aa r X i v : . [ m a t h . R T ] A ug FUSION PROCEDURE FOR CYCLOTOMIC BMW ALGEBRAS
WEIDENG CUI
Abstract.
Inspired by the work [IMOg2], in this note, we prove that the pairwiseorthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl alge-bras can be constructed by consecutive evaluations of a certain rational function. In theappendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras. Introduction S n , showed by Jucys [Juc], canbe obtained by taking a certain limiting process on a rational function. The process isnow commonly known as the fusion procedure, which has been further developed in thesituation of Hecke algebras [Ch]; see also [Na2-4]. In [Mo], Molev has presented anotherapproach of the fusion procedure for S n , which depends on the existence of a maximalcommutative subalgebra generated by the Jucys-Murphy elements. In his approach, theprimitive idempotents are obtained by consecutive evaluations of a certain rational func-tion. The new version of the fusion procedure has been generalized to the Hecke algebrasof type A [IMO], to the Brauer algebras [IM, IMOg1], to the Birman-Murakami-Wenzlalgebras [IMOg2], to the complex reflection groups of type G ( d, , n ) [OgPA1], to theAriki-Koike algebras [OgPA2], to the wreath products of finite groups by the symmetricgroup [PA], to the degenerate cyclotomic Hecke algebras [ZL], to the Yokonuma-Heckealgebras [C1], to the cyclotomic Yokonuma-Hecke algebras [C2, Appendix] and to thedegenerate cyclotomic Yokonuma-Hecke algebras [C3].1.2. The Birman-Murakami-Wenzl (for brevity, BMW) algebra was algebraically definedby Birman and Wenzl [BW], and independently Murakami [Mu], which is an algebragenerated by some elements satisfying certain particular relations. These relations arein fact implicitly modeled on the ones of certain algebra of tangles studied by Kauffman[Ka] and Morton and Traczyk [MT], which is known as a Kauffman tangle algebra. BMWalgebra are closely related to Artin braid groups of type A, Iwahori-Hecke algebras of type A, quantum groups, Brauer algebras and other diagram algebras; see [Eny1-2, HuXi, Hu,LeRa, MW, RuSi4-6, RuSo, Xi] and the references therein.Motivated by studying link invariants, H¨aring-Oldenburg [HO] introduced a class offinite dimensional associative algebras called cyclotomic Birman-Murakami-Wenzl (forbrevity, BMW) algebras, generalizing the notions of BMW algebras. Such algebras areclosely related to Artin braid groups of type B, cyclotomic Hecke algebras and otherresearch objects, and have been studied by a lot of authors from different perspectives;see [Go1-4, GoHM1-2, HO, OrRa, RuSi2-3, RuXu, Si, WiYu1-3, Xu, Yu] and so on. E T of cyclotomic BMW algebras. In Section 3,we establish the fusion formula for the primitive idempotent E T . In Section 4 (Appendix),we develop the fusion formulas for the primitive idempotents of cyclotomic Nazarov-Wenzlalgebras. 2.
Preliminaries
Cyclotomic Birman-Murakami-Wenzl algebras.Definition 2.1.
Assume that K is an algebraically closed field containing δ j , ≤ j ≤ d − , and some nonzero elements ρ, q , q − q − and v i , ≤ i ≤ d , and that they satisfy therelation ρ − ρ − = ( q − q − )( δ − . Fix n ≥ . The cyclotomic Birman-Murakami-Wenzl algebra B d,n is the K -algebragenerated by the elements X ± , T ± i and E i (1 ≤ i ≤ n −
1) with the following relations:(1) (Inverses) T i T − i = T − i T i = 1 and X X − = X − X = 1 . (2) (Idempotent relations) E i = δ E i for 1 ≤ i ≤ n − . (3) (Affine braid relations)(a) T i T i +1 T i = T i +1 T i T i +1 and T i T j = T j T i if | i − j | ≥ . (b) X T X T = T X T X and X T j = T j X if j ≥ . (4) (Tangle relations)(a) E i E i ± E i = E i . (b) T i T i ± E i = E i ± E i and E i T i ± T i = E i E i ± . (c) For 1 ≤ j ≤ d − , E X j E = δ j E . (5) (Kauffman skein relations) T i − T − i = ( q − q − )(1 − E i ) for 1 ≤ i ≤ n − . (6) (Untwisting relations) T i E i = E i T i = ρ − E i for 1 ≤ i ≤ n − . (7) (Unwrapping relations) E X T X = ρE = X T X E . (8) (Cyclotomic relation) ( X − v )( X − v ) · · · ( X − v d ) = 0 . In B d,n , We define inductively the following elements: X i +1 := T i X i T i for i = 1 , . . . , n − . (2.1)It can be easily checked that the elements X , . . . , X n commute with each other, andmoreover, we have E i X i X i +1 = X i X i +1 E i = E i for i = 1 , . . . , n − . (2.2)We now define the following elements (see [IMOg2, (2.15)]): T i ( u, v ) = T i + ( q − q − ) uv − u + ( q − q − ) uu + ρqv E i for i = 1 , . . . , n − . (2.3) USION PROCEDURE 3
Note that E i = δ E i , where δ = ( q − + ρ − )( ρq − q − q − . By using this, it can be easily checkedthat (see [IMOg2, (2.17-18)]) T i ( u, v ) T i ( v, u ) = f ( u, v ) for i = 1 , . . . , n − , (2.4)where f ( u, v ) = f ( v, u ) = ( u − q v )( u − q − v )( u − v ) . (2.5)2.2. Combinatorics. λ = ( λ , . . . , λ k ) is called a partition of n if it is a finite sequence ofweakly decreasing nonnegative integers whose sum is n. We set | λ | := n. We shall identifya partition λ with a Young diagram, which is the set[ λ ] := { ( i, j ) | i ≥ ≤ j ≤ λ i } . We shall regard λ as a left-justified array of boxes such that there exist λ j boxes in the j -th row for j = 1 , . . . , k. We write θ = ( a, b ) if the box θ lies in row a and column b. Similarly, a d -partition of n is an ordered d -tuple λ = ( λ (1) , λ (2) , . . . , λ ( d ) ) of partitions λ ( k ) such that P dk =1 | λ ( k ) | = n. We denote by P d ( n ) the set of d -partitions of n. We shallidentify a d -partition λ with its Young