Categorical braid group actions and cactus groups
aa r X i v : . [ m a t h . R T ] J a n CATEGORICAL BRAID GROUP ACTIONSAND CACTUS GROUPS
IVA HALACHEVA, ANTHONY LICATA, IVAN LOSEV, AND ODED YACOBI
Abstract.
Let g be a semisimple simply-laced Lie algebra of finite type. Let C be an abeliancategorical representation of the quantum group U q ( g ) categorifying an integrable representation V . The Artin braid group B of g acts on D b ( C ) by Rickard complexes, providing a triangulatedequivalence Θ w : D b ( C µ ) → D b ( C w ( µ ) ) , where µ is a weight of V and Θ w is a positive lift of thelongest element of the Weyl group.We prove that this equivalence is t-exact up to shift when V is isotypic, generalising a funda-mental result of Chuang and Rouquier in the case g = sl . For general V , we prove that Θ w is aperverse equivalence with respect to a Jordan-H¨older filtration of C .Using these results we construct, from the action of B on V , an action of the cactus group onthe crystal of V . This recovers the cactus group action on V defined via generalised Sch¨utzenbergerinvolutions, and provides a new connection between categorical representation theory and crystalbases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades,and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Spechtmodules. Introduction
In their seminal work, Chuang and Rouquier introduced sl categorifications on abelian categories[15]. Their definition mirrors the notion of an sl representation on a vector space: weight spacesare replaced by weight categories, Chevalley generators acting on them are replaced by Chevalleyfunctors, and Lie algebra relations are replaced by isomorphisms of functors. But, crucially, theseisomorphisms are part of the “higher data” of categorification.The richness of this theory was immediately evident. As a corollary of an sl categorificationon representations of symmetric groups in positive characteristic, Chuang and Rouquier provedBroue’s abelian defect conjecture in that case. The essential tool allowing them to do this is theRickard complex, which is a categorical lifting of the reflection matrix in SL , and provides aderived equivalence between opposite weight categories.Subsequently, Rouquier and Khovanov-Lauda vastly generalised this theory to quantum sym-metrisable Kac-Moody algebras U q ( g ) [25, 26, 35]. Let k be any field. A graded abelian k -linearcategory C endowed with a categorical representation of U q ( g ) possesses a family of Rickard com-plexes Θ i , indexed by the simple roots of g , acting on the derived category D b ( C ).Henceforth let g be a semisimple simply-laced Lie algebra of finite type, W its Weyl group, and B its Artin braid group. Let C be a categorical representation of U q ( g ) as in the previous paragraph.Cautis and Kamnitzer proved that Rickard complexes satisfy the braid relations, as conjectured byRouquier [10]. This defines an action of B on D b ( C ), and is our main object of study.Categorical braid group actions defined via Rickard complexes have many significant applications.For example, in low dimensional topology, the type A link homology theories (in particular Kho-vanov homology) emerge as a byproduct of these types of categorical braid group actions [8, 9, 28].In mirror symmetry, the theory of spherical twists plays an important role, and these all arise fromcategorical sl representations [37].To describe our first theorem, recall that minimal categorifications are certain distinguishedcategorifications of simple representations. On these the Rickard complex Θ i is t-exact up to shift [15, Theorem 6.6]. Notice that this is a result about sl categorifications, and in fact, this is one ofChuang-Rouquier’s key technical results which they use to prove the derived equivalence.We generalise this result to U q ( g ), where we show that the composition of Rickard complexescorresponding to a positive lift of the longest element w ∈ W is t-exact up to shift on any isotypiccategorification. More precisely: Theorem A. [Theorem 6.4 & Corollary 6.7] Let C be a categorical representation of U q ( g ) cate-gorifying an isotypic representation of type λ , where λ is a dominant integral weight. Let µ be anyweight, and let n be the height of µ − w ( λ ) . Then the derived equivalence Θ w µ [ n ] : D b ( C µ ) → D b ( C w ( µ ) ) is t-exact. This theorem is the technical heart of the paper. In order to prove it we introduce a newcombinatorial notion of “marked words” (Section 5). This allows us to use relations between Θ i and Chevalley functors established by Cautis and Kamnitzer to deduce the commutation relationsinvolving Θ w (Proposition 5.9). We then use these relations to prove the theorem by induction on n . Our second theorem describes Θ w on an arbitrary categorical representation of U q ( g ), alsogeneralising a result of Chuang-Rouquier in the case g = sl . Indeed, their study of the Rickardcomplex on an sl categorification led them to define the notion of a “perverse equivalence” [14].Consider an equivalence of triangulated categories F : T → T ′ with t-structures [3]. Supposefurther that T (respectively T ′ ) is filtered by thick triangulated subcategories0 ⊂ T ⊂ · · · ⊂ T r = T , ⊂ T ′ ⊂ · · · ⊂ T ′ r = T ′ , and F is compatible with these filtrations (cf. Section 4.1 for precise definitions). Then, roughlyspeaking, F is a perverse equivalence if on each subquotient F : T i / T i − → T ′ i / T ′ i − is t-exact up toshift.Since their introduction, perverse equivalences have proven useful in various contexts (e.g. rep-resentations of finite groups [13], geometric representation theory and mirror symmetry [1], andalgebraic combinatorics [40]). Our second theorem shows that perverse equivalences are ubiquitousin categorical representation theory: Theorem B. [Theorem 6.8] Let C be a categorical representation of U q ( g ) , and let µ be any weight.The derived equivalence Θ w µ : D b ( C µ ) → D b ( C w ( µ ) ) is a perverse equivalence with respect to aJordan-H¨older filtration of C . Note that if J ⊆ I is a subdiagram, and w J is corresponding longest element, then this theoremimplies that Θ w J µ is a perverse equivalence for any J .Let us explain the filtration arising in Theorem B more precisely. We apply Rouquier’s Jordan-H¨older theory for representations of 2-Kac-Moody algebras to our setting [35]. We thus obtain afiltration of C by Serre subcategories, 0 ⊂ C ⊂ · · · ⊂ C r = C , such that each factor C i is a subrepresentation, and each subquotient C i / C i − categorifies a simplemodule (Theorem 3.5). Then Θ w µ is a perverse equivalence with respect to the filtration whose i -th filtered component consists of complexes in D b ( C µ ) with cohomology supported in C i .We remark that in the case g = sl this gives a more conceptual proof of a result of Chuang-Rouquier [14, Proposition 8.4]. If C is the tensor product categorification of the n -fold tensorproduct of the standard representation of sl n , we recover a theorem of the third author [29].In fact, the third author and Bezrukavnikov formulated a principle that suitable categorical braidgroup representations should have a “crystal limit” [5, Section 9]. As an application of our resultswe can make this precise in the setting of categorical representations of U q ( g ). ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 3
Recall that to an integrable representation V of U q ( g ), Kashiwara associated its crystal basis B V [24], which is closely related to Lusztig’s canonical basis [17]. If V is categorified by C then thereis a natural identification B V = Irr ( C ), the set of isomorphism classes of simple objects in C up toshift (cf. Proposition 7.3) .One of the most important features of the theory is the existence of a tensor product, endowingthe category of crystals with a monoidal structure. The commutator of crystals is controlled by agroup called the cactus group, just as B controls the commutator in the category of representationsof U q ( g ) [19]. There is also an internal cactus group action, mirroring Lusztig’s internal braid groupaction on V . Indeed, there is a cactus group C associated to g (or rather to its Dynkin diagram I ), which can be presented by generators c J indexed by connected subdiagrams J ⊆ I (cf. Section7.1). Then C acts on B V via the so-called Sch¨utzenberger involutions (cf. Theorem 7.5).So, starting with an integrable representation V of the quantum group we obtain: an action of B on V, a g -crystal B V , and an action of C on B V . We schematically picture this situation asfollows: U q ( g ) ñ V B ñ V g -crystal B V C ñ B V ? Naturally one asks: can we “crystallise” the braid group action on V directly to obtain the cactusgroup action on B V ? Our results allow us to answer this in the affirmative.The key point is that a perverse equivalence F : T → T ′ induces a bijection Irr ( T ♥ ) ↔ Irr (( T ′ ) ♥ ),where T ♥ denotes the heart of the t-structure. In the setting of Theorem B, we obtain a bijection ϕ I : Irr ( C ) → Irr ( C ). In fact, if J ⊆ I is a subdiagram and g J ⊆ g I is the corresponding Liesubalgebra, we can regard C as a categorical representation of U q ( g J ) by restriction. By TheoremB we also obtain a bijection ϕ J : Irr ( C ) → Irr ( C ). Theorem C. [Theorem 7.9 & Theorem 7.14] Let C be a categorical representation of U q ( g ) , cat-egorifying the integrable representation V . The assignment c J ϕ J defines an action of C on B V = Irr ( C ) , and this agrees with the combinatorial action arising from Sch¨utzenberger involutions. We thus obtain the sought-after crystalisation process for braid groups: B ñ V ù B ñ D b ( C ) ù C ñ Irr ( C ) , which associates a cactus group set Irr ( C ) to the braid group representation of B on V . The firstappearance of such a crystalisation process is in the work of the third author, where a cactus groupaction on W is constructed [29]. It’s an interesting question to crystallise the braid group actionwithout appealing to categorical representation theory.Finally we remark that perversity of Rickard complexes, and more specifically the t-exactness ofΘ w on isotypic categorifications as in Theorem A, is a fruitful vantage from which to view resultsin algebraic combinatorics. As an example, we show in Section 8 how to use this to easily recovertheorems of Berenstein-Zelevinsky [4] and Stembridge [38] (respectively Rhoades [34] ), namelythat the action of w ∈ S n (respectively the long cycle (1 , , . . . , n ) ∈ S n ) on the Kazhdan-Lusztigbasis of a Specht module of S n is governed by the evacuation operator (respectively the promotionopertor) on standard Young tableaux. We expect that other similar results can be obtained usingthis approach, especially outside of type A. Acknowledgement
We would like to thank Sabin Cautis, Ian Grojnowski, Joel Kamnitzer, Aaron Lauda, AndrewMathas, Peter McNamara, Bregje Pauwels, Raph¨ael Rouquier, and Geordie Williamson for insight-ful discussions. We also thank the Sydney Mathematical Research Institute for supporting visits ofA.L. and I.L. to Sydney, where much of this research took place. A.L. is partially supported by the
IVA HALACHEVA, ANTHONY LICATA, IVAN LOSEV, AND ODED YACOBI
Australian Research Council (ARC) under grants DP180103150 and FT180100069. I.L. is partiallysupported by the NSF under grant DMS-2001139. O.Y. is partially supported by the ARC undergrant DP180102563. 2.
