Representations of cyclotomic oriented Brauer categories
aa r X i v : . [ m a t h . R T ] F e b REPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES
MENGMENG GAO, HEBING RUI, LINLIANG SONG
Abstract.
Let A be the locally unital algebra associated to a cyclotomic oriented Brauer categoryover an arbitrary algebraically closed field k of characteristic p ≥
0. The category of locally finitedimensional representations of A is used to give the tensor product categorification (in the generalsense of Losev and Webster) for an integrable lowest weight with an integrable highest weightrepresentation of the same level for the Lie algebra g , where g is a direct sum of sl ∞ (resp., ˆ sl p ) if p = 0 (resp., p > k = C and proved previously in [2]when the level is 1. Introduction
Throughout this paper, k is an arbitrary algebraically closed field of characteristic p ≥
0. Unlessotherwise stated, all algebras, categories and functors are assumed to be k -linear.The aim of this paper is to give a tensor product categorification of any integrable lowest weightwith any integrable highest weight module of the same level for g by using locally finite dimensionalrepresentations of cyclotomic oriented Brauer categories, where g is a direct sum of sl ∞ (resp., b sl p ) if p = 0 (resp., p > ℓ ∈ Z > , u , u ′ ∈ k ℓ and set u = ( u , . . . , u ℓ ) , u ′ = ( u ′ , . . . , u ′ ℓ ) . (1.1)In [3], Brundan etc. introduced the notion of the cyclotomic oriented Brauer category OB ( u , u ′ ).When ℓ = 1, Reynolds proved that A admits a triangular decomposition, where A is the locally unitalalgebra associated to OB ( u , u ′ ) [17]. Brundan further showed that the locally unital algebra associatedto the oriented skein category also admits a triangular decomposition [2]. Assume that B is the locallyunital algebra associated to either an oriented skein category or an oriented Brauer category. Thetriangular decomposition property is the key in the proof of that the category B -lfdmod of locallyfinite dimensional left B -modules is an upper finite fully stratified category in the sense of Brundanand Stroppel [2, 8]. Based on [17], Brundan also constructed certain endofunctors of B -lfdmod andits associated graded categories so as to give a tensor product categorification of an integrable lowestweight with an integrable highest weight module of the same level 1 for g .In [11], a class of categories called upper finite weakly triangular categories was introduced andstudied. Clearly, it is easy to see that any category which admits an upper finite triangular decompo-sition is an upper finite weakly triangular category. Moreover, cyclotomic oriented Brauer categories,cyclotomic Brauer categories [20] and cyclotomic Kauffman categories [10] are upper finite weakly tri-angular categories [11], but it is not clear whether these categories admit triangular decompositions.It is proved in [11] that C -lfdmod is an upper finite fully stratified category if C is the locally unitalalgebra associated to an upper finite weakly triangular category. In particular, A -lfdmod is an upperfinite fully stratified category. Set I = I u [ I u ′ , (1.2)where I u = { u j + n | ≤ j ≤ ℓ, n ∈ Z } and I u ′ = { u ′ j + n | ≤ j ≤ ℓ, n ∈ Z } , and u , . . . , u ℓ and u ′ , . . . , u ′ ℓ are given in (1.1). Let g be the complex Kac-Moody Lie algebra associated to Cartanmatrix ( a i,j ) i,j ∈ I , where a i,j = , if i = j ; − , if i = j ± p = 2; − , if i = j − p = 2;0 , otherwise. (1.3) M. Gao and H. Rui is supported partially by NSFC (grant No. 11971351). L. Song is supported partially by NSFC(grant No. 11501368).
Then g is isomorphic to a direct sum of certain sl ∞ (resp., ˆ sl p ) if p = 0 (resp., p >
0) depending on thenumber of Z -orbits of { u , . . . , u ℓ , u ′ , . . . , u ′ ℓ } in the sense that, for any x, y ∈ { u , . . . , u ℓ , u ′ , . . . , u ′ ℓ } , x and y are in the same Z -orbit if and only if x − y ∈ Z k . Let V ( ω u ) be the integrable highest weight g -module of weight ω u = ℓ X i =1 ω u i , (1.4)where ω u i ’s are fundamental weights and ℓ is said to be the level. Similarly, let ˜ V ( − ω u ′ ) be theintegrable lowest weight g -module of weight − ω u ′ . The following is the main result of this paper. Itwas obtained previously in [2] when ℓ = 1. Theorem 1.1.
Let A be the locally unital k -algebra associated to the cyclotomic oriented Brauercategory OB ( u , u ′ ) . Then A -lfdmod admits the structure of a tensor product categorification of the g -module ˜ V ( − ω u ′ ) ⊗ V ( ω u ) in the general sense of Losev and Webster (e.g., Definition 4.3). If k = C and u S u ′ consists of a unique orbit, then the category A -lfdmod admits the structure ofa tensor product categorification of sl ∞ -module ˜ V ( − ω u ′ ) ⊗ V ( ω u ). Such a result was expected in [2].In order to prove Theorem 1.1, we study representations of cyclotomic oriented Brauer categories.Our results include a classification of simple A -modules, a criterion on the complete reducibility of thecategory of left A -modules, certain partial results on blocks, and certain endofunctors of A -lfdmodand its associated graded categories and so on.We organize this paper as follows. In section 2, we recall some elementary results on cyclotomicoriented Brauer categories in [3,11]. We study representations of cyclotomic oriented Brauer categoriesin section 3 and prove Theorem 1.1 in section 4.2. Cyclotomic oriented Brauer categories
In this section, we recall some of results on cyclotomic oriented Brauer categories. In the usualstring calculus for strict monoidal categories, the composition g ◦ h of two morphisms g and h is givenby vertical stacking and the tensor product g ⊗ h is given by horizontal concatenation. More explicitly, g ◦ h = gh , g ⊗ h = g h . (2.1)2.1. The category OB . The oriented Brauer category OB was introduced in [3]. It is the strictmonoidal category generated by two generating objects ↑ , ↓ and four elementary morphisms: ∅ →↑ ⊗ ↓ , : ↓ ⊗ ↑→ ∅ , : ↑ ⊗ ↑→↑ ⊗ ↑ , : ↑ ⊗ ↓→↓ ⊗ ↑ , (2.2)which satisfy the relations [3, (1.4)-(1.8)] : = , (2.3)= , (2.4)= , (2.5)= , (2.6)is the two-sided inverse to . (2.7)Here, is the identity morphism from to , and is the identity morphism from to . In the following,the inverse of in (2.7) will be denoted by . For any δ ∈ k , let OB ( δ ) be the category obtainedfrom OB by adding the additional relation = δ . (2.8) EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES The category
AOB . The affine oriented Brauer category
AOB was also introduced in [3]. Itis the strict monoidal category generated by two generating objects ↑ , ↓ and the morphism togetherwith four morphisms in (2.2) subject to the relations in (2.3)–(2.7) and one extra relation:= + . (2.9)Thanks to [3, (3.3)-(3.6), (3.8)], = , (2.10)= , (2.11)= , (2.12)= , (2.13)= − , (2.14)where := , := , := , := . (2.15) Lemma 2.1.
There is a k -linear monoidal contravariant functor τ : AOB → AOB , which fixesgenerating objects and both and • and switches (resp., , resp., ) to (resp., ,resp., ). Furthermore, τ = Id.Proof.
The result follows immediately from the defining relations of
AOB above. (cid:3)
Lemma 2.2.
As morphisms in
AOB , we have (1) = − . (2) = − . (3) = + . (4) = + . (5) = . (6) = . (7) = . (8) = .Proof. Easy exercise. (cid:3)
The category OB ( u , u ′ ) . Recall u , u ′ in (1.1) and define f ( u ) = ℓ Y i =1 ( u − u i ) , f ′ ( u ) = ℓ Y i =1 ( u − u ′ i ) . (2.16)Thanks to [3, (1.13)], there are scalars δ i ∈ k , i ∈ N \ { } , such that1 + X i ≥ δ i u − i = f ′ ( u ) /f ( u ) ∈ k [[ u − ]] . (2.17)In [3], Brundan et. al consider the right tensor ideal K of AOB generated by f ( ) together with i − δ i +1 for all i ∈ N , where i is the i th power of . The cyclotomic oriented Brauer category OB ( u , u ′ ) = AOB /K, (2.18)the quotient category of
AOB [3].
MENGMENG GAO, HEBING RUI, LINLIANG SONG
Remark 2.3.
By [3, (1.14)], there are δ ′ j ∈ k , j ∈ N \ { } , such that(1 + X i ≥ δ i u − i )(1 − X j ≥ δ ′ j u − j ) = 1 . It is proved in [3, Remark 1.6] that the previous K is also generated by f ′ ( ) together with i − δ ′ i +1 for all i ∈ N .By (2.17), δ = u − u ′ and OB ( u , u ′ ) is OB ( δ ) if ℓ = 1. We are going to discuss OB ( u , u ′ ) nomatter whether ℓ = 1 or not. In any case, the set of objects in OB ( u , u ′ ) is J = h↑ , ↓i , (2.19)the set of finite sequence of the symbols ↑ , ↓ , including the empty word ∅ . Let A = M a,b ∈ J Hom OB ( u , u ′ ) ( a, b ) . (2.20)Since the contravariant functor τ in Lemma 2.1 stabilizes the right tensor ideal K , it induces ananti-involution τ A on A .For any subspace B ⊆ A and any a, b ∈ J , let B a,b = 1 a B b , where 1 a is the identity morphismfrom a to a . When b = a , B a,b is also denoted by B a . Then A a,b = Hom OB ( u , u ′ ) ( b, a ) and A = M a,b ∈ J A a,b . (2.21)So, A is a locally unital k -algebra and the set { a | a ∈ J } serves as the system of mutually orthogonalidempotents of A . In this paper C -mod is the category of left C -modules M such that M = L a ∈ H a M for any locally unital algebra C = L a,b ∈ H a C b . Let C -lfdmod be the subcategory of C -mod consistsof modules M such that any 1 a M is of finite dimensional. Let C -fdmod (resp., C -pmod) be thecategory of finite dimensional (resp., finitely generated projective) left C -modules.We are going to recall the notion of normally ordered oriented Brauer diagrams in [3]. For any a, b ∈ N , an ( a, b )- Brauer diagram is a diagram on which a + b points are placed on two parallelhorizontal lines, and a points on the lower line and b points on the upper line, and each point joinsprecisely to one other point. If two points at the upper (resp., lower) line join each other, then thisstrand is called a cup (resp., cap). Otherwise, it is called a vertical strand.Any ( a, b )-Brauer diagram can also be considered as a partitioning of the set { , , . . . , a + b } intodisjoint union of pairs. If a + b is odd, then there is no ( a, b )-Brauer diagram. Two ( a, b )-Brauerdiagrams are said to be equivalent if they give the same partitioning of { , , . . . , a + b } into disjointunion of pairs.An oriented Brauer diagram is obtained by adding consistent orientation to each strand in a Brauerdiagram as above. Given any oriented Brauer diagram d , let a (resp., b ) be the element in J which isindicated from the orientation of the endpoints of the lower line (resp., upper line) and d is called oftype a → b . For example, the following diagram is of type ↑↑↓↓→↓↑ :.Two oriented Brauer diagrams of type a → b are said to be equivalent if their underlying Brauerdiagrams are equivalent.A dotted oriented Brauer diagram of type a → b is an oriented Brauer diagram of type a → b suchthat each segment is decorated with some non-negative number of • ’s (called dots), where a segmentmeans a connected component of the diagram obtained when all crossings are deleted. A normallyordered dotted oriented Brauer diagram of type a → b is a dotted oriented Brauer diagram of type a → b such that: • whenever a dot appears on a strand, it is on the outward-pointing boundary, • there are at most ℓ − ℓ = 4. The right one is a normally ordered dotted oriented Brauer diagram of type ↑↑↓↓→↓↑ and the left one is not: EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES , 3 .Two normally ordered dotted oriented Brauer diagrams are said to be equivalent if the underlyingoriented Brauer diagrams are equivalent and there are the same number of dots on their correspondingstrands. Theorem 2.4. [3, Theorem 1.5]
Suppose a, b ∈ J . (1) Two equivalent normally ordered dotted oriented diagrams represent the same morphism in OB ( u , u ′ ) . (2) Hom OB ( u , u ′ ) ( a, b ) has basis given by the set of all equivalence classes of normally ordered dottedoriented Brauer diagrams of type a → b . Thanks to Theorem 2.4, A is locally finite dimensional in the sense thatdim 1 a A b < ∞ for all a, b ∈ J . We are going to explain that A admits an upper finite weakly triangular decomposi-tion [11, Proposition 2.2].Suppose a = a a · · · a h , where a i ∈ {↑ , ↓} , 1 ≤ i ≤ h . Define ℓ ↓ ( a ) = |{ i : a i = ↓}| , ℓ ↑ ( a ) = |{ i : a i = ↑}| , where | D | is the cardinality of a set D . When h = 0, i.e., a is the empty word, ℓ ↓ ( a ) = ℓ ↑ ( a ) = 0. Forany a, b ∈ J , write a ∼ b if ( ℓ ↓ ( a ) , ℓ ↑ ( a )) = ( ℓ ↓ ( b ) , ℓ ↑ ( b )). Then ∼ is an equivalence relation on J . Let I = J/ ∼ . As sets, I ∼ = N . (2.22) Definition 2.5.
