Monoidal categories associated with strata of flag manifolds

Transcription

1 Monoidal categories associated with strata of flag manifolds 16 Dec 2017 This is a joint work with Masaki Kashiwara, Myungho Kim and Se-jin Oh (arxiv: )

2 Quantum groups 1.1. Quantum groups I := an index set A := (a ij ) i,j I (generalized Cartan matrix) Q := i I Zα i (root lattice) Q + := i I Z 0α i (positive root lattice) P := weight lattice P := dual weight lattice Λ i P + : fundamental weight (i.e., Λ i (h j ) = δ ij ) Definition The quantum group U q (g) associated with A is the associative algebra over Q(q) with 1 generated by e i, f i (i I ) and q h (h P ) satisfying certain defining relations.

3 Quantum groups We consider the unipotent quantum coordinate ring A q (n) = A q (n) β, where A q (n) β := Hom Q(q) (U q + (g) β, Q(q)). β Q Note A q (n) U q (g) as a Q(q)-algebra. For Λ P + and µ, ζ WΛ with µ ζ, D(µ, ζ) := unipotent quantum minor associated with µ, ζ It was known that D(µ, ζ) is a member of the upper global basis (dual canonical basis) of A q (n). Let w, v W with v w. Then D(wΛ, vλ)d(wλ, vλ ) = q (vλ,vλ wλ ) D(w(Λ+Λ ), v(λ+λ )).

4 Quantum groups If n := h i, µ 0, then ε i (D(µ, ζ)) = 0 and e (n) i D(s i µ, ζ) = D(µ, ζ). If h i, µ 0 and s i µ ζ, then ε i (D(µ, ζ)) = h i, µ. If m := h i, ζ 0, then ε i (D(µ, ζ)) = 0 and e (m) i D(µ, s i ζ) = D(µ, ζ). If h i, ζ 0 and µ s i ζ, then ε i (D(µ, ζ)) = h i, ζ.

5 Quantum groups 1.2. Quiver Hecke algebras For α Q + with α = m, let I α = {ν = (ν 1,..., ν m ) I m α ν1 + + α νm = α} For i, j I, we take a homogeneous polynomial { 2(α Q i,j (u, v) = i α j ) 2d i p 2d j q=0 t i,j;p,qu p v q if i j, 0 if i = j, such that Q i,j (u, v) = Q j,i (v, u) and t i,j; aij,0 k 0.

6 Quantum groups Let α Q + with height m. Definition The quiver Hecke algebra R(α) is the associative graded k-algebra generated by e(ν) (ν I α ), x k (1 k m), τ t (1 t m 1) satisfying the following defining relations: e(ν)e(ν ) = δ ν,ν e(ν), e(ν) = 1, x k e(ν) = e(ν)x k, x k x l = x l x k, ν I α τ t e(ν) = e(s t (ν))τ t, τ t τ s = τ s τ t if t s > 1, { τt 2 0 if νt = ν e(ν) = t+1, Q νt,νt+1 (x t, x t+1 )e(ν) if ν t ν t+1, (τ t x k x st(k)τ t )e(ν) = e(ν) if k = t and ν t = ν t+1, e(ν) if k = t + 1 and ν t = ν t+1, 0 otherwise,

7 Quantum groups where (τ t+1 τ t τ t+1 τ t τ t+1 τ t )e(ν) { Qνt,ν = t+1 (x t, x t+1, x t+2 )e(ν) if ν t = ν t+2 ν t+1, 0 otherwise, Q i,j (u, v, w) := Q i,j(u, v) Q i,j (w, v). u w The Z-grading on R(α) is given by deg(e(ν)) = 0, deg(x k e(ν)) = (α νk α νk ), deg(τ t e(ν)) = (α νt α νt+1 ).