Background on quantum groups
The quantum group.
In this article we work with a simply-laced quantum group U q ( g ) offinite type. Recall that we have an associated Cartan datum and a root datum, which consists of: • A finite set I , • a symmetric bilinear form ( · , · ) on Z I satisfying ( i, i ) = 2 and ( i, j ) ∈ { , − } for all i = j ∈ I , • a free Z -module X , called the weight lattice, and • a choice of simple roots { α i } i ∈ I ⊂ X and simple coroots { h i } i ∈ I ⊂ X ∨ = Hom( X, Z )satisfying h h i , α j i = ( i, j ), where h· , ·i : X ∨ × X → Z is the natural pairing.The quantum group U q ( g ) is the unital, associative, C ( q ) algebra generated by E i , F i , K h , ( i ∈ I, h ∈ X ∨ ) subject to relations:(1) K = 1 and K h K h ′ = K h + h ′ for any h, h ′ ∈ X ∨ ,(2) K h E i = q h h,α i i E i K h for any i ∈ I, h ∈ X ∨ ,(3) K h F i = q −h h,α i i F i K h for any i ∈ I, h ∈ X ∨ ,(4) E i F j − F j E i = δ ij K i − K − i q − q − , where we set K i = K h i , and(5) for all i = j , X a + b = −h h i ,α j i +1 ( − a E ( a ) i E j E ( b ) i = 0 and X a + b = −h h i ,α j i +1 ( − a F ( a ) i F j F ( b ) i = 0 , where E ( a ) i = E ai / [ a ]! , F ( a ) i = F ai / [ a ]!, and [ a ]! = Q ai =1 q i − q − i q − q − .We let a ij = ( i, j ), so that ( a ij ) i,j ∈ I is a Cartan matrix. Given λ ∈ X we abbreviate λ i = h h i , λ i ,and let X + = { λ ∈ X : λ i ≥ i ∈ I } be the set of dominant weights.Let R ⊂ X be the root lattice, defined as the Z -span of the simple roots, and let R + ⊂ R be the N -span of the simple roots. We define the usual partial order ≻ on X by λ ≻ µ if λ − µ ∈ R + . For µ ∈ R let ht ( µ ) denote the height of µ , i.e. ht ( P i a i α i ) = P i a i .When convenient, we also view I as the Dynkin diagram of g , and make reference to subdiagramsor diagram automorphisms of I .2.2. Braid group actions on integrable representations of U q ( g ) . Given a U q ( g )-module V and µ ∈ X we let V µ denote the µ weight space of V . For λ ∈ X + we let L ( λ ) be the irreduciblerepresentation of U q ( g ) of highest weight λ . Let Iso λ ( V ) denotes the λ -isotypic component of V .We say that V is isotypic if there exists λ ∈ X + such that V = Iso λ ( V ).The representation L ( λ ) has a canonical basis, which we denote by B ( λ ) [31]. We let v λ (respec-tively v lowλ ) denote the unique highest weight (respectively lowest weight) element of B ( λ ).Let B = B I denote the braid group of type I , which is generated by θ i ( i ∈ I ) subject to thebraid relations: θ i θ j = θ j θ i , if ( i, j ) = 0 , and θ i θ j θ i = θ j θ i θ j , if ( i, j ) = − . Let W = W I be the Weyl group of type I , which has generators s i ( i ∈ I ) subject to the braidrelations, and in addition the quadratic relation s i = 1. Let w ∈ W be the longest element. Recall ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 5 that W acts on X via s i · λ = λ − h λ, h i i α i . We define τ : I → I by the equality α τ ( i ) = − w ( α i )for any i ∈ I .To J ⊂ I a subdiagram, we associate W J ⊂ W the parabolic subgroup, w J ∈ W J its longestelement, and τ J : I → I the bijection given by α τ J ( i ) = ( − w J ( α i ) if i ∈ J,α i otherwise.For any w ∈ W we can consider its positive lift θ w ∈ B , where θ w = θ i · · · θ i ℓ and w = s i · · · s i ℓ is any reduced decomposition.Let V be an integrable representation of U q ( g ). A fundamental structure of V , discovered byLusztig, is that it admits (several) braid group symmetries, sometimes referred to as the “quantumWeyl group actions”. To recall this, let µ denote the projection onto the µ weight space. For each i ∈ I we define t i : V → V by: t i µ = X − a + b =( µ,α i ) ( − q ) − b E ( a ) i F ( b ) i µ . (2.1)In the notation of [31], t i = t ′′ i, − . (Note that the formula given in [31] is more complicated.This simpler form was initialy observed in [15] in the non-quantum setting, and generalised in [11]to the quantum setting.). The assignment θ i t i defines an action of B on V , and so we canunambiguously write t w for any w ∈ W .3. Categorical representations of U q ( g )3.1. Notation.
Fix a field k of any characteristic. In this paper we will be concerned mostly withabelian k -linear categories C . We will assume throughout each block of C is a finite abelian category[33, Definition 1.8.6].Recall that a category A is graded if it is equipped with an auto-equivalence h i : A → A calledthe “shift functor”. We let h ℓ i be the auto-equivalence obtained by applying the shift functor ℓ times. We denote by Irr ( A ) the set of equivalence classes of simple objects of A up to shift.A functor F : A → A ′ between graded categories is graded if it commutes with the shift functors.We denote by [ A ] Z the Grothendieck group of an abelian caegory A . If A is graded we denote by[ A ] Z [ q,q − ] the quotient of [ A ] Z by the additional relation q [ M ] = [ M h− i ], which is naturally a Z [ q, q − ]-module. Set [ A ] C = [ A ] Z ⊗ Z C , [ A ] C ( q ) = [ A ] Z [ q,q − ] ⊗ Z [ q,q − ] C ( q ) . For an k -algebra A , we let A − mod be the category of finitely generated A -modules, and if A is Z -graded, we let A − mod Z be the category of finitely generated Z -graded modules. These arenaturally k -linear abelian categories.3.2. Definition.
In this section we introduce our main objects of study: categorical representationsof the quantum group. The definition we use is originally due to Rouquier [35], and we use theformulation due to Cautis-Lauda [12].
Definition 3.1. A categorical representation of U q ( g ) consists of the following data: • A family of graded abelian k -linear categories C µ indexed by µ ∈ X . We refer to C µ as a weight category . • Exact graded functors E i µ : C µ → C µ + α i and F i µ : C µ → C µ − α i , for i ∈ I and µ ∈ X . Weassume that E i µ admits a left and right adjoint, and F i µ ∼ = ( E i µ − α i ) R h− µ i − i . We refer to E i , F i as Chevalley functors . IVA HALACHEVA, ANTHONY LICATA, IVAN LOSEV, AND ODED YACOBI
This data is subject to the following conditions:(1) For any µ ∈ X, M ∈ C µ and i ∈ I , E ri ( M ) = F ri ( M ) = 0 for r >>
0. (We suppress the µ swhen it’s clear from context.)(2) We have the following isomorphisms: F i E i µ ∼ = E i F i µ ⊕ [ −h h i ,µ i ] µ , if h h i , µ i ≤ , E i F i µ ∼ = F i E i µ ⊕ [ h h i ,µ i ] µ , if h h i , µ i ≥ , E i F j µ ∼ = F j E i µ . (3) The powers of E i carry an action of the KLR algebra associated to Q , where Q denotes achoice of units ( t ij ) i = j ∈ I in k × . These units satisfy some restrictions which are not relevantfor us. We refer the reader to [12] for further details.Note that by condition (1) for us a categorical representation is by definition integrable. Usuallywe just say that C = L µ C µ is a categorical representation of U q ( g ) (the remaining data is implicit).Given an integrable U q ( g )-module V , we say that C is a categorification of V if C is a categoricalrepresentation of U q ( g ) such that [ C ] C ( q ) ∼ = V as U q ( g )-modules.An additive categorification is a categorical representation on a graded additive k -linearcategory V satisfying the same conditions as Definition 3.1, except the Chevalley functors are ofcourse only required to be additive. We let V i be the idempotent completion of V .Given an abelian category C , we consider the additive category C− proj defined as the full sub-category of projective objects in C . Note that if C is a categorical representation of U q ( g ), then C− proj naturally inherits the structure of an additive categorification.As a consequence of this definition, and in particular condition (3), there exist divided powerfunctors E ( r ) i λ ⊂ E ri λ , F ( r ) i λ ⊂ F ri λ which categorify the usual divided powers on the level ofthe quantum group. Again, we refer the reader to [12] and references therein for further details.We note that their adjoints are related as follows:( E ( r ) i λ ) R ∼ = F ( r ) i λ + rα i h r ( λ i + r ) i , (3.2)( E ( r ) i λ ) L ∼ = F ( r ) i λ + rα i h− r ( λ i + r ) i . (3.3)3.3. Jordan-H¨older series.