For any ( r, s ) , ( r , s ) ∈ I , define ( r, s ) (cid:22) ( r , s ) if r = r + k and s = s + k forsome k ∈ N .Then (cid:22) is a partial order on I . Later on, we also use a , b etc. to denote elements in I . The partialorder (cid:22) on I is upper finite in the sense that { b ∈ I | a (cid:22) b } is finite for all a ∈ I . It induces a partialorder on J such that a ≺ b if a ∈ a , b ∈ b for a , b ∈ I and a ≺ b . If a ∈ I , define1 a = X b ∈ a b and B a , b = L c ∈ a ,d ∈ b c B d for any a , b ∈ I and any subspace B of A . Then B a , b = 1 a B b . Definition 2.6.
For any a, b, c ∈ J and b ∼ c , define(a) Y ( a, b ) : the set of all normally ordered dotted oriented Brauer diagrams of type b → a onwhich there are neither caps nor crossings among vertical strands, and there are no dots onvertical strands,(b) H ( b, c ) : the set of all normally ordered dotted oriented Brauer diagrams of type c → b onwhich there are neither cups nor caps,(c) X ( b, a ) : the set of all normally ordered dotted oriented Brauer diagrams of type a → b onwhich there are neither cups nor crossings among vertical strands, and there are no dots onvertical strands. Lemma 2.7.
Suppose a, b ∈ J . Then X ( b, a ) = ∅ if and only if Y ( a, b ) = ∅ , and X ( b, a ) = Y ( a, b ) = ∅ unless a (cid:22) b . Furthermore, (1) if a = ↑ s ↓ r , then X ( b, a ) = ∅ if and only if b = ↑ m ↓ n and a (cid:22) b ; (2) if a = ↓ r ↑ s , then X ( b, a ) = ∅ if and only if b = ↓ m ↑ n and a (cid:22) b .Proof. Easy exercise. (cid:3)
In [11], we define three subspaces A + , A − and A ◦ (not subalgebras!) of A such that A ± = M b,c ∈ J A ± b,c , A ◦ = M a ∈ I M b,c ∈ a A ◦ b,c , where A − b,c (resp., A + b,c , resp., A ◦ b,c ) is the k -space with basis Y ( b, c ) (resp., X ( b, c ), resp., H ( b, c )). Proposition 2.8. [11, Proposition 2.2]
The data ( I, A − , A ◦ , A + ) satisfies the following conditions: MENGMENG GAO, HEBING RUI, LINLIANG SONG (1) ( I, (cid:22) ) is upper finite, where I is given in Definition 2.5. (2) A − a , b = 0 and A + b , a = 0 unless a (cid:22) b . Furthermore, A − a = A + a = L c ∈ a k c . (3) A − ⊗ K A ◦ ⊗ K A + ∼ = A as k -spaces where K = ⊕ b ∈ J k b . The required isomorphism is given bythe multiplication on A . In other words, (
I, A − , A ◦ , A + ) is an upper finite weakly triangular decomposition in the senseof [11, Definition 2.1]. When ℓ = 1, ( I, A − , A ◦ , A + ) is the same as that for the oriented Brauercategory OB ( δ ) in [17]. In fact, it gives a triangular decomposition in the sense of [8].2.4. Quotient algebras.
For any a ∈ I , define A (cid:22) a = A/A a , A a = 1 a A (cid:22) a a , where A ⋄ a is the two-sided ideal of A generated by I ⋄ a = { b | b ⋄ a } , ⋄ ∈ {≻ , , ⊀ } . For any x ∈ A ,let x be its image in any quotient algebra of A . Lemma 2.9. [11, Lemma 2.6, Proposition 2.10]
Suppose e, c ∈ J and a ∈ I . (1) A e,c has basis { yhx | ( y, h, x ) ∈ S d (cid:23) c,b (cid:23) e,b ∼ d Y ( e, b ) × H ( b, d ) × X ( d, c ) } . (2) 1 e A (cid:22) a c has basis { yhx | ( y, h, x ) ∈ S b,d ∈ b , b (cid:22) a Y ( e, b ) × H ( b, d ) × X ( d, c ) } . Suppose a, b ∈ J and a ∼ b . Following [17], let a σ b be the unique element in H ( a, b ) on which thereare neither crossings among strands of the same orientation (up or down) nor dots on each strand.For example, ↓↓↑ σ ↑↓↓ = . The following result follows immediately from the definition of a σ b . Lemma 2.10.
Suppose a, b, c ∈ J such that a ∼ b ∼ c . Then a σ b ◦ b σ c = a σ c and a σ a = 1 a . For any a, b ∈ a , let A a,b = 1 a A a b . Then A a,a , which will be denoted by A a , is a subalgebra of A a . For any a, b, c, d ∈ a , thanks to Lemma 2.10, there is a k -linear isomorphism A a,b ∼ −→ A c,d , g ( c σ a ) g ( b σ d ) , ∀ g ∈ H ( a, b ) . (2.23)When a = b and c = d , the isomorphism in (2.23) is an algebra isomorphism. Lemma 2.11.
Let A ◦ := L a ∈ I A a . (1) If a = ( r, s ) ∈ I , then there is an algebra isomorphism φ : Mat ( r + sr ) ( A ↓ r ↑ s ) ∼ = A a , where therows and columns of matrices are indexed by b ∈ a . (2) A ◦ -fdmod Morita ∼ L r,s ∈ N A ↓ r ↑ s -fdmod.Proof. Suppose a = ( r, s ). Thanks to (2.23), the required algebra isomorphism φ in (1) satisfies φ ( X a,b ∈ a τ a,b e a,b ) = X a,b ∈ a ( a σ ↓ r ↑ s ) τ a,b ( ↓ r ↑ s σ b ) , ∀ τ a,b ∈ A ↓ r ↑ s , where e a,b ’s are the corresponding matrix units. Now, (2) immediately follows from (1). The requiredfunctor α : A ◦ -fdmod → L r,s ∈ N A ↓ r ↑ s -fdmod and its inverse β are α = M r,s ∈ N ↓ r ↑ s (?) , β = M r,s ∈ N ( A ◦ ↓ r ↑ s ⊗ A ↓ r ↑ s ?) (2.24) (cid:3) For any endofunctor F of A ◦ -fdmod and any endofunctor G of L r,s ∈ N A ↓ r ↑ s -fdmod, set F ∼ G (2.25)if there is a natural isomorphism between F and β G α where α and β are given (2.24). Obviously, F is exact if and only if G is exact. EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES Degenerate cyclotomic Hecke algebras.
Given e = ( e , . . . , e ℓ ), the degenerate cyclotomicHecke algebra H ℓ,n ( e ) is the associative k -algebra generated by L , S , . . . , S n − subject to the rela-tions: S i = 1 , ≤ i ≤ n − S i S j = S j S i , ≤ i < j − ≤ n − S i S i +1 S i = S i +1 S i S i +1 , ≤ i ≤ n − L S i = S i L , ≤ i ≤ n − S L S + S ) L = L ( S L S + S ) , ( L − e )( L − e ) · · · ( L − e ℓ ) = 0 . Let L i = S i − L i − S i − + S i − for 2 ≤ i ≤ n . Then { L i | ≤ i ≤ n } generates a commutativesubalgebra of H ℓ,n ( e ). Thanks to the results on representations of H ℓ,n ( e ), all generalized eigenvaluesof L i , 1 ≤ i ≤ n , are of forms e j + k , 1 ≤ j ≤ ℓ and 1 − n ≤ k ≤ n − Lemma 2.12.