8 Quantum groups Convolution product M N := R(β + β )e(β, β ) R(β) R(β ) (M N) Functor E i and F i ( i I ) E i : R(β + α i )-Mod R(β)-Mod F i : R(β)-Mod R(β + α i )-Mod defined by E i (N) = e(α i, β)n and F i (M) = R(α i ) M. We set R(β)-proj := category of f. g. projective graded R(β)-modules R(β)-mod := category of f. d. graded R(β)-modules

9 Quantum groups [Khovanov-Lauda, Rouquier, Kang-Kashiwara, Webster] [R-proj] U Z[q,q 1 ] (g), [R-mod] A q(n) Z[q,q 1 ] [R Λ -proj] V Z[q,q 1 ](Λ), [R Λ -mod] V Z[q,q 1 ](Λ) [Brundan-Kleshchev, Rouquier] In type A, R Λ (n) cyclotomic Hecke algebras. [Brundan, Hu, Kleshchev, Mathas, Ram, Wang,...] In type A, Specht modules, graded cellular bases for R Λ, [Ariki-P.-Speyer] Specht modules for type C, [Brundan, Kleshchev, McNamara, Ram, Tingley, Webster,...] PBW bases theory, Convex orders, Cuspudal modules theory [Kang-Kashiwara-Kim-Oh] Quantum cluster algebras many results Remark Quiver Hecke algebras are a vast generalization of Hecke algebras in the direction of categorification.

10 Quantum groups [Khovanov-Lauda, Rouquier, Kang-Kashiwara, Webster] [R-proj] U Z[q,q 1 ] (g), [R-mod] A q(n) Z[q,q 1 ] [R Λ -proj] V Z[q,q 1 ](Λ), [R Λ -mod] V Z[q,q 1 ](Λ) [Brundan-Kleshchev, Rouquier] In type A, R Λ (n) cyclotomic Hecke algebras. [Brundan, Hu, Kleshchev, Mathas, Ram, Wang,...] In type A, Specht modules, graded cellular bases for R Λ, [Ariki-P.-Speyer] Specht modules for type C, [Brundan, Kleshchev, McNamara, Ram, Tingley, Webster,...] PBW bases theory, Convex orders, Cuspudal modules theory [Kang-Kashiwara-Kim-Oh] Quantum cluster algebras many results Remark Quiver Hecke algebras are a vast generalization of Hecke algebras in the direction of categorification.

11 Quantum groups 1.3. Convex preorders We review results in [Tingley-Webster, MV ploytopes and KLR algebras, Compos. Math. 152 (2016)]. Definition A face is a decomposition of a subset X of V into three disjoint subsets X = A A 0 A + such that (span R 0 A + + span R A 0 ) span R 0 A = {0}, (span R 0 A + span R A 0 ) span R 0 A + = {0}.

12 Quantum groups 1.3. Convex preorders We review results in [Tingley-Webster, MV ploytopes and KLR algebras, Compos. Math. 152 (2016)]. Definition A face is a decomposition of a subset X of V into three disjoint subsets X = A A 0 A + such that (span R 0 A + + span R A 0 ) span R 0 A = {0}, (span R 0 A + span R A 0 ) span R 0 A + = {0}. Definition Let X be a subset of V \ {0}. (i) A convex preorder on X is a total preorder on X such that, for any -equivalence class C, the triple ({x X x C}, C, {x X x C}) is a face. (ii) A convex preorder on X is called a convex order if every -equivalence class is of the form X l for some line l.

13 Quantum groups Note : a convex preorder on X V \ {0}. If α, β, γ X with α + β = γ and α γ, then γ β. If α, β, γ X with α + β = γ and γ β, then α γ.

14 Quantum groups Note : a convex preorder on X V \ {0}. If α, β, γ X with α + β = γ and α γ, then γ β. If α, β, γ X with α + β = γ and γ β, then α γ. We set w := s i1 s i2 s il W, β k := s i1 s ik 1 (α ik ) for k = 1,..., l. Then we have + w = {β 1,..., β l }. Proposition There is a convex preorder on + such that for any γ + w +. β 1 β 2 β l γ Notation w := a convex order which refines the above convex preorder

15 Quantum groups Example type A 3, j = s 2 s 1 s 3 s 2 s 3 s 1 W α 3 j α 1 j α 1 + α 2 + α 3 j α 2 + α 3 j α 1 + α 2 j α 2 i = s 3 s 2 s 1 s 3 s 2 s 3 W α 1 i α 1 + α 2 i α 2 i α 1 + α 2 + α 3 i α 2 + α 3 i α 3

16 Categories C w,v 2. Categories C w,v We assume that A is arbitrary Definition For M R(β)-Mod, we define W(M) := {γ Q + (β Q + ) e(γ, β γ)m 0}, W (M) := {γ Q + (β Q + ) e(β γ, γ)m 0}. Then we have W (M) = β W(M) For R-modules M and N, W(M N) = W(M)+W(N), W (M N) = W (M)+W (N).