diagram, which is the ordered d -tuple of the Youngdiagrams of its components. We write θ = ( θ, s ) if the box θ lies in the component λ ( s ) . Assume that λ and µ are two d -partitions. We say that λ is obtained from µ by addinga box if there exists a pair ( j, t ) such that λ ( t ) j = µ ( t ) j + 1 and λ ( s ) i = µ ( s ) i for ( i, s ) = ( j, t ) . In this case, we will also say that µ is obtained from λ by removing a box.Set Λ + d,n := { ( l, λ ) | ≤ l ≤ ⌊ n/ ⌋ , λ ∈ P d ( n − l ) } . The combinatorial objects appearing in the representation theory of B d,n will be up-down tableaux. For ( f, λ ) ∈ Λ + d,n , an n -updown λ -tableau, or more simply an updown λ -tableau, is a sequence T = ( T , T , . . . , T n ) of d -partitions such that T n = λ and T i isobtained from T i − by either adding or removing a box, for i = 1 , . . . , n , where we set T = ∅ . Let T udn ( λ ) be the set of updown λ -tableaux of n. Suppose that ( f, λ ) ∈ Λ + d,n and U = ( U , . . . , U n ) ∈ T udn ( λ ) . Letc( U | k ) = ( v s q j − i ) if U k = U k − ∪ (( i, j ) , s ) ,v − s q i − j ) if U k − = U k ∪ (( i, j ) , s ) . (2.6)Given a box α = (( i, j ) , s ) , we define the content of it byc( U | α ) = ( v s q j − i ) if α is an addable box of U ,v − s q i − j ) if α is a removable box of U . (2.7)We give the generalizations of some constructions in [IM, Section 3]. Suppose that( f, λ ) ∈ Λ + d,n and T = ( T , . . . , T n ) ∈ T udn ( λ ) . Set µ = T n − and consider the updown µ -tableau U = ( T , . . . , T n − ) . We now define two d -tuples of infinite matrices M ( U ) = ( m ( U ) , . . . , m d ( U )) and M ( U ) = ( m ( U ) , . . . , m d ( U )) , here the rows and columns of each m s ( U ) or m s ( U ) are labelled by positive integers andonly a finite number of entries in each of the matrices are nonzero. The entry m sij of the WEIDENG CUI matrix m s ( U ) (respectively, the entry m sij of the matrix m s ( U )) equals the number of timesthat the box (( i, j ) , s ) is added (respectively, removed) in the sequence ( ∅ , T , . . . , T n − ) . For each k ∈ Z and 1 ≤ s ≤ d, we define two nonnegative integers d sk = d k ( m s ( U )) and d sk = d k ( m s ( U )) as the sums of the entries of the matrices m s ( U ) and m s ( U ) on the k -thdiagonal, that is, d sk = X j − i = k m sij and d sk = X j − i = k m sij . (2.8)Furthermore, we define the indexes g sk = g k ( m s ( U )) and g sk = g k ( m s ( U )) as follows: g sk = δ k + d sk − + d sk +1 − d sk and g sk = d sk − + d sk +1 − d sk . (2.9)Finally, we define some integer p , . . . , p n associated to T inductively such that p k depends only on the first k d -partitions ( T , . . . , T k ) of T . Therefore, it is enough to define p n . We set p n = 1 − g k n ( m s n ( U )) (2.10)if T n is obtained from T n − by adding a box (( i n , j n ) , s n ), where k n = j n − i n ; p n = 1 − g k ′ n ( m s ′ n ( U )) (2.11)if T n is obtained from T n − by removing a box (( i ′ n , j ′ n ) , s ′ n ), where k ′ n = j ′ n − i ′ n . Assume that ( f, λ ) ∈ Λ + d,n , T = ( T , . . . , T n ) is an n -updown λ -tableau and that U =( T , . . . , T n − ) . We then define the element f ( T ) inductively by f ( T ) = f ( U ) ϕ ( U , T ) , (2.12)where ϕ ( U , T ) = Y k = k n k ∈ Z ( q k n − q k ) g snk Y ≤ t ≤ d ; t = s n k ∈ Z ( v s n q k n − v t q k ) g tk Y ≤ r ≤ dk ∈ Z ( v s n q k n − v − r q − k ) g rk if T n is obtained from T n − by adding a box (( i n , j n ) , s n ), where k n = j n − i n ; ϕ ( U , T ) = Y k = k ′ n k ∈ Z ( q − k ′ n − q − k ) g s ′ nk Y ≤ t ≤ d ; t = s ′ n k ∈ Z ( v − s ′ n q − k ′ n − v − t q − k ) g tk Y ≤ r ≤ dk ∈ Z ( v − s ′ n q − k ′ n − v r q k ) g rk if T n is obtained from T n − by removing a box (( i ′ n , j ′ n ) , s ′ n ), where k ′ n = j ′ n − i ′ n . In the special situation when f = 0 , that is, λ is a d -partition of n, there is a naturalbijection between the set of n -updown λ -tableaux and the set of standard λ -tableauxdefined in [DJM, Definition (3.10)]. The following proposition is inspired by [IM, Propo-sition 3.3] and can be proved similarly. Proposition 2.2. If λ is a d -partition of n and T = ( T , . . . , T n ) is an n -updown λ -tableau, then p , . . . , p n are all equal to zero, and f ( T ) is exactly equal to F − λ defined in [OgPA2 , Section 2 . when d = m. USION PROCEDURE 5
Idempotents of B d,n . Following [RuXu, Definition 3.4], we say that B d,n is genericif the parameters v i , 1 ≤ i ≤ d , and q satisfy the conditions (1) the order o ( q ) of q satisfies o ( q ) > n ; (2) | r | ≥ n whenever there exists r ∈ Z such that either v i v ± j = q r for i = j, or v i = ± q r . Following [WiYu1, Corollary 4.5], we say that B d,n is admissibleif the set { E , E X , . . . , E X d − } is linearly independent in B d, . It has been provedby Goodman [Go2, Theorem 4.4] that this admissible condition coincides with the u -admissible condition defined in [RuXu, Definition 2.27].From now on, we always assume that B d,n is generic and admissible. Thus, by [RuXu,Lemma 3.5], we have S = T if and only if c( S | k ) = c( T | k ) for all 1 ≤ k ≤ n. Therefore,the set { X , . . . , X n } , as a family of JM-elements for B d,n in the abstract sense definedin [Ma, Definition 2.4], satisfies the separation condition associated to the weakly cellularbasis of B d,n constructed in [RuXu, Theorem 4.19]. Note that the results in [Ma] alsohold for B d,n with respect to the weakly cellular basis. In particular, we can constructthe primitive idempotents of B d,n following the arguments in [Ma, Section 3].For each 1 ≤ k ≤ n, we define the following set: R ( k ) := { c( S | k ) | S ∈ T udn ( λ ) for some ( f, λ ) ∈ Λ + d,n } . Suppose that ( f, λ ) ∈ Λ + d,n and T ∈ T udn ( λ ) . We set E T = n Y k =1 (cid:18) Y c ∈ R ( k ) c =c( T | k ) X k − c c( T | k ) − c (cid:19) . (2.13)By standard arguments in [Ma, Section 3], the elements { E T | T ∈ T udn ( λ ) for some ( f, λ ) ∈ Λ + d,n } form a complete set of pairwise orthogonal primitive idempotents of B d,n . More-over, the elements X , . . . , X n generate a maximal commutative subalgebra of B d,n . Wealso have X k E T = E T X k = c( T | k ) E T . (2.14)3. Fusion procedure for cyclotomic BMW algebras