For any r, s ∈ N , A ↑ r ↓ s ∼ = H ℓ,r ( u ) N H ℓ,s ( − u ′ ) and A ↓ s ↑ r ∼ = H ℓ,s ( − u ′ ) N H ℓ,r ( u ) .Proof. Define the k -algebra homomorphism φ : H ℓ,r ( u ) N H ℓ,s ( − u ′ ) → A ↑ r ↓ s such that1 ⊗ g ⊗ g ↓ , h ⊗ h ↑ ⊗ , for any generators 1 ⊗ g and h ⊗ H ℓ,r ( u ) N H ℓ,s ( − u ′ ), where1 ⊗ L ↓ := − · · ·· · · r · · ·· · · r + s , L ↑ ⊗ · · ·· · · r · · ·· · · r + s ,S ↑ i ⊗ · · ·· · · i i + 1 · · ·· · · r · · ·· · · r + s , 1 ⊗ S ↓ j := · · ·· · · r + jr + j + 1 · · ·· · · · · ·· · · r . In order to verify that φ is well-defined, it suffices to verify that the images of generators of H ℓ,r ( u ) N H ℓ,s ( − u ′ ) above satisfy the defining relations for H ℓ,r ( u ) N H ℓ,s ( − u ′ ). This can be verifieddirectly by using (2.5)-(2.6), (2.9) and (2.12)-(2.14) together with the fact that dots can be slide freelyin any dotted oriented Brauer diagram if we consider it as element in A ↑ r ↓ s (see Lemma 2.2). We givean example and leave other details to the reader. By Lemma 2.2(1)-(2),1 ⊗ L ↓ = − · · ·· · · r · · ·· · · r + s . So, by Remark 2.3, ˜ f ′ (1 ⊗ L ↓ ) = 0, where ˜ f ′ ( u ) = f ′ ( − u ), and f ′ ( u ) is given in (2.16).Thanks to Lemma 2.9, H ( ↑ r ↓ s , ↑ r ↓ s ) = { g ⊗ h | ( g, h ) ∈ H ( ↑ r , ↑ r ) × H ( ↓ s , ↓ s ) } is a basis of A ↑ r ↓ s .Since φ sends the well-known basis of H ℓ,r ( u ) N H ℓ,s ( − u ′ ) to H ( ↑ r ↓ s , ↑ r ↓ s ), it is an isomorphism.The last isomorphism can be proved similarly. In fact, the required k -algebra isomorphism is φ ′ : H ℓ,s ( − u ′ ) N H ℓ,r ( u ) → A ↓ s ↑ r such that1 ⊗ g ⊗ g ↑ , h ⊗ h ↓ ⊗ , MENGMENG GAO, HEBING RUI, LINLIANG SONG for any generators 1 ⊗ g and h ⊗ H ℓ,s ( − u ′ ) N H ℓ,r ( u ), where1 ⊗ L ↑ := · · ·· · · s · · ·· · · r + s , L ↓ ⊗ − · · ·· · · s · · ·· · · r + s ,S ↓ i ⊗ · · ·· · · i i + 1 s · · ·· · · · · ·· · · r + s and 1 ⊗ S ↑ j := · · ·· · · s + js + j + 1 · · ·· · · · · ·· · · s . (2.26) (cid:3) The algebra H ℓ,n ( e ) is a cellular algebra in the sense of [12] with certain cellular basis givenin [1, Theorem 6.3]. The corresponding cell modules are denoted by S ( λ ), λ ∈ Λ ℓ,n , where Λ ℓ,n is theset of all ℓ -partitions ( λ (1) , λ (2) , . . . , λ ( ℓ ) ) of n . When all e j ’s are in the same Z -orbit in the sense that e i − e j ∈ Z k for all 1 ≤ i < j ≤ ℓ , the complete set of pairwise inequivalent irreducible modules aregiven by { D ( λ ) | λ ∈ Λ ℓ,n } , where Λ ℓ,n is the set of e -restricted ℓ -partitions in the sense of [14, (3.14)]. Moreover, D ( λ ) appearsas the simple head of S ( λ ) for all λ ∈ Λ ℓ,n (e.g., [14]). If e is a disjoint union of certain orbits, thenthe above result on the classification of simple modules is still available (see [11, Remark 6.2]).2.6. Irreducible A a -modules. Suppose a = ( r, s ) ∈ I , where r, s ∈ N . Recall u , u ′ in (1.1). LetΛ ↑ ℓ,s (resp., Λ ↓ ℓ,r , resp., Λ ↑ ℓ,s , resp., Λ ↓ ℓ,r ) be the set of ℓ -partitions of s (resp., ℓ -partitions of r , resp., u -restricted ℓ -partitions of s , resp., − u ′ -restricted ℓ -partitions of r ). DefineΛ a = Λ ↓ ℓ,r × Λ ↑ ℓ,s , Λ a = Λ ↓ ℓ,r × Λ ↑ ℓ,s . (2.27)Thanks to Lemma 2.12, A ↓ r ↑ s is a cellular algebra with a cellular basis given by those of H ℓ,r ( − u ′ ) N H ℓ,s ( u ). In this case, the cell modules can be considered as S ( λ ↓ ) ⊠ S ( λ ↑ )’s, where λ = ( λ ↓ , λ ↑ ) ∈ Λ a . Furthermore, { D ( λ ↓ ) ⊠ D ( λ ↑ ) | λ ∈ Λ a } gives a complete set of pairwise inequiv-alent irreducible A ↓ r ↑ s -modules. As proved in [11, § P ( λ ↓ ) ⊠ P ( λ ↑ ) be the projective cover (injective hull) of D ( λ ↓ ) ⊠ D ( λ ↑ ). For any N ∈ A ↓ r ↑ s -fdmod, β ( N ) is an A a -module where β is the functor given in (2.24). For any λ = ( λ ↓ , λ ↑ ) ∈ Λ a and µ = ( µ ↓ , µ ↑ ) ∈ Λ a , define S ( λ ) = β ( S ( λ ↓ ) ⊠ S ( λ ↑ )) , D ( µ ) = β ( D ( µ ↓ ) ⊠ D ( µ ↑ )) , P ( µ ) = β ( P ( µ ↓ ) ⊠ P ( µ ↑ )) . Then P ( µ ) is the projective cover ( and injective hull) of the irreducible A a -module D ( µ ).Recall the anti-involution τ A in subsection 2.3. Mimicking arguments in [11, 17], we see that thereis an exact contravariant duality functor ⊛ on A -lfdmod (resp., A a -fdmod) such that for any V ∈ A -lfdmod and W ∈ A a -fdmod, V ⊛ = M a ∈ J Hom k (1 a V, k ) , W ⊛ = Hom k ( W, k ) . (2.28)Thanks to Lemmas 2.11(1), 2.12, it is not difficult to verify that A a is a cellular algebra with a suitablecellular basis such that S ( λ )’s are the corresponding cell modules. By [16, Chapter 2, Exercise 7], D ( λ ) ⊛ ∼ = D ( λ ) (2.29)for all λ ∈ Λ a . Since P ( λ ) is the projective cover and injective hull of D ( λ ), P ( λ ) ⊛ ∼ = P ( λ ) . (2.30) EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES Induction and restriction functors.
Suppose a = ↓ r ↑ s , b = ↓ r +1 ↑ s and c = ↓ r ↑ s +1 , where r, s ∈ N . For 2 ≤ i ≤ r and 2 ≤ j ≤ s , define L ↓ i ⊗ S ↓ i − ⊗ L ↓ i − ⊗ S ↓ i − ⊗
1) + S ↓ i − ⊗ , ⊗ L ↑ j = (1 ⊗ S ↑ j − )(1 ⊗ L ↑ j − )(1 ⊗ S ↑ j − ) + 1 ⊗ S ↑ j − , where S ↓ i − ⊗ L ↓ ⊗
1, 1 ⊗ S ↑ j − and 1 ⊗ L ↑ are given in (2.26). Then { L ↓ i ⊗ , ⊗ L ↑ j | ≤ i ≤ r, ≤ j ≤ s } generates a commutative subalgebra of A ↓ r ↑ s . By Lemma 2.12 and the well-known results on basesof cyclotomic Hecke algebras, we have four exact functors:res r +1 ,sr,s := A b ⊗ A b ? , ind r +1 ,sr,s := A b ⊗ A a ? , res r,s +1 r,s := A c ⊗ A c ? , ind r,s +1 r,s := A c ⊗ A a ? . Suppose φ : M → M is a k -linear map. For any i ∈ k , the i -generalized eigenspace of φ is M i = { m ∈ M | ( φ − i ) n m = 0 , ∀ n ≫ } . (2.31)Then res r +1 ,sr,s = M i ∈ k i -res r +1 ,sr,s , ind r +1 ,sr,s = M i ∈ k i -ind r +1 ,sr,s , res r,s +1 r,s = M i ∈ k i -res r,s +1 r,s ind r,s +1 r,s = M i ∈ k i -ind r,s +1 r,s , (2.32)where i -res r +1 ,sr,s ( M ) and i -ind r +1 ,sr,s ( N ) are the generalized i -eigenspaces of − L ↓ r +1 ⊗ M and N,i -res r,s +1 r,s ( M ′ ) and i -ind r,s +1 r,s ( N ) are the generalized i -eigenspaces of 1 ⊗ L ↑ s +1 on M ′ and N , for any (
M, M ′ , N ) ∈ A b -fdmod × A c -fdmod × A a -fdmod.2.8. Irreducible A -modules. Following [11, (3.1)-(3.2)], there are exact functors∆ = M a ∈ I j a ! , ∇ = M a ∈ I j a ∗ , (2.33)from L a ∈ I A a -fdmod to A -lfdmod where j a ! := A (cid:22) a a ⊗ A a ? and j a ∗ := L b ∈ I Hom A a (1 a A (cid:22) a b , ?). LetΛ = [ b ∈ I Λ b , Λ = [ b ∈ I Λ b . (2.34) Definition 2.13.
For any λ ∈ Λ and µ ∈ Λ, let ˜∆( λ ) = ∆( S ( λ )), ∆( µ ) = ∆( P ( µ )), ∆( µ ) = ∆( D ( µ )), ∇ ( µ ) = ∇ ( P ( µ )) and ∇ ( µ ) = ∇ ( D ( µ )).Following [15], ∆( µ ), ∆( µ ), ∇ ( µ ) and ∇ ( µ ) are called the standard, proper standard, costandardand proper costandard modules, respectively. Corollary 2.14.
Suppose a = ( r, s ) and λ ∈ Λ a . (1) ∆( λ ) has a unique irreducible quotient L ( λ ) such that a L ( λ ) = D ( λ ) as A a -modules. (2) { L ( µ ) | µ ∈ Λ } is a complete set of pairwise inequivalent irreducible A -modules.Proof. This result is a special case of [11, Theorem 3.4(2)-(3)] which is available for any locally unital k -algebra admitting an upper finite weakly triangular decomposition. Now, Proposition 2.8 says that A admits such a decomposition and hence (1)-(2) follow from Lemmas 2.11(2) and 2.12. (cid:3) Stratified categories.
A left A -module V has a finite ∆-flag if it has a finite filtration suchthat its sections are isomorphic to ∆( λ ) for various λ ∈ Λ. Let ρ : Λ → I (2.35)such that ρ ( λ ) = a for any λ ∈ Λ a , where I is given in Definition 2.5. Following [8], define∆ ε ( λ ) = (cid:26) ∆( λ ) , if ε ( ρ ( λ )) = +,∆( λ ) , if ε ( ρ ( λ )) = − , (2.36)for any given sign function ε : I → {±} . Similarly we have the notion of finite ∆ ε -flag. Theorem 2.15. [11, Theorem 3.7]
The A -lfdmod is an upper finite fully stratified category in thesense of [8, Definition 3.36] with respect to the stratification ρ in (2.35) . In other words, for each λ ∈ Λ, there exists a projective object P λ admitting a finite ∆ ε -flag with∆ ε ( λ ) at the top and other sections ∆ ε ( µ ) for µ ∈ Λ with ρ ( µ ) (cid:23) ρ ( λ ). Let A -mod ∆ be the categoryof all left A -modules with a finite ∆-flag. Since ∆( λ ) ∈ A -lfdmod for any λ ∈ Λ, A -mod ∆ is asubcategory of A -lfdmod. For any V ∈ A -mod ∆ , let ( V : ∆( λ )) be the multiplicity of ∆( λ ) in a ∆-flagof V . For any simple A -module L and any A -module V , define[ V : L ] = sup |{ i | V i +1 /V i ∼ = L }| the supremum being taken over all filtrations by submodules 0 = V ⊂ · · · ⊂ V n = V . Corollary 2.16.