17 Categories C w,v We fix a convex order on Z >0 + := {kβ k Z >0, β + }. Definition A simple R(β)-module L is -cuspidal if (a) β Z >0 +, (b) W(L) span R 0 {γ + γ β}.

18 Categories C w,v We fix a convex order on Z >0 + := {kβ k Z >0, β + }. Definition A simple R(β)-module L is -cuspidal if (a) β Z >0 +, (b) W(L) span R 0 {γ + γ β}. It was shown in [Tingley-Webster] that, for a simple R-module L, there exists a unique sequence (L 1, L 2,..., L h ) of -cuspidal modules (up to isomorphisms) such that wt(l 1 ) wt(l 2 ) wt(l h ), L hd(l 1 L 2 L h ). The sequence d(l) := (L 1, L 2,..., L h ) is called the -cuspidal decomposition of L.

19 Categories C w,v Example type A 3, j = s 2 s 1 s 3 s 2 s 3 s 1 α 3 j α 1 j α 1 + α 2 + α 3 j α 2 + α 3 j α 1 + α 2 j α 2 j -cuspidal modules corresponding to + L(3) L(1) L(213) L(23) L(21) L(2) i = s 3 s 2 s 1 s 3 s 2 s 3 α 1 i α 1 + α 2 i α 2 i α 1 + α 2 + α 3 i α 2 + α 3 i α 3 i -cuspidal modules corresponding to + L(1) L(21) L(2) L(321) L(32) L(3) (Here, L(ijk) := f i fj fk 1 R-mod)

20 Categories C w,v Definition w, v W. C w := full subcategory of R-mod whose objects M satisfy W(M) span R 0 ( + w ) C,v := full subcategory of R-mod whose objects N satisfy W (N) span R 0 ( + v + ) C w,v = C w C,v

21 Categories C w,v Definition w, v W. C w := full subcategory of R-mod whose objects M satisfy W(M) span R 0 ( + w ) C,v := full subcategory of R-mod whose objects N satisfy W (N) span R 0 ( + v + ) C w,v = C w C,v Remark C w, C,v and C w,v are stable under taking subquotients, extensions, convolution products and grading shifts. In particular, K 0 (C w ), K 0 (C,v ) and K 0 (C w,v ) are Z[q, q 1 ]-algebras.

22 Categories C w,v w := s i1 s i2 s il w := a convex order on + associated with w L := a simple R-module with d(l) := (L 1, L 2,..., L h ), γ k := wt(l k ) for k = 1,..., h. β l := s i1 s il 1 (α il ) Proposition L C w β l γ k for any k, L C,w γ k β l for any k.

23 Categories C w,v w := s i1 s i2 s il w := a convex order on + associated with w L := a simple R-module with d(l) := (L 1, L 2,..., L h ), γ k := wt(l k ) for k = 1,..., h. β l := s i1 s il 1 (α il ) Proposition L C w β l γ k for any k, L C,w γ k β l for any k. Note K 0 (C w ) = A q (n(w)) Z[q,q 1 ]. i.e. C w = the monoidal category giving a monoidal categorification of A q (n(w)) Z[q,q 1 ] given by Kang-Kashiwara-Kim-Oh.