Assume that ( f, λ ) ∈ Λ + d,n and that T = ( T , . . . , T n ) is an n -updown λ -tableau. Set µ = T n − and U = ( T , . . . , T n − ) as an updown µ -tableau. Let θ be the box that isaddable to or removable from µ to get λ . For simplicity, we set c k := c( T | k ) . By (2.13),we can rewrite E T inductively as follows: E T = E U ( X n − b ) · · · ( X n − b k )(c n − b ) · · · (c n − b k ) , (3.1)where b , . . . , b k are the contents of all boxes except θ , which can be addable to or remov-able from µ to get a d -partition.We denote by { Λ , . . . , Λ h } the set of all d -partitions obtained from µ by adding a boxor removing one. Set T j := ( T , . . . , T n − , Λ j ) for 1 ≤ j ≤ h. Note that T ∈ { T , . . . , T h } . Since B d,n is generic, hence it is semisimple. By [RuSi3, (4.16)] we have E U = h X j =1 E T j . (3.2) WEIDENG CUI
The property (2.14) implies that the following rational function E U u − c n u − X n (3.3)is regular at u = c n , and by (3.2), we get E U u − c n u − X n (cid:12)(cid:12)(cid:12) u =c n = E T . (3.4)For 1 ≤ i ≤ n − , we set Q i ( u, v ; c ) := T i + q − q − ρ − cuv − q − q − qcuv E i . (3.5)Let φ ( u ) := cuX − ρu − X . For k = 2 , . . . , n , we set φ k ( u , . . . , u k − , u ) := Q k − ( u k − , u ; c ) φ k − ( u , . . . , u k − , u ) T k − ( u k − , u )= Q k − ( u k − , u ; c ) · · · Q ( u , u ; c ) φ ( u ) T ( u , u ) · · · T k − ( u k − , u ) . (3.6)From now on, we always set c := − q − . The following lemma is inspired by [IMOg2,Lemma 1] and can be proved similarly.
Lemma 3.1.
Assume that n ≥ . We have E U φ n (c , . . . , c n − , u ) n − Y r =1 f ( u, c r ) − = E U cuX n − ρu − X n . (3.7) Proof.
We shall prove (3.7) by induction on n. For n = 1 , the situation is trivial.We set φ ′ n (c , . . . , c n − , u )= Q n − (c n − , u ; c ) · · · Q (c , u ; c ) φ ( u ) T ( u, c ) − · · · T n − ( u, c n − ) − . (3.8)By (2.4) and (3.8), in order to show (3.7), it suffices to prove that E U φ ′ n (c , . . . , c n − , u ) = E U cuX n − ρu − X n . (3.9)By the induction hypothesis, it boils down to proving the following equality: E U Q n − (c n − , u ; c ) cuX n − − ρu − X n − T n − ( u, c n − ) − = E U cuX n − ρu − X n . (3.10)Since X n commutes with E U , we can rewrite (3.10) as follows: E U ( u − X n ) Q n − (c n − , u ; c )( cuX n − − ρ )= E U ( cuX n − ρ ) T n − ( u, c n − )( u − X n − ) . (3.11)By (2.3) and (3.5), the equality (3.11) becomes E U ( u − X n ) (cid:16) T n − + q − q − ρ − cu c n − − q − q − qcu c n − E n − (cid:17) ( cuX n − − ρ )= E U ( cuX n − ρ ) (cid:16) T n − + ( q − q − ) u c n − − u + ( q − q − ) uu + ρq c n − E n − (cid:17) ( u − X n − ) . (3.12) USION PROCEDURE 7
By definition, we have T n − X n − = X n T n − − ( q − q − ) X n + ( q − q − ) X n E n − . Thus,we get that (3.12) is equivalent to E U ( u − X n ) (cid:16) cu ( X n T n − − ( q − q − ) X n + ( q − q − ) X n E n − ) − ρT n − + ( q − q − ) ρ + q − q − qcu c n − E n − ( cuX n − − ρ ) (cid:17) = E U ( cuX n − ρ ) (cid:16) − X n T n − + ( q − q − ) X n − ( q − q − ) X n E n − + uT n − − ( q − q − ) u + ( q − q − ) uu + ρq c n − E n − ( u − X n − ) (cid:17) . (3.13)Since we have cu X n T n − − cuX n T n − − ( q − q − ) cuX n ( u − X n ) − ρ ( u − X n )( T n − − ( q − q − ))= − cuX n T n − + ρX n T n − + ( q − q − )( cuX n − ρ ) X n + u ( cuX n − ρ )( T n − − ( q − q − )) , it is easy to see that the equality (3.13) comes down to the following equality: E U ( u − X n ) (cid:16) cuX n E n − + 11 + qcu c n − E n − ( cuX n − − ρ ) (cid:17) = E U ( cuX n − ρ ) (cid:16) − X n E n − + uu + ρq c n − E n − ( u − X n − ) (cid:17) . (3.14)By definition, we have E U X n − = c n − E U . Hence, we get E U X n E n − = n − E U E n − by(2.2). According to this, by comparing the coefficients of the terms involving E U E n − X n − ,we see that it suffices to show that11 + qcu c n − · cu c n − − cu c n − = uu + ρq c n − · ρ c n − − cu c n − . (3.15)By comparing the coefficients of the terms involving E U E n − , it suffices to show that cu − ρ c n − + 11 + qcu c n − · ρ − ρu c n − c n − = uu + ρq c n − · cu − ρu c n − c n − . (3.16)Noting that c = − q − , it is easy to verify that (3.15) and (3.16) are true. Thus, (3.14)holds. The lemma is proved. (cid:3) Let φ ( u ) := ( u − v ) · · · ( u − v d ) cuX − ρu − X . For k = 2 , . . . , n , we set φ k ( u , . . . , u k − , u ) := Q k − ( u k − , u ; c ) φ k − ( u , . . . , u k − , u ) T k − ( u k − , u )= Q k − ( u k − , u ; c ) · · · Q ( u , u ; c ) φ ( u ) T ( u , u ) · · · T k − ( u k − , u ) . (3.17)We also define the following rational function:Φ( u , . . . , u n ) := φ ( u ) · · · φ n − ( u , . . . , u n − ) φ n ( u , . . . , u n ) . (3.18)Recall that the integers p , . . . , p n associated to T have been defined as in (2.10) or (2.11).Now we can state the main result of this paper. WEIDENG CUI
Theorem 3.2.
