) F ∨ i ( M, V ) ≤ λ ( h i ) for all V ∈ S ∨ i (Q) and for all i ∈ I .Moreover we have the equality ε ∗ i ( M ) = min x ∈ X M dim( \ out( h )= i ker x h ) (4.3) Proof.
By (3.6) we have that Y λM ∈ Irr L ( v, λ ) if and only if Y λM contains a stable point. Notethat stability for a point ( x, t ) ∈ Λ( v, λ ) means that (cid:16) \ out( h )= i ker x h (cid:17) ∩ ker t i = 0 ∀ i ∈ I. This is equivalent to the fact that the restriction of t i to T out( h )= i ker x h is injective for all i ∈ I . Hence Y λM = ∅ ⇔ ∃ x ∈ X M with dim( \ out( h )= i ker x h ) ≤ λ ( h i ) ∀ i ∈ I. Let X ∗ M the closure of the conormal bundle to the G v -orbit to D ( M ) in Rep V ( Q ∗ ) (recall thedefinition of D ( M ) from Section 2.3). By Lemma 3.1 we have that X ∗ M ∈ B g ( ∞ ) . Note that min x ∈ X M dim( \ out( h )= i ker x h ) = ε i ( X ∗ M ) . Hence we obtain ε i ( X ∗ M ) ≤ λ ( h i ) for all i ∈ I ⇔ ε i ( X D ( M ) ) ≤ λ ( h i ) for all i ∈ I ⇔ F i ( D ( M ) , V ) ≤ λ ( h i ) for all V ∈ S i (Q ∗ ) and i ∈ I, where the last equivalence follows from [16, Proposition 4.7] and the fact that Q is cospecial ifand only if Q ∗ is special. We can identify the elements of S ∨ i (Q) canonically with the elementsof S i (Q ∗ ) via the standard duality D ( M ) , since the Auslander-Reiten quiver of C Q ∗ can beobtained by reversing each arrow in the Auslander-Reiten quiver of C Q and interchanging theroles of τ and τ − In particular, D ( V ) ∈ S ∨ i (Q) ⇔ V ∈ S i (Q ∗ ) Moreover, for V and V ′ in S ∨ i (Q) , we have V E V ′ if and only if D ( V ) E ∨ D ( V ′ ) . This shows that Y λM = ∅ ⇔ F ∨ i ( M, V ) ≤ λ ( h i ) for all V ∈ S ∨ i (Q) and i ∈ I since for all V ∈ S ∨ i (Q) we have F i ( D ( M ) , D ( V )) = F ∨ i ( M, V ) . This finishes the proof. (cid:3) In this section we shall prove the following theorem.
Theorem 4.
Let Q be special and cospecial. The map B g ( λ ) → B h ( λ ) , Y λM M is well-defined and an isomorphism of crystals. We have a commutative diagram B g ( ∞ ) / / B h ( ∞ ) B g ( λ ) i O O / / B h ( λ ) O O Before we are able to prove Theorem 4, we need some preparatory work to show that thecrystal operators are well-defined.
Lemma.
Let M be in C Q − mod . Then we have ε ∗ i ( ˜ f j ( M )) = ε ∗ i ( M ) , i = jε ∗ i ( M ) + 1 if i = j , V M = S ( i ) and max V ∈S ∨ i (Q) F ∨ i ( M, V ) = F ∨ i ( M, S ( i )) ε ∗ i ( M ) else.Proof. The case i = j is clear by (4.3) and the definition of ˜ f i (see Section 3.2).Let i = j . We consider two cases. AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 15
Case 1:
Assume that ˜ f i ( M ) = ( M ⊕ S ( i )) . Hence F ∨ i ( M, V ) = F ∨ i ( M ⊕ S ( i ) , V ) for all V ∈ S ∨ i (Q) \{ S ( i ) } and we get F ∨ i ( M, S ( i )) = F ∨ i ( M ⊕ S ( i ) , S ( i )) + 1 . We conclude for this case ε ∗ i ( ˜ f i ( M )) = ε ∗ i ( M ) + 1 if max V ∈S ∨ i (Q) F ∨ i ( M, V ) = F ∨ i ( M, S ( i )) ε ∗ i ( M ) else. Case 2:
In this case we suppose that ˜ f i ( M ) = N with N ≇ M ⊕ S ( i ) . Then N ∼ = M ′ ⊕ V M where M = M ′ ⊕ U M with U M and V M as in Section 4.1. Let B be an indecomposable C Q -module in P i (Q) ∩ P ∨ i (Q) . Then we have the following homomorphisms: B ։ S ( i ) ֒ → B which implies P i (Q) ∩ P ∨ i (Q) = { S ( i ) } . So there is no indecomposable direct summand of V M in P ∨ i (Q) which gives ε ∗ i ( ˜ f i ( M )) ≤ ε ∗ i ( M ) . Assume that there is an indecomposable direct summand C of U M in P ∨ i (Q) . Then, from thedefinition, there is a B ∈ P i (Q) such that C = τ B and we have homomorphisms B ։ S ( i ) ֒ → C = τ B. Hence
Hom C Q ( B, τ B ) = 0 , which is a contradiction. An analog argument shows that therecannot be an indecomposable direct summand B of V M such that τ − B ∈ P ∨ i (Q) . This yieldsin this case ε ∗ i ( ˜ f i ( M )) = ε ∗ i ( M ) . (cid:3) Proposition.