A categorification of a simple representation (respectively an isotypicrepresentation) is called a simple categorification (respectively an isotypic categorification ) .There is a distinguished categorification of L ( λ ) called the minimal categorification and denoted L ( λ ) [15, 23, 25, 35, 42]. It is characterized by the fact that L ( λ ) λ ∼ = L ( λ ) w ( λ ) ∼ = k − mod Z . We let k low ∈ L ( λ ) w ( λ ) , k high ∈ L ( λ ) λ be the generators.The Jordan-H¨older Theorem for categorical representations will play an important role in ourwork. To set this up, recall that given finite abelian k -linear categories A , B the Deligne tensorproduct
A ⊗ k B is universal for the functor assigning to every such abelian category C the categoryof bilinear bifunctors A × B → C right exact in both variables [33, Definition 1.11.1]. The tensorproduct is again a finite abelian k -linear category, and there is a bifunctor A×B → A⊗ k B , ( X, Y ) X ⊗ Y . This construction enjoys the following properties:(1) The tensor product is unique up to unique equivalence,(2) for finite k -algebras A, B we have that ( A − mod) ⊗ k ( B − mod) ∼ = ( A ⊗ k B ) − mod, and(3) Hom A⊗ k B ( X ⊗ Y , X ⊗ Y ) ∼ = Hom A ( X , Y ) ⊗ Hom B ( X , Y ).Let C be a categorical representation, and let A be a finite k -linear abelian category. We canendow C ⊗ k A with a structure of a categorical representation, by setting ( C ⊗ k A ) µ = C µ ⊗ k A ,defining Chevalley functors E i µ ⊗ A , etc... If C is a simple categorification then clearly C ⊗ k A isan isotypic categorification. Conversely, we have: ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 7
Lemma 3.4.
Let C be an isotypic categorification of type λ ∈ X + . Then there exists an abeliancategory A such that C ∼ = L ( λ ) ⊗ k A .Proof. Since C is an isotypic categorification, so is C− proj. By Rouquier’s Jordan-H¨older series foradditive categorifications [35, Theorem 5.8], there exists an additive k -linear category M such that C− proj ∼ = ( L ( λ ) − proj ⊗ k M ) i . Note that no filtration appears here since, in the notation of [35], ( C− proj) lw − λ = ( C− proj) − λ .Let { P i } (respectively { Q j } ) be a complete list of the projective indecomposable objects of L ( λ )(respectively M ). Let P = L i,j P i ⊗ Q j , and let B = End C ( P ) op . By Morita theory, B − mod ∼ = C .On the other hand, B ∼ = End L ( λ ) ( M i P i ) op ⊗ End M ( M j Q j ) op . Since L ( λ ) ∼ = End L ( λ ) ( L i P i ) op − mod we have the desired result with A = End M ( L j Q j ) op − mod. (cid:3) Theorem 3.5.
Let C be a categorical representation of U q ( g ) . Then there exists a filtration bySerre subcategories C ⊂ C ⊂ · · · ⊂ C n = C , (3.6) such that each C i is a subrepresentation of C , and C i / C i − ∼ = L ( λ i ) ⊗ k A i where λ i ∈ X + , A i a k -linear graded abelian category, and C i / C i − is a simple categorification.Proof. Choose a dominant highest weight λ occurring in the isotypic decomposition of [ C ] C ( q ) , whichis minimal among all highest weights occurring in [ C ] C ( q ) . Let L ∈ C λ be a simple highest object,and let C be the Serre subcategory of C generated by objects { F i · · · F i ℓ ( L ) | i j ∈ I, ℓ ≥ } . (3.7) C is clearly a subrepresentation of C . Moreover it categorifies L ( λ ). Indeed, by our choice of λ there cannot be any highest weight objects < λ occuring in C , and by construction the only simpleobject in ( C ) λ is L .Next consider the categorical representation C / C and repeat this construction. This producesa Serre subcategory C ′ ⊂ C / C which is again a simple categorification. Let π : C → C / C be thenatural quotient functor, and define C = π − ( C ′ ). Clearly, we have that C ⊂ C , C is Serre, it isa subrepresentation, and C / C ∼ = C ′ is a simple categorification.Iterating this process produces a filtration of C such that each composition factor is a simplecategorification. By Lemma 3.4 any simple categorification is of the form L ( λ ) ⊗ k A , with A a k -linear abelian category with one simple object (up to isomorphism). (cid:3) The categorical braid group action.
Let C be a categorical representation of U q ( g ) , µ ∈ X and i ∈ I . We define a complex of functors Θ i µ , supported in nonpositive cohomological degrees,where for s ≥ − s component is(Θ i µ ) − s = ( E ( − µ i + s ) i F ( s ) i µ h− s i if µ i ≤ , F ( µ i + s ) i E ( s ) i µ h− s i if µ i ≥ . The differential d s : (Θ i µ ) − s → (Θ i µ ) − s +1 is defined using the counits of the bi-adjunctionsrelating E i and F i (see [7, Section 4] for details). This produces a functor Θ i µ : D b ( C µ ) → D b ( C s i ( µ ) ), which following Chuang and Rouquier we call the Rickard complex .It’s straightforward to verify that the Rickard complex Θ i µ categorifies Lusztig’s braid groupoperators t i µ ([7, Section 2]). On the level of categories we have the following two theorems ofChuang-Rouquier and Cautis-Kamnitzer, which are the fundamental results about Rickard com-plexes. Note that the latter theorem was conjectured in [35, Conjecture 5.19]. IVA HALACHEVA, ANTHONY LICATA, IVAN LOSEV, AND ODED YACOBI
Theorem 3.8. [15, Theorem 6.4]
For any µ, i , Θ i µ : D b ( C µ ) → D b ( C s i ( µ ) ) is an equivalence oftriangulated categories. Theorem 3.9. [10, Theorem 6.3]
The Rickard complexes satisfy the braid relations: Θ i Θ j µ ∼ = Θ j Θ i µ if ( i, j ) = 0 , Θ i Θ j Θ i µ ∼ = Θ j Θ i Θ j µ if ( i, j ) = − , thereby defining an action of B on D b ( C ) . As a consequence of their proof of Theorem 3.8, Chuang and Rouquier show that the inverseof Θ i µ is its right adjoint. We denote this functor by Θ ′ i µ : D b ( C µ ) → D b ( C s i ( µ ) ), so thatΘ i Θ ′ i µ ∼ = Θ ′ i Θ i µ ∼ = µ . As a complex of functors, Θ ′ i µ is supported in nonnegative cohomologicaldegrees, where for s ≥ s component is(Θ ′ i µ ) s = ( E ( s ) i F ( µ i + s ) i µ h s ( − µ i − s + 1) i if µ i ≥ , F ( s ) i E ( − µ i + s ) i µ h s ( − µ i + 2 s + 1) i if µ i ≤ . Perverse equivalences
General definition.
Let T be a triangulated category with t-structure t = ( T ≤ , T ≥ ), andheart T ♥ = T ≤ ∩ T ≥ [3]. Recall that a triangulated functor F : T → S between triangulatedcategories with t -structure is t -exact if F ( T ≤ ) ⊆ S ≤ and F ( T ≥ ) ⊆ S ≥ .Now let S ⊂ T be a thick triangulated subcategory, and consider the quotient functor Q : T → T / S . Following [14], we say that t is compatible with S if t T / S = ( Q ( T ≤ ) , Q ( T ≥ )) is a t-structure on T / S . By [14, Lemmas 3.3 & 3.9], if t is compatible with S then ( T / S ) ♥ = T ♥ / T ♥ ∩ S ,and t S = ( S ∩ T ≤ , S ∩ T ≥ ) is a t-structure on S such that S ♥ = T ♥ ∩ S .Now suppose that T , T ′ are two triangulated categories with t-structures t, t ′ . Suppose furtherthat we have filtrations by thick triangulated subcategories:0 ⊂ T ⊂ T ⊂ · · · ⊂ T r = T , ⊂ T ′ ⊂ T ′ ⊂ · · · ⊂ T ′ r = T ′ , such that for every i , t is compatible with T i and t ′ is compatible with T ′ i . By [14, Lemma 3.11], t T i isalso compatible with T i − , and hence T i / T i − inherits a natural t-structure. Let p : { , . . . , r } → Z .The data ( T • , T ′• , p ) is termed a perversity triple .Although Chuang and Rouquier didn’t formulate perverse equivalences for graded categories, itis straightforward to extend their definitions to this setting. Definition 4.1.