The idempotent E T of B d,n corresponding to an n -updown λ -tableau T can be derived by the following consecutive evaluations : E T = 1 f ( T ) (cid:16) n Y k =1 ( u k − c k ) p k cu k c k − ρ (cid:17) Φ( u , . . . , u n ) (cid:12)(cid:12)(cid:12) u =c · · · (cid:12)(cid:12)(cid:12) u n =c n . (3.19) Proof.
We shall prove the theorem by induction on n. For n = 1 , we have p = 0 byProposition 2.2. Thus, we get that the right-hand side of (3.19) is equal to1 f ( T ) ( u − v ) · · · ( u − v d ) cu c − ρ cu X − ρu − X (cid:12)(cid:12)(cid:12) u =c = 1 f ( T ) ( u − v ) · · · ( u − v d ) u − c u − c cu c − ρ cu X − ρu − X (cid:12)(cid:12)(cid:12) u =c . (3.20)Moreover, by (2.12), we have f ( T ) = Y ≤ k ≤ d ; v k =c (c − v k ) . Therefore, it is easy to see that (3.20) is equal to E T by (2.14) and (3.4).For n ≥ , by the induction hypothesis we can write the right-hand side of (3.19) asfollows: f ( U ) f ( T ) ( u n − c n ) p n cu n c n − ρ E U φ n (c , . . . , c n − , u n ) (cid:12)(cid:12)(cid:12) u n =c n . (3.21)Note that φ n (c , . . . , c n − , u n ) = ( u n − v ) · · · ( u n − v d ) φ n (c , . . . , c n − , u n ) . By (3.7), wecan rewrite the expression (3.21) as f ( U ) f ( T ) ( u n − c n ) p n cu n c n − ρ ( u n − v ) · · · ( u n − v d ) n − Y r =1 f ( u n , c r ) E U cu n X n − ρu n − X n (cid:12)(cid:12)(cid:12) u n =c n . (3.22)By (2.12), we see that f ( U ) f ( T ) ( u n − v ) · · · ( u n − v d ) n − Y r =1 f ( u n , c r )( u n − c n ) p n − = f ( U ) f ( T ) ( u n − v ) · · · ( u n − v d ) n − Y r =1 ( u n − q c r )( u n − q − c r )( u n − c r ) ( u n − c n ) p n − is regular at u n = c n and is equal to 1 . Thus, the expression (3.22) equals E U u n − c n u n − X n cu n X n − ρcu n c n − ρ (cid:12)(cid:12)(cid:12) u n =c n . (3.23)By (3.4), we see that (3.23) is equal to E T cu n X n − ρcu n c n − ρ (cid:12)(cid:12)(cid:12) u n =c n . (3.24)By (2.14), we have E T X n = c n E T . Thus, we get that the expression (3.24), that is, theright-hand side of (3.19) equals E T . (cid:3) USION PROCEDURE 9
Remark 3.3.
Let H d,n be the cyclotomic Hecke algebra defined in [AK]. It has beenproved in [RuXu, Proposition 4.1] that H d,n is isomorphic to the quotient of B d,n by thetwo-sided ideal generated by all E i . In the process of taking quotient, the parameter ρ disappears; however, the parameter c is reserved and can be arbitrary. If we replace the T i ( u, v ) , Q i ( u, v ; c ) , φ ( u ) in (3.6) with T i ( u, v ) = T i + ( q − q − ) uv − u , Q i ( u, v ; c ) := T i + q − q − cuv − , ψ ( u ) := cuX − u − X , it is easy to see that the analogue of Lemma 3.1 holds.Let ψ ( u ) := ( u − v ) · · · ( u − v d ) cuX − u − X , and for k = 2 , . . . , n , set ψ k ( u , . . . , u k − , u ) := Q k − ( u k − , u ; c ) ψ k − ( u , . . . , u k − , u ) T k − ( u k − , u ) . We also define a rational function byΥ( u , . . . , u n ) := ψ ( u ) · · · ψ n − ( u , . . . , u n − ) ψ n ( u , . . . , u n ) . Then it is easy to see that the analogue of Theorem 3.2 is true. Thus, we get a one-parameter family of the fusion procedures for cyclotomic Hecke algebras, generalizing theresults obtained in [OgPA2].4.