Let λ ∈ P + and M be an element of B h ( λ ) . Then for all i ∈ I : ∃ j ∈ I : ε ∗ j ( ˜ f i ( M )) > λ ( h j ) ⇐⇒ ϕ i ( X M ) = 0 . Proof.
Since ε ∗ i ( M ) ≤ λ ( h i ) for all i ∈ I , it follows from Lemma 4.3 that we only need to showthe equivalence ε ∗ i ( ˜ f i ( M )) > λ ( h i ) ⇐⇒ ϕ i ( X M ) = 0 . In the case of ε ∗ i ( ˜ f i ( M )) > λ ( h i ) we obtain with Lemma 4.3 that ε ∗ i ( M ) = λ ( h i ) , ε i ( M ) = F i ( M, S ( i )) . Since Q is special and cospecial, we have dim Hom C Q ( M, S ( i )) = X B ∈P i (Q) µ B ( M ) dim Hom C Q ( B, S ( i )) = X B ∈P i (Q) µ B ( M ) . dim Hom C Q ( S ( i ) , M ) = X B ∈P ∨ i (Q) µ B ( M ) dim Hom C Q ( S ( i ) , B ) = X B ∈P ∨ i (Q) µ B ( M ) . This implies F i ( M, S ( i )) + F ∨ i ( M, S ( i )) = dim Hom C Q ( M, S ( i )) − dim Hom C Q ( τ − M, S ( i ))+ dim Hom C Q ( S ( i ) , M ) − dim Hom C Q ( S ( i ) , τ M )= h M, S ( i ) i R + h S ( i ) , M i R , where the last equation follow from the Auslander-Reiten formulas (see [1, Corollary 2.14]).Hence we have ϕ i ( X M ) = ε i ( X M ) + wt( X M )( h i )= F i ( M, S ( i )) + λ ( h i ) − ( M, S ( i )) R = F i ( M, S ( i )) + λ ( h i ) − h M, S ( i ) i R − h S ( i ) , M i R = F i ( M, S ( i )) + λ ( h i ) − F i ( M, S ( i )) − F ∨ i ( M, S ( i )) = 0 . where the second equality follows from the crystal isomorphism in [16, Theorem 3.26]. Con-versely, assume that ϕ i ( X M ) = 0 , i.e. ε i ( X M ) + wt( X M )( h i ) = ε i ( X M ) + λ ( h i ) − F i ( M, S ( i )) − F ∨ i ( M, S ( i )) . Since M ∈ B h ( λ ) we have F ∨ i ( M, S ( i )) ≤ λ ( h i ) and F i ( M, S ( i )) ≤ ε i ( X M ) . Thus ε i ( X M ) = F i ( M, S ( i )) , λ ( h i ) = F ∨ i ( M, S ( i )) = ε ∗ i ( M ) . Using Lemma 4.3, we get ε ∗ i ( ˜ f i ( M )) > ε ∗ i ( M ) = λ ( h i ) . (cid:3) Proof of Theorem 4
From Lemma 4.3 and Proposition 4.3 we obtain that the restriction ofthe crystal isomorphism B g ( ∞ ) → B h ( ∞ ) , X M M (see [16, Theorem 3.26]) induces anisomorphism of crystals B g ( λ ) → B h ( λ ) . Applications to affine crystals and the promotion operator
In this section we consider the type A n quiver Q with index set I = { , . . . , n } and orientation (cid:8)(cid:15)(cid:9)(cid:14)(cid:10)(cid:13)(cid:11)(cid:12) n (cid:8)(cid:15)(cid:9)(cid:14)(cid:10)(cid:13)(cid:11)(cid:12) n − o o (cid:8)(cid:15)(cid:9)(cid:14)(cid:10)(cid:13)(cid:11)(cid:12) n − o o . . . o o (cid:8)(cid:15)(cid:9)(cid:14)(cid:10)(cid:13)(cid:11)(cid:12) o o (cid:8)(cid:15)(cid:9)(cid:14)(cid:10)(cid:13)(cid:11)(cid:12) o o Our aim is to describe the promotion operator on the crystel B h ( λ ) and hence on B g ( λ ) for a fixed rectangular weight λ = m̟ j , j ∈ I . This will allow us to realize Kirillov-Reshetikhincrystals geometrically.For an indecomposable module M ( r, s ) (corresponding to the root α r + · · · + α s ) we abbreviatethe multiplicity of M ( r, s ) in a C Q -module M simply by µ r,s ( M ) . We fix for the rest of thissection an element M ∈ B h ( λ ) . Note that the choice of the orientation gives F ∨ i ( M, M ( i, s )) = n X k = s µ i,k ( M ) − n − X k = s µ i +1 ,k +1 ( M ) . (5.1)Since F ∨ i ( M, V ) ≤ λ ( h i ) for all i ∈ I and V ∈ S ∨ i (Q) we get that k i := m − dim Hom C Q ( S ( i ) , M ) ≥ . AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 17