A graded equivalence of graded triangulated categories F : T → T ′ is a (graded)perverse equivalence with respect to ( T • , T ′• , p ) if for every i ,(1) F ( T i ) = T ′ i , and(2) the induced equivalence F [ − p ( i )] : T i / T i − → T ′ i / T ′ i − is t-exact.For brevity, we say F is perverse if it is a graded perverse equivalence with respect to someperversity datum. Since we will be working exclusively in the graded setting, a perverse equivalencefor us will always mean a graded perverse equivalence.A perverse equivalence F : T → T ′ induces a bijection ϕ F : Irr ( T ♥ ) → Irr ( T ′♥ ). Indeed, by (2) F [ − p ( i )] induces a bijection Irr ( T ♥ i ) \ Irr ( T ♥ i − ) → Irr ( T ′♥ i ) \ Irr ( T ′♥ i − ), and these yield ϕ F .Although the construction of ϕ F depends on a choice of perversity triple, the resulting bijectiondoes not when T ♥ , T ′♥ have finitely many simple objects. This follows from the following lemma. Lemma 4.2 ([29], Lemma 2.4) . Suppose that T ♥ , T ′♥ have finitely many simple objects, and for i = 1 , let F i : T → T ′ be a perverse equivalence with respect to the perversity datum ( T i, • , T ′ i, • , p i ) .If the induced maps [ F ] , [ F ] : [ T ] Z → [ T ′ ] Z coincide then ϕ F = ϕ F . ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 9
Corollary 4.3.
Suppose that T ♥ , T ′♥ have finitely many simple objects, and F : T → T ′ isperverse. Then ϕ F is independent of the choice of perversity triple.Proof. Suppose F is a graded perverse equivalence with respect to two choices of perversity triples( T i, • , T ′ i, • , p i ), i = 1 ,
2. Now apply the lemma. (cid:3)
The proofs of the following lemmas are straightforward.
Lemma 4.4.
Suppose F : T → T ′ is perverse. Then for any ℓ ∈ Z , F [ ℓ ] is also perverse and ϕ F [ ℓ ] = ϕ F . Lemma 4.5.
Suppose F : T → T ′ is a perverse equivalence with respect to ( T • , T ′• , p ) , and G : T ′ →T ′′ is a perverse equivalence with respect to ( T ′• , T ′′• , q ) . Then G ◦ F is a perverse equivalence withrespect to ( T • , T ′′• , p + q ) , and ϕ G ◦ F = ϕ G ◦ ϕ F . Lemma 4.6.
Let T , T ′ be triangulated with t-structures t, t ′ , and let S ⊂ T , S ′ ⊂ T ′ be thicktriangulated subcategories such that t is compatible with S and t ′ is compatible with S ′ . Supposefurther that F : T → T ′ is a perverse equivalence with respect to ( T • , T ′• , p ) , and F ( S ) = S ′ .Define S i = S ∩ T i , S ′ i = S ′ ∩ T ′ i and ( T / S ) i = Q ( T i ) , ( T ′ / S ′ ) i = Q ′ ( T ′ i ) . Let G : S → S ′ and H : T / S → T ′ / S ′ be the induced equivalences. Then:(1) G is a perverse equivalence with respect to ( S • , S ′• , p ) , and ϕ G = ϕ F | Irr S ♥ .(2) H is a perverse equivalence with respect to (( T / S ) • , ( T ′ / S ′ ) • , p ) , and ϕ H = ϕ F | Irr T ♥ \ Irr S ♥ . Lemma 4.7 ([29], Lemma 2.4) . Suppose F : T → T ′ is perverse, and G (respectively G ′ ) is anautoequivalence of T (respectively T ′ ) which is t-exact up to shift. Then G ′ ◦ F ◦ G is perverse, and ϕ G ′ ◦ F ◦ G = ϕ G ′ ◦ ϕ F ◦ ϕ G ′ . Note that in [29, Lemma 2.4] is stated for functors which are t-exact. Our formulation forfunctors which are t-exact up to shift follows by Lemma 4.4.4.2.
Derived categories of graded abelian categories.
We now specialise to the case of derivedcategories. We recall that if A is an abelian category, then the bounded derived category D b ( A )has a standard t -structure whose heart is A .Given a B ⊂ A a Serre subcategory, we let D b B ( A ) ⊂ D b ( A ) denote the thick subcategoryconsisting of complexes with cohomology supported in B . It inherits a natural t-structure from thestandard t-structure on D b ( A ) whose heart is B . Moreover, if C ⊂ B is another Serre subcategorythen the t -structure on D b B ( A ) is compatible with D b C ( A ). In particular, the quotient D b B ( A ) /D b C ( A )inherits a natural t-structure whose heart is B / C .For the remainder of this section let A , A ′ be graded abelian categories. In the setting of derivedcategories of graded abelian categories, a perverse equivalence can be packaged as follows. We canencode a perversity triple ( A • , A ′• , p ) using filtrations on the abelian categories: A • and A ′• arefiltrations by shift-invariant Serre subcategories:0 = A − ⊂ A ⊂ A ⊂ . . . ⊂ A r = A , A ′− ⊂ A ′ ⊂ A ′ ⊂ . . . ⊂ A ′ r = A ′ . Then a graded equivalence F : D b ( A ) → D b ( A ′ ) is a perverse with respect to ( A • , A ′• , p ) if conditions(1) and (2) of Definition 4.1 hold for T i = D b A i ( A ) and T ′ i = D b A ′ i ( A ′ )As above, a graded perverse equivalence F : D b ( A ) → D b ( A ′ ) induces a bijection ϕ F : Irr ( A ) → Irr ( A ′ ).The following standard lemma will be useful in the proof of our main result. Lemma 4.8.
Let A , A ′ be abelian categories, and B , B ′ Serre subcategories. Let a ≤ b be integers,and F i : A → A ′ be exact functors for a ≤ i ≤ b . Suppose these functors fit into a complex F = ( F a → F a +1 → · · · → F b ) , defining a functor F : D b ( A ) → D b ( A ′ ) . If F i ( B ) ⊂ B ′ for all a ≤ i ≤ b , then F ( D b B ( A )) ⊆ D b B ′ ( A ′ ) . Some commutation relations
We fix throughout a categorical representation C of U q ( g ). Recall that w ∈ W is the longestword, and let w = s i s i · · · s i r be a reduced expression. We consider the composition of Rickardcomplexes which categorifies the positive lift in B of w :Θ w λ = Θ i · · · Θ i r µ : D b ( C λ ) → D b ( C w ( λ ) ) . In preparation for the proofs our main results in the next section, we prove some commutationrelations between Θ w and the Chevalley functors.5.1. Cautis’ relations.
To begin, we recall some relations of Cautis (building on work with Kam-nitzer [10]). Although they are stated only for type A, their proofs apply to any simply-laced Liealgebra.
Lemma 5.1 (Lemma 4.6, [7]) . For any i ∈ I, λ ∈ X we have the following relations: Θ i E i λ ∼ = F i Θ i λ [1] h λ i i , Θ i F i λ ∼ = E i Θ i λ [1] h− λ i i . Remark 5.2.
The careful reader will notice that actually Cautis proves Lemma 5.1 under certainconditions on λ . For instance, the first relation is only proven in the case when λ i ≤
0. To deducethe general case from this, one can rewrite the relation as E i Θ − i ∼ = Θ − i F i [1] h λ i i . Now recall thatthere is an anti-automorphism r σ on the sl n
7→ − n [27, Section5.6]. This anti-automorphism maps Θ − i to Θ i , and hence applying it to the relation above wededuce the desired relation in the case when λ i ≥ sl E i , E j ] acting on representations of U q ( g ). Given nodes i, j ∈ I such that ( i, j ) = − λ ∈ X , define complexes of functors E ij λ : D b ( C λ ) → D b ( C λ + α i + α j ) , E ij λ = E i E j λ h− i → E j E i λ , F ij λ : D b ( C λ ) → D b ( C λ − α i − α j ) , F ij λ = F i F j λ → F j F i λ h i . In both instances the differential is given by the element T ij arising from the KLR algebra, and theleft term of the complex is in homological degree zero [7]. Lemma 5.3 (Lemma 5.2, [7]) . Let i, j ∈ I, λ ∈ X and suppose ( i, j ) = − . We have the followingisomorphisms: E ij Θ i λ ∼ = ( Θ i E j if λ i > , Θ i E j [1] h− i if λ i ≤ F ij Θ i λ ∼ = ( Θ i F j if λ i ≥ , Θ i F j [ − h i if λ i < λ Θ j E ij ∼ = ( E i Θ j if λ j < , E i Θ j [1] h− i if λ j ≥ λ Θ j F ij ∼ = ( F i Θ j if λ j ≤ , F i Θ j [ − h i if λ j > ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 11
Marked words.
We now introduce a combinatorial set-up which we’ll use to prove Propo-sition 5.9 below. A marked word is a word in the elements of I with one letter marked: a = ( i , i , ..., i ℓ , ..., i n ). From a we can define a functor and an element of W :Φ( a ) = Θ i · · · Θ i ℓ − F i ℓ Θ i ℓ · · · Θ i n λ ,w ( a ) = s i · · · s i n . Note that unlike Φ( a ), w ( a ) forgets the location of the marked letter. We say that a is reduced ,if the corresponding unmarked word is a reduced expression for w ( a ).We will apply braid relations to marked words. Away from the marked letter these operate asusual, and at the marked letter we have:( . . . , k, ℓ, . . . ) ↔ ( . . . , ℓ, k, . . . ) , if ( k, ℓ ) = 0 and, (5.4)( . . . , ℓ, k, ℓ, . . . ) ↔ ( . . . , k, ℓ, k, . . . ) , if ( k, ℓ ) = − . (5.5)For marked words a , b we write a ∼ b if they are related by a sequence of braid relations. Lemma 5.6.