Appendix. Fusion procedure for cyclotomic Nazarov-Wenzl algebras
When studying the representations of Brauer algebras, Nazarov [Na1] introduced aclass of infinite dimensional algebras under the name affine Wenzl algebras. In orderto study finite dimensional irreducible representations of affine Wenzl algebras, Ariki,Mathas and Rui [AMR] defined the finite dimensional quotients of them, known as thecyclotomic Nazarov-Wenzl algebras. Cyclotomic Nazarov-Wenzl algebras are related todegenerate cyclotomic Hecke algebras just in the same way that cyclotomic BMW algebrasare connected with cyclotomic Hecke algebras. Cyclotomic Nazarov-Wenzl algebras havebeen studied by many authors; see [Go3-4, RuSi1-2, Xu] and so on.4.1.
Cyclotomic Nazarov-Wenzl algebras.Definition 4.1.
Suppose that K is an algebraically closed field containing ω j (0 ≤ j ≤ d − v i (1 ≤ i ≤ d ), and the invertible element 2 . Fix n ≥ . The cyclotomic Nazarov-Wenzl algebra W d,n is the K -algebra generated bythe elements S i , E i (1 ≤ i ≤ n −
1) and X j (1 ≤ j ≤ n ) satisfying the following relations:(1) (Involutions) S i = 1 for 1 ≤ i ≤ n − . (2) (Idempotent relations) E i = ω E i for 1 ≤ i ≤ n − . (3) (Affine braid relations)(a) S i S i +1 S i = S i +1 S i S i +1 and S i S j = S j S i if | i − j | ≥ . (b) S i X j = X j S i if j = i, i + 1 . (4) (Tangle relations)(a) E i E i ± E i = E i . (b) S i S i ± E i = E i ± E i and E i S i ± S i = E i E i ± . (c) For 1 ≤ k ≤ d − , E X k E = ω k E . (5) (Untwisting relations) S i E i = E i S i = E i for 1 ≤ i ≤ n − . (6) (Skein relations) S i X i − X i +1 S i = E i − ≤ i ≤ n − . (7) (Anti-symmetry relations) E i ( X i + X i +1 ) = ( X i + X i +1 ) E i = 0 for 1 ≤ i ≤ n − . (8) (Commutative relations)(a) S i E j = E j S i and E i E j = E j E i if | i − j | ≥ . (b) E i X j = X j E i if j = i, i + 1 . (c) X i X j = X j X i for 1 ≤ i, j ≤ n. (9) (Cyclotomic relation) ( X − v )( X − v ) · · · ( X − v d ) = 0 . We define the following elements: S i ( u, v ) = S i + 1 v − u − v − u + ω − E i for 1 ≤ i ≤ n − . (4.1)By using the fact that E i = ω E i , we can easily get S i ( u, v ) S i ( v, u ) = g ( u, v ) for 1 ≤ i ≤ n − , (4.2)where g ( u, v ) = g ( v, u ) = ( u − v + 1)( u − v − u − v ) . (4.3)4.2. Combinatorics.
Suppose that ( f, λ ) ∈ Λ + d,n and s = ( s , . . . , s n ) ∈ T udn ( λ ) . We candefine the integers d sk , d sk , g sk , g sk and some integers p , . . . , p n associated to s in exactlythe same way as those related to some T defined in Subsection 2.2. We shall follow thenotations and only emphasize the differences.Set c( s | k ) = ( v s + j − i if s k = s k − ∪ (( i, j ) , s ) , − v s + i − j if s k − = s k ∪ (( i, j ) , s ) . (4.4)Given a box β = (( i, j ) , s ) , we define the content of it byc( U | β ) = ( v s + j − i if β is an addable box of s , − v s + i − j if β is a removable box of s . (4.5)Assume that ( f, λ ) ∈ Λ + d,n , t = ( t , . . . , t n ) is an n -updown λ -tableau and that u =( t , . . . , t n − ) . We then define the element g ( t ) inductively by g ( t ) = g ( u ) ψ ( u , t ) , (4.6)where ψ ( u , t ) = Y k = k n k ∈ Z ( k n − k ) g snk Y ≤ t ≤ d ; t = s n k ∈ Z ( v s n − v t + k n − k ) g tk Y ≤ r ≤ dk ∈ Z ( v s n + v r + k n + k ) g rk if t n is obtained from t n − by adding a box (( i n , j n ) , s n ), where k n = j n − i n ; ψ ( u , t ) = Y k = k ′ n k ∈ Z ( − k ′ n + k ) g s ′ nk Y ≤ t ≤ d ; t = s ′ n k ∈ Z ( − v s ′ n + v t − k ′ n + k ) g tk Y ≤ r ≤ dk ∈ Z ( − v s ′ n − v r − k ′ n − k ) g rk if t n is obtained from t n − by removing a box (( i ′ n , j ′ n ) , s ′ n ), where k ′ n = j ′ n − i ′ n . USION PROCEDURE 11
The following proposition is inspired by [IM, Proposition 3.3] and can be proved simi-larly.
Proposition 4.2. If λ is a d -partition of n and t = ( t , . . . , t n ) is an n -updown λ -tableau, then p , . . . , p n are all equal to zero, and g ( t ) is exactly equal to Θ λ ( Q ) − definedin [ZL , (3 . when d = m and v s = q s for ≤ s ≤ m. Idempotents of W d,n . Following [AMR, Definition 4.3], we say that W d,n is genericif the parameters v i , 1 ≤ i ≤ d , satisfy the conditions (1) the characteristic p of K satisfies p = 0 or p > n ; (2) | r | ≥ n whenever there exists r ∈ Z such that either v i ± v j = r and i = j, or 2 v i = r. Following [Go3, Definition 4.2], we say that W d,n isadmissible if the set { E , E X , . . . , E X d − } is linearly independent in B d, . It has beenproved by Goodman [Go3, Theorem 5.2] that this admissible condition coincides with the u -admissible condition defined in [AMR, Definition 3.6].From now on, we always assume that W d,n is generic and admissible. Thus, by [AMR,Lemma 4.4], we have s = t if and only if c( s | k ) = c( t | k ) for all 1 ≤ k ≤ n. Therefore,the set { X , . . . , X n } , as a family of JM-elements for W d,n in the abstract sense defined in[Ma, Definition 2.4], satisfies the separation condition associated to the cellular basis of W d,n constructed in [AMR, Theorem 7.17]. In particular, we can construct the primitiveidempotents of W d,n following the arguments in [Ma, Section 3].For each 1 ≤ k ≤ n, we define the following set: R ( k ) := { c( s | k ) | s ∈ T udn ( λ ) for some ( f, λ ) ∈ Λ + d,n } . Suppose that ( f, λ ) ∈ Λ + d,n and t ∈ T udn ( λ ) . We set E t = n Y k =1 (cid:18) Y a ∈ R ( k ) a =c( t | k ) X k − a c( t | k ) − a (cid:19) . (4.7)By standard arguments in [Ma, Section 3], the elements { E t | t ∈ T udn ( λ ) for some ( f, λ ) ∈ Λ + d,n } form a complete set of pairwise orthogonal primitive idempotents of W d,n . Moreover,the elements X , . . . , X n generate a maximal commutative subalgebra of W d,n . We alsohave X k E t = E t X k = c( t | k ) E t . (4.8)4.4. Fusion procedure for cyclotomic Nazarov-Wenzl algebras.