For any λ ∈ Λ a , let P ( λ ) be the projective cover of L ( λ ) . (1) P ( λ ) ∈ A -mod ∆ , and ( P ( λ ) : ∆( µ )) = [∆( µ ) : L ( λ )] , which is non-zero for µ = λ only if µ ∈ S c ≻ a Λ c . In particular, ( P ( λ ) : ∆( λ )) = 1 . (2) P ( λ ) has a finite ˜∆ -flag. If ˜∆( µ ) appears as a section, then µ ∈ S b (cid:23) a Λ b . Furthermore, themultiplicity of ˜∆( µ ) in this flag is [ ˜∆( µ ) : L ( λ )] .Proof. Thanks to (2.29) and (2.30) (i.e. [11, Assumption 3.12] holds for A ), (1) is a special caseof [11, Proposition 3.9(2), Lemma 3.13(2)] and (2) is a special case of [11, Corollary 4.4(2)]. (cid:3) Endofunctors and categorical actions
Motivated by [2,11,17], we study certain endofunctors so as to give a categorical action on A -lfdmod.3.1. Endofunctors.
For any ⋄ ∈ {↑ , ↓} , define A ⋄ = L a,b ∈ J a A ⋄ b and ⋄ A = L a,b ∈ J a ( ⋄ A )1 b ,where 1 a ( ⋄ A )1 b = (1 a ⋄ ) A b , a A ⋄ b = 1 a A b ⋄ . (3.1)Then both ⋄ A and A ⋄ are ( A, A )-bimodules such that the right (resp., left) action of A on ⋄ A (resp., A ⋄ ) is given by the usual multiplication, whereas the left (resp., right) action of A on ⋄ A (resp., A ⋄ )is given as follows: a · m = m · · · a , g · a = a · · · g , for all ( m, g, a ) ∈ b ↓ A × A c ↓ × c A b ; a · m = m · · · a , g · a = a · · · g , for all ( m, g, a ) ∈ b ↑ A × A c ↑ × c A b . (3.2)Similarly, we have ( A ◦ , A ◦ )-bimodules A ◦⋄ = L a,b ∈ J a A ◦⋄ b and ⋄ A ◦ = L a,b ∈ J a ( ⋄ A ◦ )1 b such that1 a ( ⋄ A ◦ )1 b = (1 a ⋄ ) A ◦ b , a A ◦⋄ b = 1 a A ◦ b ⋄ , (3.3)where A ◦ is given in Lemma 2.11. Proposition 3.1. As ( A, A ) -bimodules, A ↑ ∼ = ↓ A and A ↓ ∼ = ↑ A .Proof. Thanks to Lemma 2.9(1) and (3.1), there are four k -linear maps φ : ↑ A → A ↓ , ψ : A ↓ → ↑ A , φ ′ : ↓ A → A ↑ and ψ ′ : A ↑ → ↓ A such that φ ( m ) = · · · m , ψ ( g ) = · · · g , φ ′ ( m ′ ) = · · · m ′ , ψ ′ ( g ′ ) = · · · g ′ , for all basis elements m, g, m ′ , g ′ of ↑ A , A ↓ , ↓ A and A ↑ given in Lemma 2.9(1), respectively. By(2.10)-(2.11) (resp., (2.3)-(2.4)), φ − = ψ (resp., φ ′− = ψ ′ ). Finally, it is easy to verify that both φ and φ ′ are ( A, A )-homomorphisms. (cid:3)
Thanks to Proposition 3.1, ↑ A ⊗ A ? ∼ = A ↓ ⊗ A ? and ↓ A ⊗ A ? ∼ = A ↑ ⊗ A ? as functors. Define E = ↑ A ⊗ A ? , F = ↓ A ⊗ A ? . (3.4) Definition 3.2.
Suppose that E and F are two functors in (3.4). Define four natural transformations η : Id A -mod → F E, η ′ : Id A -mod → EF, ε : EF → Id A -mod , ε ′ : F E → Id A -mod such that EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES (1) η and η ′ are induced by the ( A, A )-homomorphisms α : A → ↓ A ⊗ A ( ↑ A ) and α ′ : A → ↑ A ⊗ A ( ↓ A ), respectively, where α ( f ) = f ⊗ a , α ′ ( f ) = f ⊗ a , ∀ f ∈ a A and a ∈ J. (3.5)(2) ε and ε ′ are induced by the ( A, A )-homomorphisms β : ↑ A ⊗ A ( ↓ A ) → A and β ′ : ↓ A ⊗ A ( ↑ A ) → A , respectively, where β ( f ⊗ g ) = g · · · f , β ′ ( f ⊗ g ) = g · · · f (3.6)for all ( f, g, f , g ) ∈ ↑ A a × a ↓ A × ↓ A a × a ↑ A and all a ∈ J . Lemma 3.3. E and F are biadjoint to each other.Proof. Using (2.10)-(2.11) (resp., (2.3)-(2.4)) yields εE ◦ Eη = Id E , F ε ◦ ηF = Id F , ε ′ F ◦ F η ′ = Id F and Eε ′ ◦ η ′ E = Id E . So, E and F are biadjoint to each other. (cid:3) Lemma 3.4.
Suppose a, b ∈ J . (1) (1 a ) ◦ ( m ) = ( m ) ◦ (1 b ) and (1 a ) ◦ ( m ) = ( m ) ◦ (1 b ) for all m ∈ a A b . (2) For any ⋄ ∈ {↑ , ↓} , the linear map x ⋄ is ( A, A ) -homomorphism, where x ⋄ ∈ End k ( ⋄ A ) suchthat x ↑ ( m ) = (1 a ) ◦ m and x ↓ ( m ′ ) = (1 a ) ◦ m ′ for all ( m, m ′ ) ∈ a ↑ A × a ↓ A . (3) For any ⋄ ∈ {↑ , ↓} , both x ⋄ L and x ⋄ R are ( A ◦ , A ◦ ) -homomorphisms, where x ⋄ L ∈ End k ( ⋄ A ◦ ) and x ⋄ R ∈ End k ( A ◦⋄ ) such that x ↑ L ( m ) = (1 a ) ◦ m , x ↓ L ( m ′ ) = (1 a ) ◦ m ′ , x ↑ R ( n ) = n ◦ (1 a ) and x ↓ R ( n ′ ) = n ′ ◦ (1 a ) for any ( m, m ′ , n, n ′ ) ∈ a ↑ A ◦ × a ↓ A ◦ × a ↑ A ◦ × A ◦ a ↓ Proof. (1) is trivial and (2)-(3) follow immediately from (1). (cid:3)
For any i ∈ k , define E i = ( ↑ A ) i ⊗ A ? , F i = ( ↓ A ) i ⊗ A ? , (3.7)where ( ⋄ A ) i is the generalized i -eigenspace of x ⋄ on ⋄ A , ⋄ ∈ {↑ , ↓} . Lemma 3.5. E = L i ∈ k E i and F = L i ∈ k F i .Proof. Thanks to Lemma 2.9(1), 1 a ( ⋄ A )1 b is a finite dimensional k -space for any a, b ∈ J and any ⋄ ∈ {↑ , ↓} . Since x ⋄ preserves any 1 a ( ⋄ A )1 b , the result follows. (cid:3) Lemma 3.6.
For any i ∈ k , E i and F i are biadjoint to each other.Proof. Recall ε, ε ′ , η and η ′ in Definition 3.2. Suppose M and N are two left A -modules. Thanks tothe proof of Lemma 3.3, there are k -linear isomorphismsHom A ( EM, N ) α M,N → Hom A ( M, F N ) , Hom A ( M, F N ) α − M,N → Hom A ( EM, N ) , Hom A ( F M, N ) β M,N → Hom A ( M, EN ) , Hom A ( M, EN ) β − M,N → Hom A ( F M, N ) , (3.8)such that α M,N ( h ) = F ( h ) ◦ η M , α − M,N ( h ′ ) = ε N ◦ E ( h ′ ) ,β M,N ( h ) = E ( h ) ◦ η ′ M , β − M,N ( h ′ ) = ε ′ N ◦ F ( h ′ ) , for all ( h, h ′ , h , h ′ ) ∈ Hom A ( EM, N ) × Hom A ( M, F N ) × Hom A ( F M, N ) × Hom A ( M, EN ).For any ( h, f, m ) ∈ Hom A ( E i M, N ) × a A × M , we have (under the isomorphism A ⊗ A M ∼ = M ) α M,N ( h )( f ⊗ m ) = F ( h ) ◦ η M ( f ⊗ m )= F ( h )(1 a ⊗ f ⊗ m ) = F ( h )( f ⊗ a ⊗ m ) . (3.9)Write f = P j ∈ k f j , where f j ∈ ( ↓ A ) j . Thanks to Lemma 2.2(7), f j ∈ ( ↑ A ) j . Note that h ( E j M ) = 0 if i = j . By (3.9), α M,N ( h )( f ⊗ m ) = f i ⊗ h (1 a ⊗ m ) ∈ F i N. So, α M,N ( h ) ∈ Hom A ( M, F i N ). For any ( h, m ) ∈ Hom A ( M, F i N ) × M , there are finite numbers of f k ⊗ n k ’s ∈ ( ↓ A ) i ⊗ N such that h ( m ) = X k f k ⊗ n k . (3.10)For all ( f, m ) ∈ ↑ A × M , we have (under the isomorphism A ⊗ A M ∼ = M ) α − M,N ( h )( f ⊗ m ) = ε N E ( h )( f ⊗ m ) = ε N ( f ⊗ X k f k ⊗ n k ) = X k f k · · · f ⊗ n k . (3.11)We claim α − M,N ( h )( f ⊗ m ) = 0 if f ∈ ( ↑ A ) j for any j = i . If so, α − M,N ( h ) ∈ Hom A ( E i M, N ) andthe restriction α M,N in (3.8) to Hom A ( E i M, N ) is an isomorphism between Hom A ( E i M, N ) andHom A ( M, F i N ). This proves that ( E i , F i ) is an adjoint pair.In fact, ( x ↑ − j ) t α − M,N ( h )( f ⊗ m ) = α − M,N ( h )(( x ↑ − j ) t f ⊗ n ) = 0for some integer t ≫ f ∈ ( ↑ A ) j and i = j . Similarly, ( x ↓ − i ) s f k = 0 for some integer s ≫ f k ’s are given (3.10). Since there are only finite number of f k , we can find an s ≫ f k such that ( x ↓ − i ) s f k = 0 for all admissible k . Thanks to (3.11) and Lemma 2.2(8),( x ↑ − i ) s α − M,N ( h )( f ⊗ m ) = α − M,N ( h )(( x ↑ − i ) s f ⊗ n ) = X k · · · f ◦ ( x ↓ − i ) s f k ⊗ n k = 0 , forcing α − M,N ( h )( f ⊗ m ) = 0. This proves our claim.Similarly, the restriction of β M,N in (3.8) to Hom A ( F i M, N ) induces an isomorphism betweenHom A ( F i M, N ) and Hom A ( M, E i N ). The only difference is that one has to replace Lemma 2.2(7)–2.2(8) by Lemma 2.2(5)–(6). This proves that ( F i , E i ) is an adjoint pair. (cid:3) Recall x ⋄ L and x ⋄ R in Lemma 3.4(3) for any ⋄ ∈ {↑ , ↓} . For any i ∈ k let ( ⋄ A ◦ ) i (resp., ( A ◦⋄ ) i ) bethe generalized i -eigenspace of x ⋄ L (resp., x ⋄ R ) on ⋄ A ◦ (resp., A ◦⋄ ). Define E ⋄ = ⋄ A ◦ ⊗ A ◦ ? , F ⋄ = A ◦⋄ ⊗ A ◦ ? . (3.12) Lemma 3.7. E ⋄ = L i ∈ k E ⋄ i and F ⋄ = L i ∈ k F ⋄ i , where E ⋄ i = ( ⋄ A ◦ ) i ⊗ A ◦ ? and F i = ( A ◦⋄ ) i ⊗ A ◦ ? .Proof. The result follows from the fact that x ⋄ L (resp., x ⋄ R ) preserves the finite dimensional k -spaces1 a ( ⋄ A ◦ )1 b ’s and 1 a ( A ◦⋄ )1 b ’s. (cid:3) Thanks to Lemma 2.11(2) and (2.25), we have E ↑ ∼ M r,s ∈ N res r,s +1 r,s , E ↓ ∼ M r,s ∈ N ind r +1 ,sr,s , F ↑ ∼ M r,s ∈ N ind r,s +1 r,s , F ↓ ∼ M r,s ∈ N res r +1 ,sr,s . (3.13)So, both E ⋄ and F ⋄ are exact. By Lemma 2.12, (2.26) and (2.32), E ↑ i ∼ M r,s ∈ N i -res r,s +1 r,s , E ↓ i ∼ M r,s ∈ N i -ind r +1 ,sr,s ,F ↑ i ∼ M r,s ∈ N i -ind r,s +1 r,s , F ↓ i ∼ M r,s ∈ N i -res r +1 ,sr,s . (3.14) Lemma 3.8.