24 Categories C w,v Example type A 3, w = s 2 s 1 s 3 s 2 s 3, v = s 3 s 2 j = s 2 s 1 s 3 s 2 s 3 s 1, β 5 = s 2 s 1 s 3 s 2 (α 3 ) = α 1 α 3 j α 1 j α 1 + α 2 + α 3 j α 2 + α 3 j α 1 + α 2 j α 2 L C w d(l) = (L(1) t 1, L(213) t 2, L(23) t 3, L(21) t 4, L(2) t 5 )

25 Categories C w,v Example type A 3, w = s 2 s 1 s 3 s 2 s 3, v = s 3 s 2 j = s 2 s 1 s 3 s 2 s 3 s 1, β 5 = s 2 s 1 s 3 s 2 (α 3 ) = α 1 α 3 j α 1 j α 1 + α 2 + α 3 j α 2 + α 3 j α 1 + α 2 j α 2 L C w d(l) = (L(1) t 1, L(213) t 2, L(23) t 3, L(21) t 4, L(2) t 5 ) i = s 3 s 2 s 1 s 3 s 2 s 3 β 2 = s 3 (α 2 ) = α 2 + α 3 α 1 i α 1 + α 2 i α 2 i α 1 + α 2 + α 3 i α 2 + α 3 i α 3 L C,v d(l) = (L(1) t 1, L(21) t 2, L(2) t 3, L(321) t 4 )

26 Categories C w,v Example type A 3, w = s 2 s 1 s 3 s 2 s 3, v = s 3 s 2 j = s 2 s 1 s 3 s 2 s 3 s 1, β 5 = s 2 s 1 s 3 s 2 (α 3 ) = α 1 α 3 j α 1 j α 1 + α 2 + α 3 j α 2 + α 3 j α 1 + α 2 j α 2 L C w d(l) = (L(1) t 1, L(213) t 2, L(23) t 3, L(21) t 4, L(2) t 5 ) i = s 3 s 2 s 1 s 3 s 2 s 3 β 2 = s 3 (α 2 ) = α 2 + α 3 α 1 i α 1 + α 2 i α 2 i α 1 + α 2 + α 3 i α 2 + α 3 i α 3 L C,v d(l) = (L(1) t 1, L(21) t 2, L(2) t 3, L(321) t 4 ) C w,v = C w C,v

27 Categories C w,v We revisit the unipotent quantum coordinate ring A q (n) Z[q,q 1 ]. Definition A w := the Z[q, q 1 ]-linear subspace of A q (n) Z[q,q 1 ] spanned by x A q (n) Z[q,q 1 ] such that e i1 e il x = 0 for any sequence (i 1,..., i l ) I β with β Q + wq + \ {0}, A,v := the Z[q, q 1 ]-linear subspace of A q (n) Z[q,q 1 ] spanned by x A q (n) Z[q,q 1 ] such that e i 1 e i l x = 0 for any sequence (i 1,..., i l ) I β with β Q + vq \ {0}, A w,v := A w A,v A q (n) Z[q,q 1 ]

28 Categories C w,v Remark G : reductive group over C N and N : unipotent radicals of B and B n and n : Lie algebras of N and N w, v W Note that C[N] is isomorphic to the dual of U(n). We set N (w) = N (wnw 1 ) and N(v) := N (vn v 1 ). Then the doubly-invariant algebra N (w) C[N] N(v) = {f f (nxm) = f (x), x N, m N (w), n N(v)} = {f U(n) β f U(n) γ = 0, for all β Q + wq + \ {0}, γ Q + vq \ {0}} Therefore, A w,v = a quantum deformation of N (w) C[N] N(v).

29 Categories C w,v Theorem Under the categorification, for w, v W, we have K 0 (C w ) = A w, K 0 (C,v ) = A,v, K 0 (C w,v ) = A w,v, (Idea of Proof) Key properties Cuspidal decomposition unmixed M C w W(M) + + w

30 Categories C w,v Theorem Under the categorification, for w, v W, we have K 0 (C w ) = A w, K 0 (C,v ) = A,v, K 0 (C w,v ) = A w,v, (Idea of Proof) Key properties Cuspidal decomposition unmixed M C w W(M) + + w Corollary The subspaces A w, A,v and A w,v are subalgebras of A q (n) Z[q,q 1 ].