Assume that ( f, λ ) ∈ Λ + d,n and that t = ( t , . . . , t n ) is an n -updown λ -tableau. Set µ = t n − and u =( t , . . . , t n − ) as an updown µ -tableau. Let θ be the box that is addable to or removablefrom µ to get λ . For simplicity, we set c k := c( t | k ) . By (4.7), we can rewrite E t inductivelyas follows: E t = E t ( X n − a ) · · · ( X n − a k )(c n − a ) · · · (c n − a k ) , (4.9)where a , . . . , a k are the contents of all boxes except θ , which can be addable to orremovable from µ to get a d -partition.We denote by { ∆ , . . . , ∆ e } the set of all d -partitions obtained from µ by adding a boxor removing one. Set S j := ( t , . . . , t n − , ∆ j ) for 1 ≤ j ≤ e. Note that t ∈ { S , . . . , S e } . Since W d,n is generic, hence it is semisimple. By [AMR, Theorem 5.3 a)] we have E u = e X j =1 E S j . (4.10)The equality (4.8) implies that the following rational function E u u − c n u − X n (4.11)is regular at u = c n , and by (4.10), we get E u u − c n u − X n (cid:12)(cid:12)(cid:12) u =c n = E t . (4.12)For 1 ≤ i ≤ n − , we set R i ( u, v ; c ) := S i + 1 u + v + c − u + v E i . (4.13)Let ϕ ( u ) := u + X + cu − X . For k = 2 , . . . , n , we set ϕ k ( u , . . . , u k − , u ) := R k − ( u k − , u ; c ) ϕ k − ( u , . . . , u k − , u ) S k − ( u k − , u )= R k − ( u k − , u ; c ) · · · R ( u , u ; c ) ϕ ( u ) S ( u , u ) · · · S k − ( u k − , u ) . (4.14)From now on, we always set c := 1 − ω . The following lemma is inspired by [IMOg2,Lemma 1] and can be proved similarly.
Lemma 4.3.
Assume that n ≥ . We have E u ϕ n (c , . . . , c n − , u ) n − Y r =1 g ( u, c r ) − = E u u + X n + cu − X n . (4.15) Proof.
We shall prove (4.15) by induction on n. For n = 1 , the situation is trivial.We set ϕ ′ n (c , . . . , c n − , u )= R n − (c n − , u ; c ) · · · R (c , u ; c ) ϕ ( u ) S ( u, c ) − · · · S n − ( u, c n − ) − . (4.16)By (4.2) and (4.16), in order to show (4.15), it suffices to prove that E u ϕ ′ n (c , . . . , c n − , u ) = E u u + X n + cu − X n . (4.17)By the induction hypothesis, it boils down to proving the following equality: E u R n − (c n − , u ; c ) u + X n − + cu − X n − S n − ( u, c n − ) − = E u u + X n + cu − X n . (4.18)Since X n commutes with E u , we can rewrite (4.18) as follows: E u ( u − X n ) R n − (c n − , u ; c )( u + X n − + c )= E u ( u + X n + c ) S n − ( u, c n − )( u − X n − ) . (4.19) USION PROCEDURE 13
By (4.1) and (4.13), the equality (4.19) becomes E u ( u − X n ) (cid:16) S n − + 1c n − + u + c − n − + u E n − (cid:17) ( u + X n − + c )= E u ( u + X n + c ) (cid:16) S n − + 1c n − − u − n − − u + ω − E n − (cid:17) ( u − X n − ) . (4.20)By definition, we have S n − X n − = X n S n − + E n − − . Thus, we get that (4.20) isequivalent to E u ( u − X n ) (cid:16) uS n − + ( X n S n − + E n − −
1) + cS n − + 1 − n − + u E n − ( u + X n − + c ) (cid:17) = E u ( u + X n + c ) (cid:16) uS n − − ( X n S n − + E n − − − − n − − u + ω − E n − ( u − X n − ) (cid:17) . (4.21)It is easy to see that the equality (4.21) comes down to the following equality:( c + 2 u ) E u − E u ( u − X n ) 1c n − + u E n − ( u + X n − + c )= − E u ( u + X n + c ) 1c n − − u + ω − E n − ( u − X n − ) . (4.22)By definition, we have E u X n − = c n − E u . Hence, we get E u X n E n − = − c n − E u E n − by definition. According to this, by comparing the coefficients of the terms involving E u E n − X n − , we see that it suffices to show that − u − c n − c n − + u = u − c n − + c c n − − u + ω − . (4.23)By comparing the coefficients of the terms involving E u E n − , it suffices to show that( c + 2 u ) + − ( c + u )(c n − + u )c n − + u = u ( − u + c n − − c )c n − − u + ω − . (4.24)Noting that c = 1 − ω , it is easy to verify that (4.23) and (4.24) are true. Thus, (4.22)holds. The lemma is proved. (cid:3) Let ϕ ( u ) := ( u − v ) · · · ( u − v d ) u + X + cu − X . For k = 2 , . . . , n , we set ϕ k ( u , . . . , u k − , u ) := R k − ( u k − , u ; c ) ϕ k − ( u , . . . , u k − , u ) S k − ( u k − , u )= R k − ( u k − , u ; c ) · · · R ( u , u ; c ) ϕ ( u ) S ( u , u ) · · · S k − ( u k − , u ) . (4.25)We also define the following rational function:Ψ( u , . . . , u n ) := ϕ ( u ) · · · ϕ n − ( u , . . . , u n − ) ϕ n ( u , . . . , u n ) . (4.26)Recall that the integers p , . . . , p n associated to t have been defined as in (2.10) or (2.11).Now we can state the main result of this paper. Theorem 4.4.