Let a , b be marked words which differ by a single braid relation. Then there exists k ∈ { , ± } such that Φ( a ) ∼ = Φ( b )[ k ] h− k i .Proof. If the relation doesn’t involve the marked letter then Φ( a ) ∼ = Φ( b ) since Rickard complexessatisfy the braid relations [10, Theorem 2.10]. Suppose then that the relation does involve themarked letter. If ( j, ℓ ) = 0 the result follows from the fact that Θ j F ℓ ∼ = F ℓ Θ j . Otherwise ( j, ℓ ) = − µ = s ℓ s j ( λ ) − α j . Applying Lemma 5.3 (twice) we deduce thatΘ ℓ Θ j F ℓ λ ∼ = ( Θ ℓ F jℓ Θ j λ , if λ j ≥ , Θ ℓ F jℓ Θ j λ [1] h− i , if λ j < . ∼ = F j Θ ℓ Θ j λ [ − h i if λ j ≥ , µ ℓ > , F j Θ ℓ Θ j λ [1] h− i if λ j < , µ ℓ ≤ , F j Θ ℓ Θ j λ otherwise.Hence the result follows. (cid:3) Corollary 5.7.
Let a , b be marked words such that a ∼ b . Then there exists an integer k suchthat Φ( a ) ∼ = Φ( b )[ k ] h− k i . Lemma 5.8.
Let a = ( i , ..., i n , ℓ ) and b = ( ℓ ′ , i , ..., i n ) be reduced marked words such that w ( a ) = w ( b ) . Then a ∼ b .Proof. We induct on n . Since w ( a ) = w ( b ) there is a Matsumoto sequence relating the unmarkedwords ( i , ..., i n , ℓ ) and ( ℓ ′ , i , ..., i n ). Consider the same sequence of relations, but now applied tothe marked words. We claim this relates a to b . Without loss of generality, we can assume thefirst step of the sequence involves ℓ . If this step is an application of (5.5) then i n − = ℓ and thesequence looks like: ( i , ..., i n , ℓ ) → ( i , ..., i n , ℓ, i n ) → · · · → ( ℓ ′ , i , ..., i n ) . The induction hypothesis applies to ( i , ..., i n , ℓ, i n ) → · · · → ( ℓ ′ , i , ..., i n ), implying that( i , ..., i n , ℓ, i n ) ∼ b . Hence we conclude that a ∼ b . If the first step of the sequence is an application of (5.4) then asimilar analysis applies. (cid:3) The relation between Θ w and Chevalley functors. We now have the machinery in placeto prove our main relation.
Proposition 5.9.
For any i ∈ I, λ ∈ X we have the following relations: Θ w E i λ ∼ = F τ ( i ) Θ w λ [1] h λ i i , Θ w F i λ ∼ = E τ ( i ) Θ w λ [1] h− λ i i . Proof.
We’ll prove the first relation, the second being entirely analogous. For two functors F , G wewrite F ≡ G if there exist integers ℓ, k such that F ∼ = G [ ℓ ] h k i .We first show that Θ w E i ≡ F τ ( i ) Θ w by induction on the rank of g . The base case, when g = sl ,follows from Lemma 5.1. For the inductive step let J ⊂ I be a strict subdiagram containing i .Recall the bijection τ J : I → I induced by the longest element w J ∈ W J . Let u = w ( w J ) − andlet u = s i · · · s i n be a reduced expression. Define two marked words: a = ( i , . . . , i n , τ J ( i )) , b = ( τ ( i ) , i , . . . , i n ) . Note that w ( a ) = w ( b ). By the inductive hypothesis we have Θ w J E i ≡ F τ J ( i ) Θ w J , and thereforeΘ w E i ≡ Θ u Θ w J E i ≡ Θ u F τ J ( i ) Θ w J ≡ Φ( a )Θ w J . Note that a , b satisfy the hypothesis of Lemma 5.8, so by Lemmas 5.7 and 5.8 we have thatΦ( a ) ≡ Φ( b ), and hence Θ w E i ≡ Φ( b )Θ w J ≡ F τ ( i ) Θ w . We now know there exist integers k, ℓ such that Θ w E i λ ∼ = F τ ( i ) Θ w λ [ ℓ ] h k i , and it remains toshow that ℓ = 1 , k = λ i . Let u = w s i and let u = s i · · · s i n be a reduced expression. Define twomarked words: a = ( i , . . . , i n , i ) , b = ( τ ( i ) , i , . . . , i n ) . Note that a , b satisfy the hypothesis of Lemma 5.8, so by Lemmas 5.7 and 5.8 there exists aninteger m such that Φ( a ) ∼ = Φ( b )[ m ] h− m i . Hence we have thatΘ w E i λ ∼ = Θ u Θ i E i λ ∼ = Θ u F i Θ i λ [1] h λ i i∼ = F τ ( i ) Θ u Θ i λ [ m + 1] h− m + λ i i∼ = F τ ( i ) Θ w λ [ m + 1] h− m + λ i i showing that ℓ + k = 1 + λ i .On the other hand, we can deduce k by inspecting the relation on the level of Grothendieckgroups. Namely, by [22, Lemma 5.4], we have that t w E i λ = − q − λ i F τ ( i ) t w λ , showing that k = λ i . (cid:3) On t-exactness and perversity of Θ w In this section we will state and prove the central results of the paper. We fix throughout acategorical representation C of U q ( g ), and let w = s i s i · · · s i n be a reduced expression. ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 13 w on isotypic categorifications. In this section we prove that Θ w is t-exact on anyisotypic categorification. Fix λ ∈ X + . We write Θ = Θ w , L = L ( λ ) and L = L ( λ ). Lemma 6.1.
Let k ∈ { , . . . , n } and set µ = s i k · · · s i n ( w ( λ )) . The weight space L ( λ ) µ − α ik − iszero.Proof. By [21, Proposition 21.3], it suffices to find u ∈ W such that u ( µ − α i k − ) č w ( λ ). Take u = s i k − and, noting that ( µ, α i k − ) <
0, the result follows. (cid:3)
Recall that v λ and v lowλ are the highest and lowest weight elements of the canonical basis B ( λ ). Lemma 6.2. [22, Comment 5.10]
We have t w ( v lowλ ) = v λ . Proposition 6.3.
The equivalence Θ w ( λ ) : D b ( L w ( λ ) ) → D b ( L λ ) satisfies k low k high , whereboth are considered as complexes concentrated in degree zero. In particular, under the equivalences L λ ∼ = L w ( λ ) ∼ = k − mod Z , Θ w ( λ ) is isomorphic to the identity autofunctor of D b ( k − mod Z ) .Proof. Consider first the case g = sl . On the minimal categorification of highest weight m , wehave that Θ − m ( k low ) = k high h ℓ i for some ℓ by [15, Theorem 6.6]. Since [Θ − m ] = t − m , byLemma 6.2 we conclude that ℓ = 0, and hence Θ − m ( k low ) = k high .For general g , suppose X ∈ L ν is simple and F i X = 0 for some i ∈ I . Consider L as a categoricalrepresentation of sl by restriction to the i -th root subalgebra. Then for some m we have a morphismof categorical sl representations R X : L ( m ) → L , such that R X ( k low ) = X [15, Theorem 5.24].The functor R X is equivariant for the categorical sl action on C determined by E i , F i (in fact it isstrongly equivariant in the sense of [30, Definition 3.1]), and hence commutes with Θ i ν . Thereforewe have that Θ i ν ( X ) ∼ = Θ i ν ( R X ( k low )) ∼ = R X (Θ i ν ( k low )) ∼ = R X ( k high ) ∈ L , and so Θ i ν ( X ) is in homological degree zero. It follows that in the case when X is not necessarilysimple (but still assume that F i X = 0), Θ i ν ( X ) is still in homological degree zero. Indeed, byinduction on the length of a Jordan-H¨older filtration of X one deduces this since L ⊂ D b ( L ) isextension closed.Now we study Θ w ( λ ) applied to k low . For k = 2 , . . . , n , by Lemma 6.1, F i k − (Θ i k · · · Θ i n w ( λ ) ( k low )) = 0 . By the previous paragraph, it follows that Θ i k − · · · Θ i n w ( λ ) ( k low ) is in homological degree zero,and in particular, Θ w ( λ ) ( k low ) is supported in homological degree zero. Since in addition[Θ w ( λ ) ] = t w w ( λ ) , by Lemma 6.2 we conclude that Θ w ( λ ) ( k low ) ∼ = k high . (cid:3) Theorem 6.4.
Let λ ∈ X + and set L = L ( λ ) . For any µ ∈ X , Θ µ [ − n ] : D b ( L µ ) → D b ( L w ( µ ) ) is t-exact, where n = ht ( µ − w ( λ )) .Proof. Consider P = E i · · · E i ℓ ( k low ) ∈ L µ . We will first prove by induction on n that there existsan integer k such that Θ( P )[ − n ] ∼ = F j · · · F j ℓ ( k high ) h k i ∈ L w ( µ ) , (6.5)where j r = τ ( i r ). The base case when n = 0 follows by Proposition 6.3. For the inductive step write P = E i ( Q ).Note that Q ∈ L µ − α i . By Proposition 5.9 we have thatΘ µ ( P )[ − n ] = Θ E i µ − α i ( Q )[ − n ] ∼ = F τ ( i ) Θ µ − α i ( Q )[ − n + 1] h ( µ − α i , α i ) i . By hypothesis Θ µ − α i ( Q )[ − n + 1] ∼ = F j · · · F j ℓ ( k high ) h k i ∈ L w ( µ − α i ) for some k , and hence Equation (6.5) follows.Since up to grading shift, any projective indecomposable object in L µ , respectively L w ( µ ) , is asummand of an object of the form E i · · · E i ℓ ( k low ) (respectively F j · · · F j ℓ ( k high )), it follows thatΘ µ [ − n ] takes projective objects in L µ to projective objects in L w ( µ ) . Since Θ µ [ − n ] is a derivedequivalence it follows that it is t-exact. (cid:3) Remark 6.6.