31 Determinantial modules 3. Determinantial modules We assume that A is arbitrary. We take Λ P + w, v W with w v w = s i1 s ip and v = s j1 s jq Definition Determinantial modules M(wΛ, Λ) := F m 1 i 1 M(wΛ, vλ) := Ẽ max j 1 Note that F mp i p 1, where m k = h k, s ik+1 s ip Λ, Ẽ j max q M(wΛ, Λ). M(wΛ, vλ) is a self-dual simple module. It does not depend on the choice of reduced expressions.

32 Determinantial modules Let Λ, Λ P +, w, v W with w v and λ, µ WΛ with λ µ. Proposition [M(λ, µ)] = unipotent quantum minor D(λ, µ). Thus, it s in the upper global basis (dual canonical basis). M(wΛ, vλ) M(wΛ, vλ ) q (vλ,vλ wλ ) M(w(Λ + Λ ), v(λ + Λ )) Thus, M(λ, µ) is real. Let λ, λ, λ WΛ with λ λ λ. Then there is an epimorphism M(λ, λ ) M(λ, λ ) M(λ, λ ).

33 Determinantial modules Lemma Λ P +, λ, µ WΛ with λ µ u λ := the extremal weight vector of weight λ in V (Λ). If β W(M(λ, µ)), then If γ W (M(λ, µ)), then λ + β wt(u + q (g)u λ ) wt(v (Λ)). µ γ wt(u + q (g)u λ ) wt(v (Λ)). Note U + q (g)u λ = Demazure module associated with λ in V (Λ)

34 Determinantial modules w, v W with v w Fix w = s i1 s i2 s il of w W. For i = 1,..., l, we set w k := s i1 s ik, w k+1 := s ik+1 s il. We also define (i) v k = (v k 1 ) 1 v, { v k 1 s ik if s ik v k < v k, (ii) v k = v k 1 if s ik v k > v k. Here we set w 0 = v 0 = id W. Note w l = w and v l = v l(v) = l(v k 1 ) + l(v k ) v k w k and v k w k

35 Determinantial modules Λ, Λ P +, w, v W with v w, w := s i1 s il. Theorem For k = 0, 1,..., l, M(w k Λ, v k Λ) C w,v (Idea of Proof) properties of wt(demazure module)

36 Determinantial modules Λ, Λ P +, w, v W with v w, w := s i1 s il. Theorem For k = 0, 1,..., l, M(w k Λ, v k Λ) C w,v (Idea of Proof) properties of wt(demazure module) Theorem Suppose that A is symmetric and R is symmetric. For 1 k j l, M(w j Λ, v j Λ) M(w k Λ, v k Λ ) is simple, For 1 k j l, Λ(M(w k Λ, v k Λ), M(w j Λ, v j Λ )) = (w k Λ + v k Λ, v j Λ w j Λ ), where Λ(M, N) = degree of the R-matrix : M N N M. (Idea of Proof) properties of R-matrices

37 Determinantial modules Remark Results on cluster algebras in [Leclerc, Cluster str. on strata of flag varieties, Adv. Math., 2016] G: ADE type, w, v W with v w, w = s i1 s il. R w,v = Schubert cell C w opposite Schubert cell C v C[R w,v ] a certain localization of N (w) C[N] N(v) Frobenious subcategory C w,v of mod(λ) such that Span C {ϕ M M C w,v } = N (w) C[N] N(v) where ϕ M C[N] : cluster character of M subalgebra of N (w) C[N] N(v) having a cluster structure with an initial cluster coming from D(w k (ϖ k+1 ), v k (ϖ k+1 )). Note A quantization of C[R w,v ] was studied by Lenagan-Yakimov in the aspect of quantum cluster algs.

38 Determinantial modules Remark Results on cluster algebras in [Leclerc, Cluster str. on strata of flag varieties, Adv. Math., 2016] G: ADE type, w, v W with v w, w = s i1 s il. R w,v = Schubert cell C w opposite Schubert cell C v C[R w,v ] a certain localization of N (w) C[N] N(v) Frobenious subcategory C w,v of mod(λ) such that Span C {ϕ M M C w,v } = N (w) C[N] N(v) where ϕ M C[N] : cluster character of M subalgebra of N (w) C[N] N(v) having a cluster structure with an initial cluster coming from D(w k (ϖ k+1 ), v k (ϖ k+1 )). Note A quantization of C[R w,v ] was studied by Lenagan-Yakimov in the aspect of quantum cluster algs. Conjecture C w,v gives a monoidal categorification of a quantization of the cluster algebra arising from R w,v.