There are two short exact sequence of functors from A ◦ -fdmod to A -lfdmod: → ∆ ◦ F ↓ → F ◦ ∆ → ∆ ◦ F ↑ → → ∆ ◦ E ↑ → E ◦ ∆ → ∆ ◦ E ↓ → . (3.15) Proof.
Suppose a = ( s, r ), b = ( s − , r ), c = ( s, r + 1), where r, s are any admissible non-negativeintegers. We claim that there are short exact sequences of ( A, A ◦ )-bimodules0 → A (cid:22) b ⊗ A ◦ b ( ↓ A ◦ )1 a ϕ → ↓ A ⊗ A A (cid:22) a a ψ → A (cid:22) c ⊗ A ◦ c A ◦↑ a → , → A (cid:22) b ⊗ A ◦ b ( ↑ A ◦ )1 a η → ↑ A ⊗ A A (cid:22) a a ε → A (cid:22) c ⊗ A ◦ c A ◦↓ a → , (3.16) EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES where the required morphisms ϕ, ψ, η and ε satisfy: ϕ ( f ⊗ g ) = ( f ) ⊗ g, ψ ( f ⊗ g ) = · · · f ⊗ ( g ) ,η ( h ⊗ k ) = ( h ) ⊗ k, ε ( h ⊗ k ) = · · · h ⊗ ( k ) , for any admissible basis diagrams f, g, f , g , h, k, h , k of A in Lemma 2.9(1). If the claim is true,then (3.15) follows immediately.It is routine to check that ϕ , ψ , ε and η are well-defined ( A, A ◦ )-homomorphisms. We are going toprove the exactness of the first sequence in (3.16). One can verify the second one similarly.Suppose d ∼ c ∈ J and a, b ∈ a . Thanks to Lemma 2.10 and (2.23), there is a commutative diagram0 → d A (cid:22) b ⊗ A ◦ b ( ↓ A ◦ )1 a ϕ a,d −→ d ( ↓ A ⊗ A A (cid:22) a )1 a ψ a,d −→ d A (cid:22) c ⊗ A ◦ c A ◦↑ a → ≀ ↓ τ ≀ ↓ τ ≀ ↓ τ → c A (cid:22) b ⊗ A ◦ b ( ↓ A ◦ )1 b ϕ b,c −→ c ( ↓ A ⊗ A A (cid:22) a )1 b ψ b,c −→ c A (cid:22) c ⊗ A ◦ c A ◦↑ b → , (3.17)where ϕ a,d and ϕ b,c (resp., ψ a,d and ψ b,c ) are restrictions of ϕ (resp., ψ ) in (3.16). Here, the k -linearisomorphisms τ i are defined so that τ ( f ⊗ g ) := c σ d ( f ⊗ g ) a σ b , τ i ( f ⊗ g ) := c σ d ( f ⊗ g ) a σ b , i ∈ { , } , for all admissible basis diagrams f, g of A , where c σ d is given in Lemma 2.10. Therefore, it’s enoughto verify the exactness of the first sequence in (3.17) as k -spaces under the assumptions a = ↑ r ↓ s and d = ↑ m ↓ n , ∀ m, n ∈ N . If so, we immediately obtain the first short exact sequence in (3.16).At moment, we define B ⊗ B = { b ⊗ b | b ∈ B , b ∈ B } for any sets B , B . For any B ⊂ A ,and any quotient of A , we denote by B = { t | t ∈ B } the corresponding subset in the quotient algebra.For convenience, H ( h, e ) will be denoted by H ( e ) if h = e .By Lemma 2.7, three k -spaces in the first sequence of (3.17) are zero if ↑ r ↓ s − (cid:15) d and hence thereis nothing to be proved. Suppose ↑ r ↓ s − (cid:23) d and define H ( ↑ r +1 ↓ s ) = ( h g · · · h · · · | ( g, h , h ) ∈ D ( ↑ r +1 ) × H ( ↑ r ) × H ( ↓ s ) ) , where D ( ↑ r +1 ) = · · ·· · · i jr + 1 · · ·· · · | ≤ j ≤ ℓ − , ≤ i ≤ r + 1 . Thanks to Lemma 2.12 and the general result on the basis of degenerate cyclotomic Hecke algebra, itis easy to check that H ( ↑ r +1 ↓ s ) is a basis of A ↑ r +1 ↓ s . Let H ( ↑ r +1 ↓ s )( σ ) := { g ◦ ↑ r +1 ↓ s σ ↑ r ↓ s ↑ | g ∈ H ( ↑ r +1 ↓ s ) } . By Lemmas 2.7, 2.9 and (2.23), we have(a) 1 d A (cid:22) b ⊗ A ◦ b ( ↓ A ◦ )1 a has basis Y ( ↑ m ↓ n , ↑ r ↓ s − ) ⊗ H ( ↑ r ↓ s ),(b) 1 d ( ↓ A ) ⊗ A A (cid:22) a a has basis Y ( ↑ m ↓ n +1 , ↑ r ↓ s ) ⊗ H ( ↑ r ↓ s ),(c) 1 d A (cid:22) c ⊗ A ◦ c A ◦↑ a has basis Y ( ↑ m ↓ n , ↑ r +1 ↓ s ) ⊗ H ( ↑ r +1 ↓ s )( σ ).Thanks to f = 0 and f = 0 in 1 d A (cid:22) c for any f ∈ Y ( ↑ m ↓ n , ↑ r ↓ s − ), we have ψ a,d ϕ a,d = 0 and ε a,d η a,d = 0. Since { g | g ∈ Y ( ↑ m ↓ n , ↑ r ↓ s − ) } ⊆ Y ( ↑ m ↓ n +1 , ↑ r ↓ s ), ϕ a,d is injective. Define Y ( ↑ m ↓ n +1 , ↑ r ↓ s ) = g · · · · · · f | ( f, g ) ∈ Y ( ↑ m ↓ n , ↑ r +1 ↓ s ) × D ( ↑ r +1 ) . So Y ( ↑ m ↓ n +1 , ↑ r ↓ s ) ⊗ H ( ↑ r ↓ s ) ⊆ d ( ↓ A ) ⊗ A A (cid:22) a a . If f ∈ Y ( ↑ m ↓ n , ↑ r +1 ↓ s ), g ∈ D ( ↑ r +1 ) and h ∈ H ( ↑ r ↓ s ), then ψ a,d g · · · · · · f ⊗ h = ψ a,d g · · · · · · jf ⊗ h by Lemma 2.2(2),= g · · · · · · jf ⊗ h = g · · · · · · f ⊗ ↑ r +1 ↓ s σ ↑ r ↓ s ↑ ◦ ( h ) by Lemma 2.2(2) and (2.3),= g · · · · · · f ⊗ ( h h ) ◦ ↑ r +1 ↓ s σ ↑ r ↓ s ↑ = f ⊗ h g · · · h · · · ! ◦ ↑ r +1 ↓ s σ ↑ r ↓ s ↑ ∈ Y ( ↑ m ↓ n , ↑ r +1 ↓ s ) ⊗ H ( ↑ r +1 ↓ s )( σ ) , where g is obtained from g by removing all dots on g , and ( h , h ) ∈ H ( ↑ r ) × H ( ↓ s ) such that h = h ⊗ h . Thanks to (c), ψ a,d is surjective. Now, the exactness of the first sequence in (3.17)follows immediately sincedim1 d A (cid:22) b ⊗ A ◦ b ↓ A ◦ a = | Y ( ↑ m ↓ n +1 , ↑ r ↓ s ) || H ( ↑ r ↓ s ) | = (cid:0) | Y ( ↑ m ↓ n , ↑ r ↓ s − ) | + | Y ( ↑ m ↓ n +1 , ↑ r ↓ s ) | (cid:1) | H ( ↑ r ↓ s ) | = | Y ( ↑ m ↓ n , ↑ r ↓ s − ) || H ( ↑ r ↓ s ) | + | Y ( ↑ m ↓ n , ↑ r +1 ↓ s ) || D ( ↑ r +1 ) || H ( ↑ r ↓ s ) | = dim1 d ( ↓ A ⊗ A A (cid:22) a )1 a + dim1 d A (cid:22) c ⊗ A ◦ c A ◦↑ a . (cid:3) Lemma 3.9.
Suppose ϕ , ψ , η and ε are ( A, A ◦ ) -homomorphisms in (3.16) . Then (1) ( x ↓ ⊗ Id ) ◦ ϕ = ϕ ◦ ( Id ⊗ x ↓ L ) , (2) ( Id ⊗ x ↑ R ) ◦ ψ = ψ ◦ ( x ↓ ⊗ Id ) , (3) ( x ↑ ⊗ Id ) ◦ η = η ◦ ( Id ⊗ x ↑ L ) , (4) ( Id ⊗ x ↓ R ) ◦ ε = ε ◦ ( x ↑ ⊗ Id ) .Proof. Suppose ( f , g ) ∈ A (cid:22) b b × b ↓ A ◦ a , where a = ( s, r ), b ∈ b = ( s − , r ) and s ≥
1. Thanks toLemma 3.4(1),( x ↓ ⊗ Id) ◦ ϕ ( f ⊗ g ) = ( x ↓ ⊗ Id)(( f ) ⊗ g ) = ( f ) ⊗ g = f ⊗ b ◦ g = ϕ ◦ (Id ⊗ x ↓ L )( f ⊗ g ) , proving (1). One can check (3) similarly. If ( f, g ) ∈ c ↓ A ⊗ A A (cid:22) a a and c ∈ J , then(Id ⊗ x ↑ R ) ◦ ψ ( f ⊗ g ) = (Id ⊗ x ↑ R )( · · · f ⊗ ( g )) = · · · f ⊗ ( g ) Lemma 2.2(6) = · · · • f ⊗ ( g )= ψ ◦ ( x ↓ ⊗ Id)( f ⊗ g ) , proving (2). Replacing Lemma 2.2(6) by Lemma 2.2(8), one can verify (4) by arguments similar tothose for (2). (cid:3) Theorem 3.10.