Theorem 6.4 is a generalisation of [15, Theorem 6.6], which covers the sl case. Notethat [15, Theorem 6.6] is crucial in the work of Chuang and Rouquier, since it’s one of the maintechnical results needed to prove that Rickard complexes are invertible. Our proof in the generalcase follows a completely different approach, but it does not give a new proof in the case of sl .Indeed we use [15, Theorem 6.6] explicitly in the proof of Proposition 6.3, and more generally weuse the fact the Θ i is invertible throughout. Corollary 6.7.
Suppose C is an isotypic categorification of type λ , for some λ ∈ X + , and let µ ∈ X . Then Θ µ [ − n ] : D b ( C µ ) → D b ( C w ( µ ) ) is a t-exact equivalence, where n = ht ( µ − w ( λ )) .Proof. By Lemma 3.4, there exists an abelian k -linear category A such that C ∼ = L ( λ ) ⊗ k A ascategorical representations. We have thatΘ µ [ − n ]( L ( λ ) µ ⊗ k A ) ∼ = Θ µ [ − n ]( L ( λ ) µ ) ⊗ k A ∼ = L ( λ ) w ( µ ) ⊗ k A , proving that Θ µ [ − n ] : D b ( C µ ) → D b ( C w ( µ ) ) is a t-exact equivalence. (cid:3) w on general categorical representations. In this section we prove that Θ w is a per-verse equivalence on an arbitrary categorical representation. Fix µ ∈ X such that C µ is nonzero.For ease of notation, set A = C µ and A ′ = C w ( µ ) .By Theorem 3.5 there exists a Jordan-H¨older filtration by Serre subcategories:0 = C ⊂ C ⊂ · · · ⊂ C r = C , where for every i , C i a subrepresentation of C , and C i / C i − is a simple categorification. Define λ i ∈ X + by requiring that [ C i / C i − ] C ( q ) ∼ = L ( λ i ) as representations of U q ( g ).Construct filtrations of A and A ′ by A i = C i ∩ A , A ′ i = C i ∩ A ′ . These are Serre subcategories of A and A ′ respectively. Let p : { , ..., r } → Z be given by p ( i ) = ht ( µ − w ( λ i )). Theorem 6.8. Θ w µ : D b ( A ) → D b ( A ′ ) is a perverse equivalence with respect to ( A • , A ′• , p ) .Proof. Since C i ⊂ C is a categorical subrepresentation, the terms of the functor Θ w µ leave C i invari-ant, and in particular take objects in A i to A ′ i . By Lemma 4.8 this implies that Θ w µ ( D b A i ( A )) ⊆ D b A ′ i ( A ′ ).Now, C i / C i − is a simple categorification (of L ( λ i )). By Corollary 6.7, Θ w µ [ − p ( i )] restricts toan abelian equivalence A i / A i − → A ′ i / A ′ i − , i.e. the functorΘ w µ [ − p ( i )] : D b A i ( A ) /D b A i − ( A ) → D b A ′ i ( A ′ ) /D b A ′ i − ( A ′ )is a t-exact equivalence. This shows that Θ w µ is a perverse equivalence with respect to ( A • , A ′• , p ). (cid:3) ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 15
Remark 6.9.
The sl case of Theorem 6.8 appears as [14, Proposition 8.4], by a different argumentrelying on a technical lemma [14, Lemma 4.12].6.3. Examples.
We now examine three examples. The first two consider minimal categorificationsof the adjoint representation, while the third studies the categorification of the n -fold tensor productof the standard representation of sl n . For ease of presentation, we ignore gradings and considernon-quantum categorical representations. Example 6.10.
Let’s consider the first non-trivial example of Theorem 6.4: the minimal categori-fication of the adjoint representation of sl . We can model this as follows: k − mod R − mod k − mod ind resres ind where R = k [ x ] / ( x ) [15, Example 5.17]. Here k − mod is the ± R − mod isthe zero weight category. The arrows describe the E , F functors (we omit the higher structure).Consider the Rickard complex Θ = Θ : D b ( R − mod) → D b ( R − mod). For M ∈ R − mod wehave: Θ( M ) = R ⊗ k M → M, where the differential is given by the action map, and M is in cohomological degree 0. It’s an exerciseto verify that Θ( M ) is quasi-isomorphic to M ′ [1], where M ′ is the twist of M by the automorphismof R : a + bx a − bx . This shows that Θ[ −
1] is the t-exact equivalence M M ′ . (cid:3) Example 6.11.
More generally, one can consider the minimal categorification of the adjoint rep-resentation of a simple simply-laced Lie algebra g . This was studied by Khovanov and Huerfano in[20], who used zigzag algebras to model this category.For a weight α of the adjoint representation of g , the weight category C α is taken to be k − modas long as α = 0. However, the zero weight category is interesting: C := A − mod, where A is thezigzag algebra associated to the Dynkin diagram of g (cf. [20] for the precise definition).The isomorphism classes of indecomposable projective left A -modules { P i } and the isomorphismclasses of indecomposable projective right A -modules { Q i } are both indexed by i ∈ I . Tensoringwith these modules defines functors E i : C −→ C α i , M Q i ⊗ A M, E i : C − α i −→ C , V P i ⊗ k V. The functors F i are defined analogously and are biadjoint to the E i .The Rickard complex Θ i is given by tensoring with the complex P i ⊗ k Q i → A of ( A, A )-bimodules, where the differential is given by the multiplication in A , and A sits is coho-mological degree zero. These functors are autoequivalences of the derived category D b ( C ).By Theorem 6.4 Θ w [ − n ] is t-exact, where n + 1 is the Coxeter number of g . This auto-equivalence can be explicitly described as follows: the automorphism of the Dynkin diagram τ : I → I induces an automorphism ψ of A . Then Θ w [ − n ] is the abelian autoequivalence of A − moddefined by twisting with ψ . (cid:3) Example 6.12.
Let g = sl n and consider the n -fold tensor power of the standard representation V ⊗ n . Categorifications of V ⊗ n have been well-studied, and a model C for this categorical represen-tation can be constructed using the BGG category O of g [32, 39]. In this model the principal block O ⊂ O appears as the zero weight category of C , and the Rickard complexes acting on D b ( O ) arethe well known shuffling functors. By Theorem 6.8 Θ = Θ w : D b ( O ) → D b ( O ) is a perverse equivalence. In fact, this recoversthe type A case of a theorem of the third named author [29], using completely different methods(in [29] the perversity of Θ is proved using the theory of W-algebras). Let us examine the filtrationof O arising from our perspective.Recall that the simple objects in O are the irreducible highest weight representations L ( w ) , w ∈ S n , where L ( w ) has highest weight wρ − ρ ( ρ is the half-sum of positive roots of sl n ).We view a partition λ of n simultaneously as a dominant integral weight for sl n , and as an indexfor the irreducible Specht module S ( λ ) of the symmetric group S n . Let SYT( λ ) denote the set ofstandard Young tableau of shape λ , and let d ( λ ) = | SYT( λ ) | . Recall that d ( λ ) = dim S ( λ ).Let λ , ..., λ r be the partitions of n ordered so that if λ i < λ j in the dominance order, then j < i . Set N = P ri =1 d ( λ i ). There is a Jordan-H¨older filtration on C , 0 ⊂ C ⊂ · · · ⊂ C N = C , where each C i / C i − is a simple categorification, and is of type λ if 0 < i ≤ d ( λ ), of type λ if d ( λ ) < i ≤ d ( λ ) + d ( λ ), and so on. Then Θ is a perverse equivalence with respect to the filtration O ,i = O ∩ C i and the perversity function p ( i ) = ht ( λ ℓ ), for P ℓ − j =1 d ( λ j ) < i ≤ P ℓj =1 d ( λ j ).We can interpret the filtration of O using the Robinson-Schensted correspondence [36]: RS : S n → r G j =1 SYT( λ j ) × SYT( λ j ) . Let Q , ..., Q N be an ordering of the standard Young tableau of size n , so that the first d ( λ ) areof shape λ , the next d ( λ ) are of shape λ , and so on. Then O ,i is the Serre subcategory of O generated by simples L ( w ) such that RS ( w ) = ( P, Q ) , where Q ∈ { Q , ..., Q i } . (cid:3) Crystalising the braid group action
Already in the work of Chuang and Rouquier, a close connection is established between categori-cal representation theory and the theory of crystals (although it is not phrased in this language, cf.Proposition 7.3 below). In this section we describe a new component of this theory. More precisely,let V be an integrable representation of U q ( g ). Recall that Lusztig has defined a braid group actionon V [31]. In this section we explain how to use our results to “crystalise” this braid group actionto obtain a cactus group action on the crystal of V , recovering the recently discovered action bygeneralised Sch¨utzenberger involutions.7.1. Cactus groups.
The cactus group associated to the Dynkin diagram I has several incarna-tions. Geometrically, it appears as the fundamental group of a space associated to the Cartansubalgebra h of g . Namely, let h reg ⊆ g denote the regular elements of h . The cactus group C = C I is the W -equivariant fundamental group of the real locus of the de Concini-Procesi wonderfulcompactification of h reg (see [16], [18, Section 2] for further details): C = π W ( P ( h reg )( R )) . There is a surjective map C → W , and the kernel of this map is called the pure cactus group. Intype A it is the fundamental group of the Deligne-Mumford compactification of the moduli spaceof real genus 0 curves with n + 1 marked points [19].The cactus group has a presentation using Dynkin diagram combinatorics. For any subdiagram J ⊆ I , recall that τ J : J → J is the diagram automorphism induced by the longest element w J ∈ W J . Definition 7.1.