Moreover, since λ is rectangular, we have µ r,s ( M ) = 0 , for all r > j or ( r ≤ j and s > n − j + r ) . So we can think of an element M ∈ B h ( λ ) as an array M = ( µ r,s ( M )) with ≤ r ≤ j and r ≤ s ≤ n − j + r in the Auslander-Reiten quiver Γ Q . We associate to such an array its extendedarray M ext = ( µ r,s ( M ext )) , ≤ r ≤ j, r ≤ s ≤ n + 1 − j + r defined by µ r,s ( M ext ) = ( µ r,s − ( M ) , if r = sk r , if r = s and view M ext as a C e Q -module, where e Q is the A n +1 quiver with same standard orientation. Example.
Let n = 3 and j = 2 . Below is the array of M ext and the array of M is highlightedin blue (we abbreviate µ r,s ( M ) = µ r,s ). µ , µ , µ , µ , k k where we have k = m − µ , − µ , and k = m − µ , − µ , . Fix an arbitrary C e Q -module N and let W N the E ∨ -maximal element of S ∨ i ( e Q) such that max V ∈S ∨ i ( ˜Q) F ∨ i ( N, V ) = F ∨ i ( N, W N ) and let E N = τ − B where B is an element of P ∨ i ( e Q) with W N ∨ B and B is maximal withthis property. Since S ∨ i ( e Q) = { M ( i, i ) , M ( i, i + 1) , . . . , M ( i, n + 1) } we have that W N and E N are indecomposable (if the latter exists). With other words W N = M ( i, s ) , E N = M ( i + 1 , s ) , where s = max { i ≤ s ≤ n + 1 : F ∨ i ( M, M ( i, s )) is maximal } . Definition. (1) Let N be a C e Q -module. We define operators T i for i < j and sh j in orderto obtain another C e Q -module as follows. Let T i ( N ) = N (resp. sh j ( N ) = N ) if W N (resp. M ( j, n + 1) ) is not a summand of N and otherwise we set T i ( N ) = N ′ ⊕ E N , where N = N ′ ⊕ W N , sh j ( N ) = N ′′ , where N = N ′′ ⊕ M ( j, n + 1) . We define further e pr( N ) = ( T ◦ · · · ◦ T j − ◦ sh j ) µ j,n +1 ( N ) ( N ) . (2) We let pr( M ) to be the C Q -module determined by the array µ r,s (pr( M )) = µ r,s ( e pr( M ext )) 1 ≤ r ≤ j, r ≤ s ≤ n − j + r. Example.
Let n = m = j = 3 and consider the element M ∈ B h ( λ ) with array M = 1 1 20 1 01 1 1 We get by applying our operators M ext = 1 1 20 1 01 1 11 0 0 T ◦ sh −−−−→ T −→ sh −−→ T −→ T −→ Hence we get pr( M ) = 0 1 11 1 20 0 0 The promotion operator (see [17] for details) is the analogue of the cyclic Dynkin diagramautomorphism on the level of crystals. On the set of all semi-standard Young tableaux
SSYT( λ ) of shape λ over the alphabet ≺ · · · ≺ n + 1 the promotion operator can be described asfollows. Let T be a Young tableaux, then we get pr( T ) by removing all letters ( n + 1) , adding1 to each letter in the remaining tableaux, using jeu-de-taquin to slide all letters up and finallyfilling the holes with 1’s. Combining [15, Theorem 6.4] and Theorem 4 we get a classical crystalisomorphism (which is known explicitly only for the standard orientation) ϕ : B h ( λ ) → SSYT( λ ) , M ϕ ( M ) where the semi-standard tableaux ϕ ( M ) for M ∈ B h ( λ ) is determined by µ r,s ( M ) = ϕ ( M ) , ≤ r ≤ j, r ≤ s ≤ n − j + r. (5.2)We claim that pr from Definition 5.2 is the promotion operator on B h ( λ ) and our strategy isto show that it commutes with the promotion operator on SSYT( λ ) under the isomorphism ϕ .It is possible to define a tableau ϕ ( M ′ ) using (5.2) for any C Q -module M ′ . Obviously we have M ′ ∈ B h ( λ ) ⇐⇒ ϕ ( M ′ ) ∈ SSYT( λ ) (5.3) Theorem 5.