For each i ∈ k , there are two short exact sequences of functors from A ◦ -fdmod to A -lfdmod: → ∆ ◦ F ↓ i → F i ◦ ∆ → ∆ ◦ F ↑ i → , → ∆ ◦ E ↑ i → E i ◦ ∆ → ∆ ◦ E ↓ i → . (3.18) Proof.
Thanks to Lemma 3.9, the short exact sequences in (3.18) follow from those in (3.15) by passingto appropriate generalized eigenspaces. (cid:3)
EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES Characters.
Suppose a = a · · · a r + s ∈ a = ( r, s ). For any i , 1 ≤ i ≤ r + s , define X i a = ( a ··· a i − a i +1 ··· a r + s , if a i = ,1 a ··· a i − a i +1 ··· a r + s , if a i = .Then { X i a | ≤ i ≤ r + s, a ∈ a } generates a finite dimensional commutative subalgebra of A a . Forany i = ( i , i , . . . , i r + s ) ∈ k r + s , there is an idempotent 1 a ; i ∈ A a which projects any M ∈ A a -fdmodonto M a ; i , the simultaneous generalized eigenspace of X a , . . . , X r + s a with respect to i . When r = s = 0, A a ∼ = k . In this case, there is a unique idempotent 1 ∅ ; ∅ . Definition 3.11.
For any V ∈ A -lfdmod, definech V = X a ∈ J, i ∈ k ℓ ( a ) (dim 1 a ; i V ) e a ; i , (3.19)where ℓ ( a ) = ℓ ↑ ( a ) + ℓ ↓ ( a ), and 1 a ; i V is the simultaneous generalized eigenspace of X a , . . . , X ℓ ( a ) a corresponding to i .Suppose a = ( r, s ) and λ = ( λ ↓ , λ ↑ ) ∈ Λ a , where Λ a is given in (2.27). Motivated by Lemma 2.12and (3.13)-(3.14), We define the content c ( x ) of a node x with respect to λ as follow: c ( x ) = ( c ↑ ( x ) , if x is in [ λ ↑ ], c ↓ ( x ) , if x is in [ λ ↓ ], (3.20)where c ↑ ( x ) = u j + k − l (resp., c ↓ ( x ) = u ′ j − k + l ) if x is at the l th row and k th column ofthe j th component of the Young diagram [ λ ↑ ] (resp., [ λ ↓ ]). Recall the cell modules S ( λ )’s of A a insubsection 2.6. The following result follows from Lemma 2.12, (3.14) and the branching rules of thecell modules for degenerate cyclotomic Hecke algebras. Lemma 3.12.
Suppose i ∈ k and λ = ( λ ↓ , λ ↑ ) ∈ Λ ( r,s ) , where r, s ∈ N . We have (1) F ↓ i S ( λ ) (resp., E ↑ i S ( λ ) ) has a multiplicity-free filtration with sections S ( µ ) , where µ ∈ Λ ( r − ,s ) (resp., µ ∈ Λ ( r,s − ) is obtained by removing a box in [ λ ↓ ] (resp., [ λ ↑ ] ) of content i . (2) E ↓ i S ( λ ) (resp., F ↑ i S ( λ ) ) has a multiplicity-free filtration with sections S ( µ ) , where µ ∈ Λ ( r +1 ,s ) (resp., µ ∈ Λ ( r,s +1) ) is obtained by adding a box in [ λ ↓ ] (resp., [ λ ↑ ] ) of content i . Recall I , I u and I u ′ in (1.2). By Lemma 3.12, E ↑ i and F ↑ i (resp., E ↓ i and F ↓ i ) are non-zero only if i ∈ I u (resp., i ∈ I u ′ ). Thanks to Theorem 3.10 and Lemma 3.5, E = M i ∈ I E i , F = M i ∈ I F i . (3.21)Recall that Λ = S a ∈ I Λ a in (2.34). Let Ξ be the graph such that the set of vertices is Λ and anyedge is of form λ — µ whenever µ is obtained from λ by either adding a box or removing a box. A path γ : λ µ in Ξ is a finite sequence of vertices λ = λ , λ , . . . , λ m = µ with each λ j — λ j +1 connected byan edge. We color the edge λ j — λ j +1 by c ( x ) if λ j +1 is obtained from λ i by either adding an addablenode x or removing a removable node x . Definition 3.13.
Suppose i = ( i , . . . , i m ). A path γ is of type ( a ; i ) if i j is the color of the j th edgeof γ , and if a = a · · · a m ∈ J such that(1) a j +1 = ↑ if λ j +1 is obtained from λ j by adding a box to λ ↑ j or removing a box from λ ↓ j ,(2) a j +1 = ↓ if λ j +1 is obtained from λ j by adding a box to λ ↓ j or removing a box from λ ↑ j .When λ = µ = ( ∅ , ∅ ), we say there is a unique path from λ to µ with type ( ∅ ; ∅ ). Proposition 3.14.
For any λ ∈ Λ , ch ˜∆( λ ) = P γ e type ( γ ) where the summation ranges over all paths γ : ( ∅ , ∅ ) λ and ˜∆( λ ) is given in Definition 2.13.Proof. Suppose λ ∈ Λ a and a ∈ I . Thanks to (2.33), ˜∆( λ ) = L d (cid:23) a d ˜∆( λ ) and 1 a ˜∆( λ ) = S ( λ ). Inorder to prove the required formula on ch ˜∆( λ ), it suffices to show thatdim 1 a ; i ˜∆( λ ) = |{ γ : ( ∅ , ∅ ) λ | γ is of type ( a ; i ) }| (3.22)for all i ∈ k m and all a = a · · · a m ∈ J and all m ∈ N . We prove (3.22) by induction on m ∈ N . Suppose that m = 0. If λ = ( ∅ , ∅ ), then 1 ∅ ; ∅ ˜∆( λ ) = 0. Otherwise, 1 ∅ ; ∅ ˜∆( λ ) = k . So, the resultholds for m = 0. In general, by (3.4),1 a ( EV ) = 1 a ↑ A ⊗ A V ∼ = 1 a ↑ V, a ( F V ) = 1 a ↓ A ⊗ A V ∼ = 1 a ↓ V for any V ∈ A -lfdmod. So,dim 1 a ; i E i V = dim 1 a ↑ ; i i V, dim 1 a ; i F i V = dim 1 a ↓ ; i i V. (3.23)Thanks to (3.18) and Lemma 3.12, E i ˜∆( λ ) (resp., F i ˜∆( λ )) has a multiplicity-free ˜∆-filtration suchthat ˜∆( µ ) appears as a section if and only if µ is obtained by either removing a box in [ λ ↑ ] (resp.,[ λ ↓ ]) of content i or adding a box in [ λ ↓ ] (resp., [ λ ↑ ]) of content i . Now (3.22) follows from (3.23) andinduction on m , immediately. (cid:3) Corollary 3.15.
Suppose ( λ, µ ) ∈ Λ × Λ a , a ∈ I . If [ ˜∆( λ ) : L ( µ )] = 0 , then there are two paths γ : ( ∅ , ∅ ) λ and δ : ( ∅ , ∅ ) µ such that γ and δ are of the same type ( a ; i ) and a ∈ a .Proof. Mimicking arguments in the proof of [11, Corollary 5.11], one can verify this result by usingCorollary 2.14(1) and Proposition 3.14, immediately. (cid:3)
Theorem 3.16.
Suppose I u T I u ′ = ∅ , where I u and I u ′ are in (1.2) . Then (1) P ( µ ) = ∆( µ ) for all µ ∈ Λ , (2) ∆ : A ◦ -mod → A -mod is an equivalence of categories.Proof. Take an arbitrary µ ∈ Λ a . If P ( µ ) = ∆( µ ), by Corollary 2.16(1), [∆( λ ) : L ( µ )] = 0 for some λ ∈ Λ b and λ = µ . Since ∆ is exact and D ( λ ) is the simple head of S ( λ ), there is an epimorphismfrom ˜∆( λ ) to ∆( λ ). So, [ ˜∆( λ ) : L ( µ )] = 0. By Corollary 3.15, there are two paths γ : ( ∅ , ∅ ) λ and δ : ( ∅ , ∅ ) µ such that γ and δ are of the same type ( a ; i ) and a ∈ a .We claim a = b . Otherwise, b ≻ a . By Definition 3.13, there is an edge, say λ j − — λ j such that λ j is obtained by removing a box x either in λ ↓ j − with c ( x ) ∈ I u or in λ ↑ j − with c ( x ) ∈ I u ′ . Inany case, c ( x ) ∈ I u T I u ′ = ∅ , a contradiction. By the definition of ∆ in (2.33), 1 a ∆( λ ) = D ( λ ).Since 1 a L ( µ ) = D ( µ ), it is a composition factor of the simple A a -module D ( λ ), forcing λ = µ , acontradiction. So, P ( µ ) = ∆( µ ) for all µ ∈ Λ. Now (2) immediately follows from [4, Corollary 2.5]and (1), since the exact functor ∆ sends projective A ◦ -modules P ( λ )’s to projective A -modules P ( λ )’sfor any λ ∈ Λ. (cid:3) Corollary 3.17.
The category A -mod is completely reducible if (1) u i − u ′ j Z k for all ≤ i, j ≤ ℓ , (2) u i − u j Z k , u ′ i − u ′ j Z k for all ≤ i < j ≤ ℓ and p = 0 .Proof. Thanks to (1) and Theorem 3.16(2), A ◦ -mod is Morita equivalent to A -mod. By (2) and [1,Theorem 6.11], both H ℓ,r ( u ) and H ℓ,s ( − u ′ ) are semisimple for all r, s ∈ N . Now, the result followsimmediately from Lemma 2.11(2) and Lemma 2.12. (cid:3) We expect that Corollary 3.17(1)-(2) are necessary and sufficient conditions for A -mod being com-pletely reducible.3.3. Categorical actions.
Let g be the complex Kac-Moody Lie algebra g associated to Cartanmatrix ( a i,j ) i,j ∈ I defined by (1.3). Then g is the Lie algebra generated by its Cartan subalgebra andChevalley generators { e i , f i | i ∈ I } subject to the usual Serre relations. Furthermore, g is isomorphicto a direct sum of certain sl ∞ (resp., ˆ sl p ) if p = 0 (resp., p >
0) depending on both u and u ′ . LetΠ = { α i | i ∈ I } , (3.24)the set of simple roots. The weight lattice is P := { λ ∈ h ∗ | h h i , λ i ∈ Z for all i ∈ I } , (3.25)where h i := [ e i , f i ]. Let P + = { λ ∈ h ∗ | h h i , λ i ∈ N for all i ∈ I } , (3.26)and { ω i | i ∈ I } be the set of fundamental weights of g . There is a usual dominance order on P in thesense that λ ≤ µ if µ − λ ∈ P i ∈ I N α i . The partial order on P induces a partial order on P × P suchthat ( λ , λ ) (cid:23) ( µ , µ ) if λ + λ = µ + µ and λ ≤ µ . (3.27) EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES Define wt : Λ → P × P, λ → (wt ↓ ( λ ) , wt ↑ ( λ )) (3.28)where wt ↓ ( λ ) = − ω u ′ + P y ∈ [ λ ↓ ] α c ( y ) , wt ↑ ( λ ) = ω u − P x ∈ [ λ ↑ ] α c ( x ) , and ω u , ω u ′ are given in (1.4). Proposition 3.18.