The cactus group C = C I is generated by c J , where J ⊆ I is a connectedsubdiagram, subject to the following relations:(i) c J = 1 for all J ⊆ I ,(ii) c J c K = c K c J , if J ∩ K = ∅ and there are no edges connecting any j ∈ J to any k ∈ K , and(iii) c J c K = c K c τ K ( J ) if J ⊆ K . ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 17
The surjective map C → W mentioned above is given by c J w J . We are interested in thecactus group in connection to the theory of crystals. We start by recalling the definition of a crystal. Definition 7.2 ([24]) . A g -crystal is a finite set B together with maps: r e i , r f i : B → B ⊔ { } , ε i , ϕ i : B → Z , wt : B → X for all i ∈ I , such that:(1) for any b, b ′ ∈ B , r e i ( b ) = b ′ if and only if b = r f i ( b ′ ) ,(2) for all b ∈ B , if r e i ( b ) ∈ B then wt( r e i ( b )) = wt( b ) + α i , and if r f i ( b ) ∈ B then wt( r f i ( b )) =wt( b ) − α i ,(3) for all b ∈ B , ε i ( b ) = max { n ∈ Z : r e ni ( b ) = 0 } , ϕ i ( b ) = max { n ∈ Z : r f ni ( b ) = 0 } ,(4) for all b ∈ B , ϕ i ( b ) − ε i ( b ) = h wt( b ) , h i i .Any U q ( g )-representation V has a corresponding g -crystal B = B V , whose elements correspondto a basis of V and the maps r e i , r f i are related to the raising and lowering Chevalley operators. Inparticular, for an integral dominant weight λ , the canonical basis B ( λ ) of L ( λ ) carries a naturalcrystal structure [17].The crystal of V naturally arises via categorical representation theory. Namely, as we describe inthe next proposition, if C is a categorification of V , then Irr ( C ) carries a crystal structure isomorphicto B V . This follows from [15, Proposition 5.20], and is explained in detail in [6]. Proposition 7.3. ( [6, Theorem 4.31] ) The set Irr ( C ) together with: • Kashiwara operators defined as r E i ( X ) = soc E i ( X ) , r F i ( X ) = soc F i ( X ) for X ∈ Irr ( C ) , • wt ( X ) = µ for X ∈ C µ , and • ε ( X ) = max { n | E ni ( X ) = 0 } , and ϕ ( X ) = max { n | F ni ( X ) = 0 } ,is a g -crystal isomorphic to the crystal B = B V . We can now describe the relation between the cactus group and crystals. A g -crystal B is called normal if it is isomorphic to a disjoint union ⊔ λ B ( λ ) for some collection of highest weights λ . Thecategory of normal g -crystals has the structure of a coboundary category analogous to the braidedtensor category structure on U q ( g )-representations. It is realized through an “external” cactusgroup action of C A n − on n -tensor products of g -crystals, described by Henriques and Kamnitzer[19, Theorems 6,7].We are interested in the “internal” cactus group action of C on any g -crystal B . Both the internaland external actions rely on the following combinatorially defined maps, which are generalisationsof the partial Sch¨utzenberger involutions in type A . Definition 7.4.
The generalised Sch¨utzenberger involution ξ on B = ⊔ λ B ( λ ) is the set map whichacts as ξ λ on each irreducible component B ( λ ), where for all b ∈ B ( λ ) and i ∈ I :(1) wt( ξ λ ( b )) = w wt( b ),(2) ξ λ r e i ( b ) = r f τ ( i ) ξ λ ( b ),(3) ξ λ r f i ( b ) = r e τ ( i ) ξ λ ( b ).For J ⊆ I , denote by B J the crystal B restricted to the subdiagram J . We denote the corre-sponding Sch¨utzenberger involution by ξ J . Theorem 7.5. ( [18, Theorem 5.19] ) For any g -crystal B , the assignment c J ξ J defines a (set-theoretic) action of C on B . The cactus group action arising from Rickard complexes.
We now explain how cactusgroup actions arise from categorical representations, analogous to the construction of the crystalon
Irr ( C ) in Proposition 7.3. Let C be a categorical representation of U q ( g ). For any weight µ ∈ X , by Theorem 6.8 Θ w µ : D b ( C µ ) → D b ( C w ( µ ) ) is a perverse equivalence, and hence it induces a bijection ϕ I µ : Irr ( C µ ) → Irr ( C w ( µ ) ). By varying µ we obtain a bijection ϕ I : Irr ( C ) → Irr ( C ) . Now let J ⊆ I be a connected subdiagram, and let g J ⊂ g be the corresponding subalgebra. Byrestriction, C is also a categorical representation of U q ( g J ), and hence by the above discussion wealso obtain a bijection ϕ J : Irr ( C ) → Irr ( C ) . We will prove that this family of bijections defines an action of the cactus group
Irr ( C ). First weneed the following technical result. The important point here is just that there exists an integer n such that t w µ = ± q n µ . Lemma 7.6.
Let λ ∈ X + , µ ∈ X , and let µ − w ( λ ) = P ℓr =1 α i r , where i r ∈ I . Set j r = τ ( i r ) anddefine n ( λ, µ ) ∈ Z by n ( λ, µ ) = 2 ¨˝ ℓ X r =1 λ j r + 1 − X ≤ r ≤ s ≤ ℓ a j r j s + ( λ, ρ ) ˛‚ Then on L ( λ ) µ we have t w µ = ( − h λ,ρ ∨ i q n ( λ,µ ) µ . (7.7) Proof.
We will prove the claim by induction on ht ( µ − w ( λ )). For µ = w ( λ ), we have that ℓ = 0so n ( λ, µ ) = ( λ, ρ ). By [22, Equation (7)] t w ( v λ ) = ( − h λ,ρ ∨ i q ( λ,ρ ) v lowλ , which, combined withLemma 6.2, implies that t w ( v lowλ ) = ( − h λ,ρ ∨ i q ( λ,ρ ) v lowλ . Since dim( L ( λ ) w ( λ ) ) = 1, this proves thebase case.Now choose any µ and suppose (7.7) holds for any weight µ ′ such that ht ( µ ′ − w ( λ )) < ht ( µ − w ( λ )). Consider v = E j · · · E j ℓ v lowλ ∈ L ( λ ) µ . First note that by [22, Lemma 5.4] we have that t w E i = q K − i E i t w . (7.8)Setting v ′ = E j · · · E j ℓ v lowλ , by induction we have t w v = q K − j E j t w v ′ = ( q K − j E j )( − h λ,ρ ∨ i q n ( λ,µ − α j ) v ′ = ( − h λ,ρ ∨ i q n ( λ,µ − α j ) − µ,α j ) v One checks easily that n ( λ, µ ) = 2+ n ( λ, µ − α j ) − µ, α j ), proving that t w v = ( − h λ,ρ ∨ i q n ( λ,µ ) v .Since this holds for any vector of the form E j · · · E j ℓ v lowλ in L ( λ ) µ , this completes the inductivestep. (cid:3) Theorem 7.9.
The assignment c J ϕ J defines an action of C on Irr ( C ) .Proof. We need to show that the bijections ϕ J satisfy the cactus group relations. Relation (i) : Without loss of generality we may assume J = I . Fix a weight µ . Our aim is toshow that ϕ I ϕ I µ = Id Irr ( C µ ) . (7.10)Since the filtration of C w ( µ ) which we use in the perversity data of Θ w µ , agrees with thefiltration of C w ( µ ) which we use in the perversity data of Θ w w ( µ ) , by Lemma 4.5 the compositionΘ w Θ w µ is a perverse autoequivalence of D b ( C µ ).The functor [ h λ, ρ ∨ i ] h n ( λ, µ ) i is also a perverse autoequivalence of D b ( C µ ). By Lemma 7.6 thesetwo perverse equivalences induce the same map on Grothendieck groups, and hence by Lemma 4.2they also induce the same bijection. Since the bijection induced by [ h λ, ρ ∨ i ] h n ( λ, µ ) i is the identity,this proves relation (i). ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 19
Relation (ii) : Let
J, K ⊂ I be disjoint subdiagrams with no connecting edges. Our aim is toshow that ϕ J ϕ K µ = ϕ K ϕ J µ . (7.11)We prove a slightly more general statement, namely that for any categorical representation C of U q ( g J × g K ), relation (7.11) holds.Note that Θ J Θ K µ ∼ = Θ K Θ J µ are isomorphic perverse equivalences, so they induce the samebijections. It remains to show that ϕ Θ J Θ K µ = ϕ J ◦ ϕ K µ . (7.12)Consider first the case when C categorifies an simple representation of U q ( g J × g K ). A minimalcategorification of U q ( g J × g K ) is of the form L ( λ ) ⊗ k L ( µ ), where λ is a highest weight for g J and µ is a highest weight for g K . Hence by Lemma 3.4, a simple categorification of U q ( g J × g K ) is ofthe form L ( λ ) ⊗ k L ( µ ) ⊗ k A for some abelian category A .This implies that as a categorical representation of U q ( g J ) (respectively U q ( g K )), C categorifiesan isotypic representation. By Corollary 6.7 Θ J w K ( µ ) and Θ K µ are t-exact up shift on isotypicscategorifications. Hence Equation (7.12) follows by Lemma 4.5.Now consider a Jordan-H¨older filtration (Theorem 3.5):0 = C ⊂ · · · ⊂ C n = C , where for every i , C i is a subrepresentation of C , and C i / C i − is a simple categorification of U q ( g J × g K ). Equation (7.12) now follows by an easy induction on i . Indeed the base case when i = 1 holdsby the paragraph above, and the inductive step by Lemma 4.6. Relation (iii) : We need to show that ϕ J ϕ K µ = ϕ K ϕ τ K ( J ) µ , where J ⊂ K . Again, we mayassume that K = I . Note that we have an isomorphism at the level of functors:Θ − w Θ w J Θ w µ ∼ = Θ w τK ( J )0 µ , which lifts the corresponding relation in B . Since this is an isomorphism of perverse equivalences,they must induce the same bijections by Lemma 4.2.It remains to show that ϕ Θ − w Θ wJ Θ w µ = ϕ − I ◦ ϕ J ◦ ϕ I . (7.13)When C is a simple categorification, by Corollary 6.7 Θ w µ is t-exact (up to shift). Hence Equation(7.13) follows by Lemma 4.7. Now apply the same reasoning as in the proof of Relation (ii) to deduceequation (7.13) in the general case. (cid:3) Reconciling the two cactus group actions.