Let λ = m̟ j be a rectangular weight of type A n . Then we have a commutativediagram B h ( λ ) pr (cid:15) (cid:15) ϕ / / SSYT( λ ) pr (cid:15) (cid:15) B h ( λ ) ϕ / / SSYT( λ ) AKAJIMA QUIVER VARIETIES, AFFINE CRYSTALS AND COMBINATORICS OF AR QUIVERS 19
Proof.
Let M ∈ B h ( λ ) and let pr( M ) the C Q -module as described in Definition 5.2. If we canshow that pr( ϕ ( M )) = ϕ (pr( M )) we get at once pr( M ) ∈ B h ( λ ) and the commutativity by(5.3). Note that M ext also encodes the tableaux ϕ ( M ) via µ r,s ( M ext ) = r ’s of ϕ ( M ) , ≤ r ≤ j, r ≤ s ≤ n − j + r. In the remaining part of the proof we will show that e pr( M ext ) encodes the skew-tableaux ofshape λ \ µ , µ = ( µ j,n ( M ) , , . . . ) , which is obtained from ϕ ( M ) as follows: remove all letters n + 1 and slide the boxes up using jeu-de-taquin. This would finish the proof by construction(see Definition 5.2(2)). So assume that the last two rows of ϕ ( M ) are given by a a · · · a m − a m b b · · · b m − b m with b m = b m − = · · · = b m − µ j,n ( M )+1 = n + 1 . So the tableaux corresponding to sh j ( M ext ) isgiven by a a · · · a m − a m b b · · · b m − If we slide the empty box to the ( j − -th row we have to move an entry a p to the j -th row.In order to figure out what p is we have to consider the sums: A r := n +1 X k = r µ j,k ( M ext ) − n X k = r − µ j − ,k ( M ext ) ≥ j ≤ r ≤ n + 1 . Then we have p = max { j ≤ r ≤ n + 1 : − A r is maximal } . By the definition of T j − thisexactly means that the tableaux corresponding to T j − sh j ( M ext ) is given by a a · · · a p − a p +1 · · · a m b b · · · b p − a p b p · · · b m − Hence T j − slides the empty box in row j to the ( j − -th row following the rules of jeu-de-taquin. Now it is clear that sliding the empty box in row ( j − -th to the top means to applythe operators T j − , . . . , T to T j − sh j ( M ext ) . Repeating the above steps yields the claim. (cid:3) Example.
We consider the same situation as in Example 5.2. Then M corresponds to theYoung tableaux ϕ ( M ) = 1 2 43 4 54 6 6 −→ pr( ϕ ( M )) = 1 1 32 4 55 5 6 By the calculations in Example 5.2 we see that pr( ϕ ( M )) coincides with ϕ (pr( M )) . Combinatorial descriptions of Kirillov-Reshetikhin crystals of type A (1) n were providedfor example in [9] and [18], where the affine crystal structure in the latter work is given withoutusing the promotion operator. The affine Kashiwara operators are given by ˜ f := pr n ◦ ˜ f ◦ pr , and ˜ e := pr n ◦ ˜ e ◦ pr . (5.4) Corollary.
Let λ = m̟ j be a rectangular weight of type A n . The following operation gives B g ( λ ) the structure of an affine crystal isomorphic to the Kirillov-Reshetikhin crystal ˜ f Y λ [ M ] = ( Y λ [pr( M ) n ◦ ˜ f ◦ pr( M )] , if k < µ j,n ( M ) + µ , (pr( M ))0 , otherwise . Proof.
We only have to check that ˜ f acts on pr( M ) if and only if k < µ j,n ( M ) + µ , (pr( M )) .The rest follows from Theorem 5 and Theorem 4. Clearly ˜ f acts if and only if dim Hom C Q ( S (1) , pr( M )) < dim Hom C Q ( S (2) , pr( M )) . Obviously in the process of obtaining pr( M ) we subtract µ j,n ( M ) entries in the first column of M (viewed in the array of M as in Example 5.2), i.e. dim Hom C Q ( S (1) , pr( M )) = m − µ j,n ( M ) .Similarly we get a dim Hom C Q ( S (2) , pr( M )) = m − µ j,n ( M ) + w, where w is the number of times we add an entry to the second column in the process of obtaining pr( M ) . Since w = µ j,n ( M ) + µ , (pr( M )) − k we get the desired result. (cid:3) References [1] Ibrahim Assem, Daniel Simson, and Andrzej Skowroński.
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