Suppose λ ∈ Λ and µ ∈ Λ . If [ ˜∆( λ ) : L ( µ )] = 0 , then wt ( µ ) (cid:22) wt ( λ ) .Proof. Suppose µ ∈ Λ ( r,s ) . Thanks to Corollary 3.15, there are two paths γ : ( ∅ , ∅ ) λ and δ : ( ∅ , ∅ ) µ such that γ and δ are of the same type ( a ; i ), where a = a · · · a r + s and i = ( i , . . . , i r + s ).We are going to prove the result by induction on r + s .If r + s = 0, then λ = µ = ( ∅ , ∅ ) and there is nothing to prove. Otherwise, r + s >
0. Removingthe last edge in both γ and δ yields two shorter paths γ ′ : ( ∅ , ∅ ) λ ′ and δ ′ : ( ∅ , ∅ ) µ ′ such that γ ′ and δ ′ are of the same type ( b ; j ), where b = a · · · a r + s − and j = ( i , . . . , i r + s − ). We deal with thecase a r + s = ↑ and leave the details on a r + s = ↓ to the reader since the proof is similar.There are two cases we need to consider. If λ is obtained from λ ′ by adding a box x in [ λ ′↑ ], thenwt ↑ ( λ ′ ) = wt ↑ ( λ ) + α c ( x ) and wt ↓ ( λ ′ ) = wt ↓ ( λ ). Since µ ∈ Λ ( r,s ) ,wt ↑ ( µ ′ ) = wt ↑ ( µ ) + α c ( x ) , wt ↓ ( µ ′ ) = wt ↓ ( µ ) . (3.29)By induction assumption, we have wt( µ ′ ) (cid:22) wt( λ ′ ), forcing wt( µ ) (cid:22) wt( λ ). Otherwise, λ is obtainedfrom λ ′ by removing a box x in [ λ ′↓ ]. So, wt ↑ ( λ ′ ) = wt ↑ ( λ ), wt ↓ ( λ ′ ) = wt ↓ ( λ ) + α c ( x ) , and (3.29) stillholds true. By induction assumption, we have wt( µ ) (cid:22) wt( λ ) sincewt ↑ ( µ ) + wt ↓ ( µ ) = wt ↑ ( λ ) + wt ↓ ( λ ) , wt ↓ ( λ ) ≤ wt ↓ ( µ ) . (cid:3) Corollary 3.19.
Suppose λ, µ ∈ Λ . (1) If L ( λ ) and L ( µ ) are in the same block of A -mod, then wt ↑ ( λ ) + wt ↓ ( λ ) = wt ↑ ( µ ) + wt ↓ ( µ ) . (2) If [∆( λ ) : L ( µ )] = 0 and λ = µ , then wt ( µ ) ≺ wt ( λ ) .Proof. Mimicking arguments in the proof of [11, Theorem 5.17] and using Proposition 3.18 and Corol-lary 2.16(2) yields (1). We leave the details to the reader. (2) follows from Proposition 3.18 andCorollary 2.16. (cid:3)
Let j a : A (cid:22) a -lfdmod → A a -fdmod be the exact idempotent truncation functor. Then j a V = 1 a V for any V ∈ A (cid:22) a -lfdmod. Recall j a ! and j a ∗ in (2.33). Then ( j a ! , j a , j a ∗ ) agree with the adjoint triplebetween A -lfdmod (cid:22) a and its Serre quotient A -lfdmod (cid:22) a /A -lfdmod ≺ a . In fact, this result is availablefor any locally unital algebra associated to an upper finite weakly triangular category (see the proofof [11, Theorem 3.5]). Theorem 3.20.
The A -lfdmod is an upper finite fully stratified category in the sense of [8, Defini-tion 3.36] with respect to the stratification wt : Λ → P × P in (3.28) with the order (cid:22) on P × P .Proof. It is clear that wt is a new stratification of A -lfdmod in the sense of [8, Definition 3.1]. Moreover,the image of wt (denoted by P ) is upper finite by its definition in (3.28). By the well-known resultson block decomposition of degenerate cyclotomic Hecke algebras, P gives a block decomposition of A ◦ -lfdmod.Suppose that A a , ( ρ,σ ) -fdmod is the block of A a indexed by ( ρ, σ ) ∈ P with a = ( r, s ) (i.e., ρ is obtained from − ω u ′ by adding r simple roots and σ is obtained from ω u by subtracting s simpleroots). Let A -lfdmod (cid:22) ( ρ,σ ) be the Serre subcategory of A -lfdmod generated by { L ( λ ) | wt( λ ) (cid:22) ( ρ, σ ) } .Note that λ ∈ Λ ( r + k,s + k ) for some k ∈ N if wt( λ ) (cid:22) ( ρ, σ ). So, A -lfdmod (cid:22) ( ρ,σ ) is actually a Serresubcategory of A (cid:22) a -lfdmod. Similarly we have A -lfdmod ≺ ( ρ,σ ) .Let j a , ( ρ,σ ) be the restriction of j a to A -lfdmod (cid:22) ( ρ,σ ) . Since j a , ( ρ,σ ) ( M ) ∈ A a , ( ρ,σ ) -fdmod for any M ∈ A (cid:22) a -lfdmod (cid:22) ( ρ,σ ) , j a , ( ρ,σ ) is actually a functor from A -lfdmod (cid:22) ( ρ,σ ) to A a , ( ρ,σ ) -fdmod. More-over, j a , ( ρ,σ ) induces an equivalence of categories between A -lfdmod (cid:22) ( ρ,σ ) /A -lfdmod ≺ ( ρ,σ ) and A a , ( ρ,σ ) -fdmod.Let j a , ( ρ,σ )! be the restriction of j a ! to A a , ( ρ,σ ) -fdmod. Then j a , ( ρ,σ )! is actually a functor from A a , ( ρ,σ ) -fdmod to A -lfdmod (cid:22) ( ρ,σ ) . In fact, for any M ∈ A a , ( ρ,σ ) -fdmod, j a , ( ρ,σ )! ( M ) has a ∆-flag, and ∆( µ )appears as a section if [ M : D ( µ )] = 0. In this case, wt( µ ) = ( ρ, σ ). By Corollary 3.19(2), we see that j a , ( ρ,σ )! ( M ) ∈ A -lfdmod (cid:22) ( ρ,σ ) . Furthermore, since ( j a ! , j a ) is an adjoint pair, so is ( j a , ( ρ,σ )! , j a , ( ρ,σ ) ).Hence the required standard and proper standard objects coincide with those in Definition 2.13. Suppose λ ∈ Λ. Thanks to Corollaries 2.16(1) and 3.19(2), P ( λ ) has a finite ∆-flag such that ∆( λ )appears as the top section and other sections ∆( µ ) with wt( λ ) (cid:22) wt( µ ). So, A -lfdmod is an upperfinite +-stratified category in the sense of [8, Definition 3.36] with respect to the stratification wt. Itis fully stratified since ∆( µ ) has a finite ∆-flag with sections ∆( ν ) such that wt( ν ) = wt( µ ). (cid:3) Let K ( A ◦ -pmod) be the Grothendieck group of A ◦ -pmod. Recall V ( ω u ) (resp., ˜ V ( − ω u ′ )) is theintegrable highest (resp., lowest) weight g -module of weight ω u (resp., − ω u ′ ). Let g ↑ = { y ↑ | y ∈ g } and g ↓ = { y ↓ | y ∈ g } be the two copies of g , where both y ↑ and y ↓ are y . Proposition 3.21. As g ↓ ⊕ g ↑ -modules, C ⊗ Z K ( A ◦ -pmod ) ∼ = ˜ V ( − ω u ′ ) ⊠ V ( ω u ) , where the Chevalley generators e ↑ i and f ↑ i (resp., e ↓ i and f ↓ i ) act on C ⊗ Z K ( A ◦ -pmod ) via the endo-morphisms induced by E ↑ i and F ↑ i (resp., E ↓ i and F ↓ i ) if i ∈ I u (resp., I u ′ ) and , otherwise.Proof. This follows from [14, § (cid:3) Let U ( t ) be the universal enveloping algebra of any Lie algebra t . There is a Lie algebra homomor-phism from g to g ↓ ⊕ g ↑ sending y to y ↓ + y ↑ . This homomorphism induces the usual comultiplicationon U ( g ) since U ( g ) ⊗ U ( g ) can be identified with U ( g ↓ ⊕ g ↑ ). So, ˜ V ( − ω u ′ ) ⊠ V ( ω u ) becomes the g -module ˜ V ( − ω u ′ ) ⊗ V ( ω u ) via the above homomorphism. Let K ( A -mod ∆ ) be the Grothendieckgroup of A -mod ∆ . Theorem 3.22. As g -modules, C ⊗ Z K ( A -mod ∆ ) ∼ = ˜ V ( − ω u ′ ) ⊗ V ( ω u ) , where the Chevalley generators e i , f i act on C ⊗ Z K ( A -mod ∆ ) via the endomorphisms induced by the E i and F i for all i ∈ I .Proof. This follows from Proposition 3.21 and (3.18). (cid:3) Proof of Theorem 1.1
Recall that g is the Kac-Moody Lie algebra in subsection 3.3. The quiver Hecke category QH associated to g is a k -linear strict monoidal category generated by objects I in (1.2) and morphisms i : i → i, i j : i ⊗ j → j ⊗ i subject to certain relations in [4, Definition 3.4], where the parameters { t ij ∈ k × | i, j ∈ I } and { s mqij ∈ k | < m < − a i,j , < q < − a j,i } (only appear when p = 2) are given as follows: t ij = (cid:26) − , i = j − , otherwise. , s ij = s ji = 2 for i = j ± . For any i = ( i d , i d − , . . . , i ) ∈ I d , d >
0, we identify i with the object i d ⊗ i d − ⊗ . . . ⊗ i ∈ ob QH .The locally unital algebra associated to QH is L d ∈ N QH d , where QH d := M i , i ′ ∈ I d Hom QH ( i , i ′ ) . When d > QH d is known as the quiver Hecke algebra associated to g [13, 18]. It is generated by { e ( i ) | i ∈ I d } [ { y , . . . , y d } [ { ψ , . . . , ψ d − } subject to the relations (1.7)–(1.15) in [7, Theorem 1.1], where e ( i ) = i d i i · · · , y r e ( i ) = i d i r i · · ·· · · , ψ r e ( i ) = i d i i r · · ·· · · , and i = ( i , i , . . . , i d ). For any µ ∈ P + given in (3.26), let QH d ( µ ) be the cyclotomic quotient of QH d by the two-sided ideal generated by { y h h i ,µ i e ( i ) | i ∈ I d } .Following [9], let AH be the k -linear strict monoidal category generated by the single object ↓ andtwo morphisms and satisfying the relations (2.12)–(2.14). Then AH , known as the degenerateaffine Hecke category, can be considered as the subcategory of AOB generated by the single object ↓ and morphisms and . Let AH d := Hom AH ( ↓ ⊗ d , ↓ ⊗ d ) , EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES the degenerate affine Hecke algebra [9]. Following [9], define x r = · · ·· · · r · · ·· · · ∈ AH d . For any µ ∈ P + , let AH d ( µ ) be the cyclotomic quotient of AH d by the two-sided ideal generated by g ( x ) = Y i ∈ I ( x − i ) h h i ,µ i . The algebra AH d ( µ ) is isomorphic to some degenerated cyclotomic Hecke algebra in subsection 2.5. Atmoment, we use AH d ( µ ) so as to compare it with QH d ( µ ). It is known that there is a set of mutuallyorthogonal idempotents of AH d ( µ ), denoted by { i | i ∈ I d } such that, for any AH d ( µ )-module M i M = d \ k =1 M i k , where M i k is the i k -generalized eigenspace of M of x k . If h h i , µ, i ≤ h h i , µ ′ i for all i ∈ I , then thereare epimorphisms QH d ( µ ′ ) ։ QH d ( µ ) , AH d ( µ ′ ) ։ AH d ( µ ) . So, { QH d ( µ ) | µ ∈ P + } and { AH d ( µ ) | µ ∈ P + } form two inverse systems of locally unital algebras.Taking inverse limits yields two completions d AH d := lim ← AH d ( µ ) , d QH d := lim ← QH d ( µ ) . It is proved in [7, Theorem 1.1] that there is an isomorphism of locally unital algebras QH d ( µ ) ∼ = AH d ( µ ) such that i d i i · · · 7→ i , i d i r i · · ·· · · 7→ ( x r − i r )1 i . (4.1)These isomorphisms induces an isomorphism of locally unital algebras d QH d ∼ = d AH d . (4.2)See also [18, § QH d ֒ → d QH d . (4.3)Now, we go on studying the cyclotomic oriented Brauer category OB ( u , u ′ ). Recall A is the locallyunital k -algebra associated to OB ( u , u ′ ) in (2.20). There are endofunctors E , F in (3.4). For any i ∈ I there are endofunctors E i , F i in (3.21). Thanks to Lemma 3.4(1), there is an ( A, A )-homomorphism x ↓ : A ↓ → A ↓ defined on A ↓ b by right multiplication by 1 b . So, x ↑ and x ↓ are intertwined by theisomorphism ↑ A ∼ = A ↓ in Proposition 3.1, where x ↑ is given in Lemma 3.4(2). Moreover, x ↓ induces anatural transformation : E → E such that (cid:18) (cid:19) M = x ↓ ⊗ Id : A ↓ ⊗ A M → A ↓ ⊗ A M. Similarly, there is a natural transformation denoted by : E → E such that (cid:18) (cid:19) M : A ↓↓ ⊗ A M → A ↓↓ ⊗ A M, y ⊗ m y ◦ (1 a ) ⊗ m, where ( y, m ) ∈ A a ↓↓ × M and A ↓↓ = A ↓ ⊗ A A ↓ in the obvious way. Lemma 4.1.