The idempotent E t of W d,n corresponding to an n -updown λ -tableau t canbe derived by the following consecutive evaluations : E t = 1 g ( t ) (cid:16) n Y k =1 ( u k − c k ) p k u k + c k + c (cid:17) Ψ( u , . . . , u n ) (cid:12)(cid:12)(cid:12) u =c · · · (cid:12)(cid:12)(cid:12) u n =c n . (4.27) Proof.
We shall prove the theorem by induction on n. For n = 1 , we have p = 0 byProposition 4.2. Thus, we get that the right-hand side of (4.27) is equal to1 g ( t ) ( u − v ) · · · ( u − v d ) u + c + c u + X + cu − X (cid:12)(cid:12)(cid:12) u =c = 1 g ( t ) ( u − v ) · · · ( u − v d ) u − c u − c u + c + c u + X + cu − X (cid:12)(cid:12)(cid:12) u =c . (4.28)Moreover, by (4.6), we have g ( t ) = Y ≤ k ≤ d ; v k =c (c − v k ) . Therefore, it is easy to see that (4.28) is equal to E t by (4.8) and (4.12).For n ≥ , by the induction hypothesis we can write the right-hand side of (4.27) asfollows: g ( u ) g ( t ) ( u n − c n ) p n u n + c n + c E u ϕ n (c , . . . , c n − , u n ) (cid:12)(cid:12)(cid:12) u n =c n . (4.29)Note that ϕ n (c , . . . , c n − , u n ) = ( u n − v ) · · · ( u n − v d ) ϕ n (c , . . . , c n − , u n ) . By (4.15), wecan rewrite the expression (4.29) as g ( u ) g ( t ) ( u n − c n ) p n u n + c n + c ( u n − v ) · · · ( u n − v d ) n − Y r =1 g ( u n , c r ) E u u n + X n + cu n − X n (cid:12)(cid:12)(cid:12) u n =c n . (4.30)By (4.6), we see that g ( u ) g ( t ) ( u n − v ) · · · ( u n − v d ) n − Y r =1 g ( u n , c r )( u n − c n ) p n − = g ( u ) g ( t ) ( u n − v ) · · · ( u n − v d ) n − Y r =1 ( u n − c r + 1)( u n − c r − u n − c r ) ( u n − c n ) p n − is regular at u n = c n and is equal to 1 . Thus, the expression (4.30) equals E u u n − c n u n − X n u n + X n + cu n + c n + c (cid:12)(cid:12)(cid:12) u n =c n . (4.31)By (4.12), we see that (4.31) is equal to E t u n + X n + cu n + c n + c (cid:12)(cid:12)(cid:12) u n =c n . (4.32)By (4.8), we have E t X n = c n E t . Thus, we get that the expression (4.32), that is, theright-hand side of (4.27) equals E t . (cid:3) USION PROCEDURE 15
Remark 4.5.
Let D d,n be the degenerate cyclotomic Hecke algebra. It has been provedin [AMR, Proposition 7.2] that D d,n is isomorphic to the quotient of W d,n by the two-sidedideal generated by all E i . In the process of taking quotient, the parameter ω disappears;however, the parameter c is reserved and can be arbitrary. If we replace the S i ( u, v ) ,R i ( u, v ; c ) , ϕ ( u ) in (4.14) with S i ( u, v ) = S i + 1 v − u , R i ( u, v ; c ) := S i + 1 u + v + c , χ ( u ) := u + X + cu − X , it is easy to see that the analogue of Lemma 4.3 holds.Let χ ( u ) := ( u − v ) · · · ( u − v d ) u + X + cu − X , and for k = 2 , . . . , n , set χ k ( u , . . . , u k − , u ) := R k − ( u k − , u ; c ) χ k − ( u , . . . , u k − , u ) S k − ( u k − , u ) . We also define a rational function byΩ( u , . . . , u n ) := χ ( u ) · · · χ n − ( u , . . . , u n − ) χ n ( u , . . . , u n ) . Then it is easy to see that the analogue of Theorem 4.4 is true. Thus, we get a one-parameter family of the fusion procedures for degenerate cyclotomic Hecke algebras, gen-eralizing the results obtained in [ZL].
Acknowledgements.
The author is deeply indebted to Dr. Shoumin Liu for posing thequestion about fusion procedures for cyclotomic Nazarov-Wenzl algebras to him.