39 Finite ADE types 4. Finite ADE types We assume that A is of finite ADE type and k is of characteristic 0. It was known that any quiver Hecke algebra is isomorphic to a symmetric quiver Hecke algebra. [Rouquier, Varagnolo-Vasserot] simple R-modules correspond to the upper global basis (dual canonical basis). We define ir-mod := the full subcategory of R-mod whose objects M satisfy ε i (M) = 0 R i -mod := the full subcategory of R-mod whose objects M satisfy ε i (M) = 0

40 Finite ADE types It was proved by S. Kato that there exist reflection functors T i : R i (β)-mod i R(s i β)-mod, Ti : i R(β)-mod R i (s i β)-mod, which are equivalences of categories. The functors T i and Ti are counterparts of the Saito crystal reflections T i : B i ( ) i B( ), b f ϕ i (b) i ẽ ε i (b) i b, i : i B( ) B i ( ), b f ϕ i (b) i ẽ ε i (b) i b. T Note T i (L(b)) L(T i (b)) and T i (L(b )) L(T i (b )) where L(b) is the simple R-module corresponding to b.

41 Finite ADE types w 0 := s i1 s il of the longest w 0 W, w 0 := a convex order corresponding to w 0 L k := w 0 -cuspidal module cor. to β k = s i1 s ik 1 (α ik ) Then, we have L k L(T i 1 T i 2 T i k 1 (f ik )) T i 1 Ti 2 T i k 1 L(i k ) Lemma Let i I, w W, and let M be a simple R-module. (i) Suppose that M C w. Then we have (a) if s i w < w and ε i (M) = 0, then T i(m) C si w, (b) if s i w > w, then T i (M) C si w. (ii) Suppose that M C,w. Then we have (a) if s i w < w, then T i (M) C,si w, (b) if s i w > w and ε i (M) = 0, then T i (M) C,si w.

42 Finite ADE types Theorem Let w, v W with v w, s i w > w and s i v > v. Then the restrictions of the functors T i Csi w,s i v : C s i w,s i v C w,v, T i Cw,v : C w,v Csi w,s i v give equivalences of the categories.

43 Finite ADE types Theorem Let w, v W with v w, s i w > w and s i v > v. Then the restrictions of the functors T i Csi w,s i v : C s i w,s i v C w,v, T i Cw,v : C w,v Csi w,s i v give equivalences of the categories. Corollary Let w, u, v W with w = vu, l(w) = l(v) + l(u). Then there is an equivalence of the categories C w,v C u. Note [Kang-Kashiwara-Kim-Oh] C u gives a monoidal categorification of A q (n(u)) Z[q,q 1 ].

44 Finite ADE types Conjecture C w,v gives a monoidal categorification of A w,v. Note If T i and Ti are monoidal functors, then it implies that the conjecture is true.

45 Finite ADE types Conjecture C w,v gives a monoidal categorification of A w,v. Note If T i and Ti are monoidal functors, then it implies that the conjecture is true. Remark S. Kato extended the reflection functors T i and Ti to symmetric cases and proved the monoidality (arxiv: ). P. McNamara also proved the monoidality independently (arxiv: ). Thus, the conjecture is proved. Thanks!!

46 Finite ADE types Conjecture C w,v gives a monoidal categorification of A w,v. Note If T i and Ti are monoidal functors, then it implies that the conjecture is true. Remark S. Kato extended the reflection functors T i and Ti to symmetric cases and proved the monoidality (arxiv: ). P. McNamara also proved the monoidality independently (arxiv: ). Thus, the conjecture is proved. Thanks!! Further Study (work in progress) We expect that C w,v gives a monoidal categorification of a quantization of the cluster algebra arising from R w,v

47 Finite ADE types THANK YOU