There is a strict monoidal functor
Ψ :
AH → E nd ( A -lfdmod ) such that Ψ( ↓ ) = E, Ψ( ) = , Ψ( ) = . Proof.
The result follows directly from (2.12)–(2.14). (cid:3)
Lemma 4.2.
For all i, j ∈ I , there are natural transformations i : E i → E i , and i j : E i ◦ E j → E j ◦ E i , which induce a strict monoidal functor from QH to E nd ( A -lfdmod ) .Proof. By Lemma 4.1, there is an algebra homomorphism Ψ d : AH d → End( E d ) for any d > E and F are biadjoint to each other. So, E is a sweet endofunctor of A -lfdmod in the sense of [4, Definition 2.10]. By [4, Theorem 2.11], E preserve the set of finitelygenerated projective modules. Therefore, E d P is finitely generated projective for any P ∈ A -pmod.In particular, E d P ∈ A -lfdmod since any finitely generated left A -module has to be locally finitedimensional (see [4, Section 2.2]). It is proved in [4, Section 2.2] that dim Hom A ( V, W ) < ∞ for anyfinitely generated left A -module V , and any locally finite dimensional left A -module W . In particular,End A ( E d P ) is finite dimensional for each P ∈ A -pmod and hence ( x ) P has a minimal polynomialwith roots in I (see (3.21)). So, the compositionev P ◦ Ψ d : AH d → End A ( E d P )of Ψ d with the evaluation at P factors through some cyclotomic quotients AH d ( µ ) and hence factorsthrough all sufficient large cyclotomic quotients AH d ( µ ′ ). Note that any two natural transformations η, ξ : E d → E d are equal if η P = ξ P for each P ∈ A -pmod. So, ev M ◦ Ψ d factors through all sufficientlarge cyclotomic quotients AH d ( µ ′ ) for any M ∈ A -lfdmod. This shows that Ψ d factors through d AH d .Let ˆΨ d : d AH d → End( E d ) . (4.4)Composing ˆΨ d with the isomorphism in (4.2) and the inclusion in (4.3) yields an algebra homomor-phism Φ d : QH d → End( E d ) . Moreover, by the argument for the proof of [5, (6.13)] (i.e., by Lemma 4.1 and the explicit formulaeof the isomorphism d QH d ∼ = d AH d induced by [7, Theorem 1.1] or the non-degenerate analogue in [21,Proposition 3.10]), these maps are compatible with the monoidal structure in the categories QH and E nd ( A -lfdmod) such that there is a monoidal functor Φ : QH 7→ E nd ( A -lfdmod) given by Φ( i ) = E i and Φ( h ) = Φ d ( h ) if h ∈ QH d . (cid:3) Definition 4.3. [15, Definition 3.2, Remark 3.6] Fix ω u and ω u ′ in (1.4). The data of a tensorproduct categorification of ˜ V ( − ω u ′ ) ⊗ V ( ω u ) consists of two parts:(TPC1) A (locally) Schurian category C such that the isomorphism classes of irreducible objects labeledby the indexing set for the basis of ˜ V ( − ω u ′ ) ⊗ V ( ω u ).(TPC2) A nilpotent categorical action (in the sense of [4, Definition 4.25]) making C into a 2-representation (e.g., [18, Definition 5.1.1] or [4, Definition 4.1]) of the associated Kac-Moody2-category U ( g ) (e.g., [18, § C is an upper finite fully stratified category in the sense of Brundan and Stroppel[8] with poset P × P . Moreover, the poset structure of P × P is given by the “inverse dominanceorder” (cid:22) on P × P in (3.27).(TPC4) There is a categorical ( g ↓ ⊕ g ↑ )-action on gr C := L ( λ,µ ) ∈ P × P C ( λ,µ ) such that C ⊗ K (proj-gr C ) ∼ = ˜ V ( − ω u ′ ) ⊠ V ( ω u )as ( g ↓ ⊕ g ↑ )-modules, where(1) C ( λ,µ ) := C (cid:22) ( λ,µ ) / C ≺ ( λ,µ ) ,(2) g ↑ = { x ↑ | x ∈ g } and g ↓ = { x ↓ | x ∈ g } are two copies of g ,(3) proj-gr C is the subcategory of finite generated projective modules of gr C ,(4) C ( − ω ′ u , ω u ) ∼ = Vec k (the category of vector space over k ).The categorification functors for g ⋄ will be denoted by E ⋄ i and F ⋄ i , where i ∈ I and ⋄ ∈ {↑ , ↓} . EPRESENTATIONS OF CYCLOTOMIC ORIENTED BRAUER CATEGORIES (TPC5) For any i ∈ I , denote by E i and F i the categorification functors for g . There is a compatibilitybetween the categorical g -action on C and the categorical ( g ↓ ⊕ g ↑ )- action on gr C in the sensethat there are short exact sequences0 → ∆ ◦ E ↑ i → E i ◦ ∆ → ∆ ◦ E ↓ i → , → ∆ ◦ F ↓ i → F i ◦ ∆ → ∆ ◦ F ↑ i → , (4.5)where ∆ = L ( λ,µ ) ∈ P × P ∆ ( λ,µ ) and ∆ ( λ,µ ) : C (cid:22) ( λ,µ ) → C ( λ,µ ) is the corresponding standardiza-tion functor of the stratified category C . Proof of Theorem 1.1 : Thanks to Definition 4.3, we need to verify that A -lfdmod admits thestructures in (TPC1)–(TPC5).By Corollary 2.14 and Lemma 3.21, we have results on the classification of irreducible objects in A -lfdmod. Now, (TPC1) follows from Theorem 3.22 which gives a bijection between labellings. (TPC3)follows from Theorem 3.20 which says that A -lfdmod is an upper finite fully stratified category inthe sense of Brundan and Stroppel [8] with respect to the stratification function in (3.28). (TPC4)follows from Lemma 3.21 and the well-known categorical g -action on the representations of degeneratecyclotomic Hecke algebras [6, Theorem 4.18] [19, Theorem 4.25]. The short exact sequences in (4.5)follow from (3.18) in Theorem 3.10 and hence (TPC5) follows. So, it remains to verify (TPC2).Thanks to [4, Theorem 4.27], it suffices to check the following conditions in [4, Definition 4.25]:(1) A weight decomposition of the category A -lfdmod= L σ ∈ P × P A -lfdmod σ .(2) Biadjoint endofunctors E = L i ∈ I E i and F = L i ∈ I F i such that ( E i , F i ) are biadjoint functors.(3) For all i, j ∈ I , there are natural transformations i : E i → E i , and i j : E i ◦ E j → E j ◦ E i , which induce a strict monoidal functor from QH to E nd ( A -lfdmod).(4) The endomorphisms [ E i ] and [ F i ] make C ⊗ Z K ( A -pmod) into a well-defined g -module with σ -weight space C ⊗ Z K ( A -pmod σ ).(5) For any i ∈ I and any finitely generated left A -module M , the endomorphism( i ) M : E i M → E i M is nilpotent.In fact, (1) follows from the partial result on the blocks of A -lfdmod in Corollary 3.19. Inthis case, we set A -lfdmod σ to be the Serre subcategory of A -lfdmod generated by L ( λ ) such thatwt ↑ ( λ )+wt ↓ ( λ ) = σ . The biadjoint functors in (2) are given in Lemma 3.6 and (3.21). (3) follows fromLemma 4.2. Thanks to Corollary 2.16, C ⊗ Z K ( A -pmod) can be identified with C ⊗ Z K ( A -mod ∆ )and hence (4) follows Theorem 3.22. Finally, by (4.1), under the isomorphism in (4.2) we have i M = (cid:18) ( ) M − i Id (cid:19) E i M . Therefore, it suffices to show that there is bound on the Jordan block sizes of x ↓ ⊗ Id = ( ) M : A ↓ ⊗ A M → A ↓ ⊗ A M for any finitely generated left A -module M . Since E is a sweet functor (e.g., Lemma 3.3) of A -lfdmodin the sense of [4, Definition 2.10], by [4, Theorem 2.11], E preserve the set of finitely generatedmodules. Therefore, EM = A ↓ ⊗ A M is finitely generated. Since A is locally finite dimensional,End A ( EM ) is finite dimensional and hence (cid:18) (cid:19) M has a minimal polynomial. So, there is bound onthe Jordan block sizes. This completes the proof of (5). (cid:3) References [1]
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M.G. School of Mathematical Science, Tongji University, Shanghai, 200092, China
Email address : [email protected] H.R. School of Mathematical Science, Tongji University, Shanghai, 200092, China
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