References [AK] S. Ariki and K. Koike, A Hecke algebra of ( Z /r Z ) ≀ S n and construction of its irreducible repre-sentations, Adv. Math. (1994) 216-243.[AMR] S. Ariki, A. Mathas and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. (2006)47-134[BW] J.S. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. (1989), 249-273.[Ch] I. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra,Funct. Anal. Appl. (1986) 76-78.[C1] W. Cui, Fusion procedure for Yokonuma-Hecke algebras, preprint, arXiv: 1405.4477.[C2] W. Cui, Cellularity of cyclotomic Yokonuma-Hecke algebras, preprint, arXiv: 1506.07321.[C3] W. Cui, A note on degenerate cyclotomic Yokonuma-Hecke algebras, preprint, arXiv: 1208.4884.[DJM] R. Dipper, G. James and A. Mathas, Cyclotomic q -Schur algebras, Math. Z. (1998) 385-416.[Go1] F.M. Goodman, Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra. (2009), 3299-3320.[Go2] F.M. Goodman, Comparison of admissible conditions for cyclotomic Birman-Wenzl-Murakamialgebras, J. Pure Appl. Algebra (2010), 2009-2016.[Go3] F.M. Goodman, Admissibility conditions for degenerate cyclotomic BMW algebras, Comm. Alge-bra (2011) 452-461.[Go4] F.M. Goodman, Remarks on cyclotomic and degenerate cyclotomic BMW algebras, J. Algebra. (2012), 13-37.[GoHM1] F.M. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I.Freeness and realization as tangle algebras, J. Knot Theory Ramifications (2009) 1089-1127.[GoHM2] F.M. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras, II:admissibility relations and freeness, Algebr. Represent. Theory (2011) 1-39.[HO] R. H¨aring-Oldenburg, Cyclotomic Birman-Murakami-Wenzl algebras, J. Pure Appl. Algebra (2001), 113-144.[Juc] A. Jucys, On the Young operators of symmetric groups, Litovsk. Fiz. Sb. (1966) 163-180. [Ka] L.H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. (2) (1990), 417-471.[IM] A. Isaev and A. Molev, Fusion procedure for the Brauer algebra, Algebra Anal. (2010) 142-154.[IMO] A. Isaev, A. Molev and A. Os’kin, On the idempotents of Hecke Algebras, Lett. Math. Phys. (2008) 79-90.[IMOg1] A. Isaev, A. Molev and O. Ogievetsky, A new fusion procedure for the Brauer algebra andevaluation homomorphisms, Int. Math. Res. Not. (2012) 2571-2606.[IMOg2] A. Isaev, A. Molev and O. Ogievetsky, Idempotents for Birman-Murakami-Wenzl algebras andreflection equation, Adv. Theor. Math. Phys. (2014) 1-25.[Ma] A. Mathas, Seminormal forms and Gram determinants for cellular algebras, J. Reine Angew. Math. (2008) 141-173.[Mo] A. Molev, On the fusion procedure for the symmetric group, Rep. Math. Phys. (2008) 181-188.[MT] H. Morton and P. Traczyk, Knots and algebras, in: E. Martin-Peindador, A. Rodez Usan (Eds.),Contribuciones Matematicas en homenaje al profesor D. Antonio Plans Sanz de Bremond, Univer-sity of Zaragoza, Zaragoza, 1990, pp. 201-220.[Mu] J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. (1987), 745-758.[Na1] M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra (1996)664-693.[Na2] M. Nazarov, Yangians and Capelli identities, In: Olshanski, G.I. (ed.) Kirillov’s seminar on rep-resentation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI(1998) pp. 139-163.[Na3] M. Nazarov, Mixed hook-length formula for degenerate affine Hecke algebras, Lect. Notes Math. (2003) 223-236.[Na4] M. Nazarov, A mixed hook-length formula for affine Hecke algebras, European J. Combin. (2004) 1345-1376.[OgPA1] O. Ogievetsky and L. Poulain d’Andecy, Fusion procedure for Coxeter groups of type B andcomplex reflection groups G ( m, , n ), Proc. Amer. Math. Soc. (2014) 2929-2941.[OgPA2] O. Ogievetsky and L. Poulain d’Andecy, Fusion procedure for cyclotomic Hecke algebras, SIGMASymmetry Integrability Geom. Methods Appl. (2014) 13 pp.[OrRa] R. Orellana and A. Ram, Affine braids, Markov traces and the category O , Algebraic groups andhomogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai,2007.[PA] L. Poulain d’Andecy, Fusion procedure for wreath products of finite groups by the symmetricgroup, Algebr. Represent. Theory (2014) 809-830.[RuSi1] H. Rui and M. Si, On the structure of cyclotomic Nazarov-Wenzl algebras, J. Pure Appl. Algebra (2008) 2209-2235.[RuSi2] H. Rui and M. Si, Non-vanishing Gram determinants for cyclotomic Nazarov-Wenzl and Birman-Murakami-Wenzl algebras, J. Algebra (2011) 188-219.[RuSi3] H. Rui and M. Si, The representations of cyclotomic BMW algebras, II. Algebr. Represent. Theory (2012) 551-579.[RuXu] H. Rui and J. Xu, The representations of cyclotomic BMW algebras, J. Pure Appl. Algebra (2009), 2262-2288.[Si] M. Si, Morita equivalence for cyclotomic BMW algebras, J. Algebra (2015) 573-591.[WiYu1] S. Wilcox and S. Yu, The cyclotomic BMW algebra associated with the two string type B braidgroup, Comm. Algebra (2011), 4428-4446.[WiYu2] S. Wilcox and S. Yu, On the freeness of the cyclotomic BMW algebras: admissibility and anisomorphism with the cyclotomic Kauffman tangle algebra, (2009) (preprint).[WiYu3] S. Wilcox and S. Yu, On the cellularity of the cyclotomic Birman-Murakami-Wenzl algebras, J.London Math. Soc. (2012), 911-929.[Xu] X. Xu, Decomposition numbers of cyclotomic NW and BMW algebras, J. Pure Appl. Algebra (2013), 1037-1053.[Yu] S. Yu, The cyclotomic Birman-Murakami-Wenzl algebras, PhD thesis, University of Sydney, 2007. USION PROCEDURE 17 [ZL] D. Zhao and Y. Li, Fusion procedure for degenerate cyclotomic Hecke algebras, Algebr. Represent.Theory (2015) 449-461.[Eny1] J. Enyang, Cellular bases for the Brauer and Birman-Murakami-Wenzl algebras, J. Algebra (2004) 413-449.[Eny2] J. Enyang, Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzlalgebras, J. Algebraic Combin. (2007) 291-341.[HuXi] J. Hu and Z. Xiao, On tensor spaces for Birman-Murakami-Wenzl algebras, J. Algebra (2010)2893-2922.[Hu] J. Hu, BMW algebra, quantized coordinate algebra and type C Schur-Weyl duality, Represent.Theory (2011), 1-62.[LeRa] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations ofcentralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras, Adv. Math. (1997) 1-94.[MW] H. Morton and A. Wassermann, A basis for the Birman-Wenzl algebra, unpublished manuscript,1989, revised 2010, arXiv:1012.3116, 1-33.[RuSi4] H. Rui and M. Si, Gram determinants and semisimplicity criteria for Birman-Wenzl algebras, J.Reine Angew. Math. (2009) 153-179.[RuSi5] H. Rui and M. Si, Blocks of Birman-Murakami-Wenzl algebras, Int. Math. Res. Not. IMRN (2011)no. 2, 452-486.[RuSi6] H. Rui and M. Si, Singular parameters for the Birman-Murakami-Wenzl algebra, J. Pure Appl.Algebra (2012) 1295-1305.[RuSo] H. Rui and L. Song, Decomposition matrices of Birman-Murakami-Wenzl algebras, J. Algebra (2015) 246-271.[Xi] C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math. (2000), 280-298. School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China.
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