Let C be a categorical representation of U q ( g ),and consider the g -crystal B = Irr ( C ). There are two actions of the cactus group on B , the firstarising combinatorially via Sch¨utzenberger involutions (Theorem 7.5) and the other categoricallyvia Theorem 7.9. Theorem 7.14.
The two actions of the cactus group on B agree.Proof. It suffices to show that ϕ I = ξ I . First, suppose C is a simple categorification of type λ ∈ X + .In this case B = B ( λ ), and ξ = ξ I is determined by: ξ ( v λ ) = v lowλ , and ξ ( r e i ( v )) = r f τ ( i ) ( ξ ( v )) for all v ∈ B , so we need to show that ϕ I satisfies these properties as well.The first is an immediate consequence of Corollary 6.7. To show that ϕ I satisfies the secondproperty, fix µ ∈ X and i ∈ I . We set n = ht ( µ − w ( λ )) , j = τ ( i ), and write Θ = Θ w . Consider the following diagram: D b ( C µ ) D b ( C w ( µ ) ) D b ( C µ + α i ) D b ( C w ( µ ) − α j ) Θ µ [ − n ] h µ i i E i µ F j w µ ) Θ µ + αi [ − n − (7.15)By Proposition 5.9 this diagram commutes (note that we shifted both sides of the equation by − n − C µ C w ( µ ) C µ + α i C w ( µ ) − α j Θ µ [ − n ] h µ i i E i µ F j w µ ) Θ µ + αi [ − n − (7.16)Let L ∈ C µ be a simple object, and let L ′ = Θ µ ( L )[ − n ] h µ i i . Note that L ′ ∈ C w ( µ ) is simpleand ϕ I ( L ) = L ′ . By the above diagram we have an isomorphismΘ µ + α i ( E i ( L ))[ − n − ∼ = F j ( L ′ ) . Now, r F j ( L ′ ) ⊂ F j ( L ′ ) is the unique simple subobject. On the other hand, since Θ µ + α i [ − n − µ + α i ( r E i ( L ))[ − n − ⊂ Θ µ + α i ( E i ( L ))[ − n −
1] is a simple subobject.Therefore Θ µ + α i ( r E i ( L ))[ − n − ∼ = r F j ( L ′ ) . Since the equivalence class of the left hand side is ϕ I ◦ r e i ( L ), this shows that ϕ I satisfies thesecond defining property, and hence the two cactus group actions agree in the case of a simplecategorification.The general case when C is not necessarily a simple categorification follows easily using theJordan-H¨older filtration (Theorem 3.5) and Lemma 4.6. (cid:3) Type A combinatorics
In this final section, we specialise to type A and discuss the combinatorics of Kazhdan-Lusztigbases and standard Young tableaux from the vantage of perverse equivalences.Set I = A n − = { , . . . , n − } . As in Example 6.12, we view a partition λ ⊢ n simultaneously asa dominant integral highest weight for sl n , and as an index of the Specht module S ( λ ). Recall thatby Schur-Weyl duality, L ( λ ) is isomorphic to S ( λ ). The Specht module has the Kazhdan-Lusztigbasis { C T } indexed by T ∈ SYT( λ ), the standard Young tableau of shape λ .Consider the minimal categorification L ( λ ), and in particular its zero weight category L ( λ ) .For convenience, we forget the grading and work in the non-quantum setting. The simple objects L ( T ) ∈ L ( λ ) are indexed by T ∈ SYT( λ ), and hence a perverse equivalence F : D b ( L ( λ ) ) → D b ( L ( λ ) ) induces a bijection ϕ F : SYT( λ ) → SYT( λ ). For instance, the bijection ϕ I studied inthe previous section specialises to the well-known Sch¨utzenberger involution on standard Youngtableau, otherwise known as the “evacuation operator” e [36].The promotion operator j : SYT( λ ) → SYT( λ ) is another important function in algebraiccombinatorics, which is closely related to the RSK correspondence and related ideas such as jeu detaquin. Letting J = { , . . . , n − } ⊂ I , we can express promotion in terms of the Sch¨utzenbergerinvolution: j = ϕ I ϕ J . We refer the reader to [36] for a detailed exposition. We can now see easilythat promotion also arises from a perverse equivalence: ATEGORICAL BRAID GROUP ACTIONS AND CACTUS GROUPS 21
Proposition 8.1.
Let c n = (1 , , . . . , n ) ∈ S n be the long cycle. Then Θ c n : D b ( L ( λ ) ) → D b ( L ( λ ) ) is a perverse equivalence whose associated bijection is the promotion operator j .Proof. Notice that c n = w w J . Now recall that Θ w is (up to shift) a t-exact autoequivalenceof D b ( L ( λ ) ) (Theorem 6.4). Since Θ w J is a perverse equivalence (Theorem 6.8), its inverse istoo. Therefore Θ c n ∼ = Θ w Θ − w J is also a perverse autoequivalence. By Lemma 4.7 we have that ϕ Θ cn = ϕ I ϕ J , and hence we recover the promotion operator. (cid:3) We can also use this set-up to extract information about the action of S n on the Kazhdan-Lusztigbasis of S ( λ ). This is based on the following elementary lemma: Lemma 8.2.
Let w ∈ S n , λ ⊢ n and suppose Θ w : D b ( L ( λ ) ) → D b ( L ( λ ) ) is t-exact up to shift.Then for any T ∈ SYT( λ ) , w · C T = ± C S , where S = ϕ Θ w ( T ) .Proof. Since Θ w is t-exact up to shift, we have:[Θ w ( L ( T ))] = ± [ L ( S )] . The result now follows since the isomorphism [ L ( λ ) ] C ∼ = S ( λ ) , L ( T ) C T , is S n equivariant, andby [2, Proposition 10], the action of B on L ( λ ) factors through S n . (cid:3) Applying this lemma to Theorem 6.4 we obtain a result of Berenstein-Zelevinsky and Stembridge:
Corollary 8.3. [4, 38]
The action of the longest element on the Kazhdan-Lusztig basis recoversthe Sch¨utzenberger evacuation operator, i.e. for w ∈ S n , λ ⊢ n and T ∈ SYT( λ ) , we have that w · C T = ± C e ( T ) . Similarly we can prove a result of Rhoades regarding the action of the long cycle c n = (1 , , . . . , n )on the Kazhdan-Lusztig basis. Note that in the statement below the significance of λ being rect-angular is that the restriction of S ( λ ) to S n − in this case remains irreducible. Proposition 8.4. (cf. [34, Proposition 3.5] ) Let λ = ( a b ) ⊢ n be a rectangular partition. Then forany T ∈ SYT( λ ) , the action of the long cycle on the Kazhdan-Lusztig basis element C T recoversthe promotion operator: c n · C T = ± C j ( T ) . Proof.
We first prove that Θ w J : D b ( L ( λ ) ) → D b ( L ( λ ) ) is t-exact up to shift. Consider the setof functors which are monomials in the Chevalley functors indexed by J : M = { G j · · · G j s | G ∈ { E , F } , j ℓ ∈ J } . Let C be the abelian category generated by { M ( X ) | X ∈ L ( λ ) , M ∈ M } , that is, C is the categoryclosed under subs and quotients of objects of the form M ( X ). Since the Chevalley functors areexact, it is easy to see that C is a categorical representation of U q ( sl n − ).Let ν ⊢ n − λ by removing a box. We claim that C is acategorification of L ( ν ). Note that this is a categorification of the fact that L ( ν ) ∼ = U q ( sl n − ) · L ( λ ) .Indeed, it’s clear that [ C ] C contains L ( ν ). On the other hand, if C µ = 0 then µ is in the root latticeof sl n − , and L ( ν ) is the unique constituent of L ( λ ) | sl n − whose weights are in the root lattice.Now observe that C = L ( λ ) . Hence, by Corollary 6.7 it follows that Θ w J : D b ( L ( λ ) ) → D b ( L ( λ ) ) is t-exact up to shift. Since Θ w is also t-exact up to shift, it follows that Θ c n is too.The result now follows by Proposition 8.1 and Lemma 8.2. (cid:3) References [1] Mina Aganagic,
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I. Halacheva: Department of Mathematics, Northeastern University, USA
Email address : [email protected] A. Licata: Mathematical Sciences Institute, Australian National University, Australia
Email address : [email protected] I. Losev: Department of Mathematics, Yale University, and School of Mathematics, IAS, USA
Email address : [email protected] O. Yacobi: School of Mathematics and Statistics, University of Sydney, Australia
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