Remarks on Cyclotomic and Degenerate Cyclotomic BMW Algebras
aa r X i v : . [ m a t h . R T ] M a y Journal of Algebra (2012) 13-37
REMARKS ON CYCLOTOMIC AND DEGENERATE CYCLOTOMIC BMWALGEBRAS
FREDERICK M. GOODMAN
Abstract.
We relate the structure of cyclotomic and degenerate cyclotomic BMW algebras, forarbitrary parameter values, to that for admissible parameter values. In particular, we show thatthese algebras are cellular. We characterize those parameter sets for affine BMW algebras over analgebraically closed field that permit the algebras to have non–trivial cyclotomic quotients. Introduction
This paper is a contribution to the study of affine and degenerate affine Birman–Wenzl–Murakami(BMW) algebras. In order to study the finite dimensional representation theory of these infinitedimensional algebras, one introduces cyclotomic quotients, which are BMW analogues of cyclotomicand degenerate cyclotomic Hecke algebras (see [2, 1, 15]).A peculiar feature of the cyclotomic algebras is that the parameters cannot be chosen arbitrarily;that is, unless the parameters satisfy certain relations, the algebras (defined over a field) collapseto cyclotomic or degenerate cyclotomic Hecke algebras. These “obligatory” conditions did not seemadequate at first to develop the representation theory. Consequently, several authors, notably Ariki,Mathas and Rui [3], Wilcox and Yu [23], and Rui and Xu [21] introduced stronger “admissibility”conditions under which the algebras could be shown to have a well–behaved representation theory.Up until now, the cyclotomic algebras have been studied only under the assumption of admissibilityof the parameters. Despite the successes achieved, this was not satisfactory, since a priori the admis-sibility requirement might be too restrictive to capture the entire finite dimensional representationtheory of the affine algebras.In this paper, we extend the analysis of cyclotomic and degenerate cyclotomic BMW algebras toinclude the case of non-admissible parameters. We show that the structure and representation theoryof the cyclotomic algebras with non-admissible parameters can be derived from that of the algebraswith admissible parameters.1.1.
Background.
Affine and cyclotomic BMW algebras and their degenerate versions arise natu-rally by several different “affinization” processes. One such process amounts to making the Jucys-Murphy elements in the ordinary BMW or Brauer algebras into variables, retaining the relationsbetween these elements and the standard generators of the BMW or Brauer algebras. This pointof view was stressed by Nazarov, in defining degenerate affine BMW algebras [17]. For the BMWalgebras, there is a geometric affinization process: The ordinary BMW algebras can be realized as al-gebras of tangles in the disc cross the interval, modulo Kauffman skein relations [16]. To affinize thesealgebras, one should replace the disc by the annulus; alternatively, one replaces the ordinary braid
Mathematics Subject Classification. mathoverflow.net . I thank the referee for some helpful suggestions. group by the affine or type B braid group. This is the motivation cited by H¨aring–Oldenburg [14] forintroducing affine and cyclotomic BMW algebras. Finally, Orellana and Ram provide an affinizationof Schur–Weyl duality [18] which produces representations of the affine braid group by ˇ R –matrices ofa quantum group; for symplectic or orthogonal quantum groups, this process yields representationsof cyclotomic BMW algebras (over the complex numbers, with special parameters).As mentioned above, degenerate affine BMW algebras were introduced by Nazarov [17] under thename affine Wenzl algebras . The cyclotomic quotients of these algebras were introduced by Ariki,Mathas, and Rui in [3] and studied further by Rui and Si in [19], under the name cyclotomic Nazarov–Wenzl algebras . Affine and cyclotomic BMW algebras were introduced by H¨aring–Oldenburg in [14]and studied by Goodman and Hauschild Mosley [10, 11, 12, 5], Rui, Xu, and Si [21, 20], Wilcox andYu [23, 24, 22, 25], and Ram, Orellana, Daugherty and Virk [18, 4].The papers cited above study the algebras under the assumption of admissibility. It has beenshown that the algebras with admissible parameters are cellular [3, 24, 25, 21, 20, 5, 8, 9]; simplemodules over a field have been classified [19, 20]; and the non-degenerate cyclotomic BMW algebrashave been shown to be isomorphic to algebras of tangles [11, 12, 24, 22, 25].1.2. Summary of results.
In this note, we show that the structure of the cyclotomic and degeneratecyclotomic BMW algebras for general parameters can be derived from the admissible case. An affine(resp. degenerate affine) BMW algebra A n contains a copy of the finite dimensional BMW algebra(resp. Brauer algebra) B n and an additional “affine” generator y , satisfying several relations withthe generators of B n . A cyclotomic quotient is determined by a polynomial relation(1.1) ( y − u ) · · · ( y − u r ) = 0 . Denote the cyclotomic quotient determined by (1.1) by A n,r ( u , . . . , u r ). Let J n,r ( u , . . . , u r ) denotethe ideal generated by the “contraction” e in A n,r ( u , . . . , u r ). Then we have a short exact sequence(1.2) 0 → J n,r ( u , . . . , u r ) → A n,r ( u , . . . , u r ) → H n ( u , . . . , u r ) → , where H n ( u , . . . , u r ) is the cyclotomic Hecke algebra (resp. degenerate cyclotomic Hecke alge-bra). Admissibility of the parameters means that { e , y e , . . . , y r − e } is linearly independent in A ,r ( u , . . . , u r ); this condition translates into specific conditions on the parameters of the algebrawhich are discussed in Sections 3 and 4. Suppose now that we are working over a field and thatadmissibility fails, but e = 0; then there exists a d with 0 < d < r such that { e , y e , . . . , y d − e } islinearly independent in A ,r ( u , . . . , u r ) but { e , y e , . . . , y d e } is linearly dependent. We say thatthe parameters are d –semi–admissible. We show that there exists a subset { v , . . . , v d } ⊂ { u , . . . , u r } such that(1) A n,d ( v , . . . , v d ) has admissible parameters, and(2) J n,d ( v , . . . , v d ) ∼ = J n,r ( u , . . . , u r ).Thus we have(1.3) 0 → J n,d ( v , . . . , v d ) → A n,r ( u , . . . , u r ) → H n ( u , . . . , u r ) → . Two consequences of this analysis are the following:(1) The cyclotomic algebras are cellular, under very mild hypotheses; in particular, when theground ring is a field, the algebras are always cellular.(2) Every finite dimensional simple module of an affine (resp. degenerate affine) BMW algebraover an algebraically closed field factors through a cyclotomic (resp. degenerate cyclotomic)BMW algebra with admissible parameters, or through a cyclotomic (resp. degenerate cyclo-tomic) Hecke algebra.
YCLOTOMIC BMW ALGEBRAS 3
The latter result is a step towards classifying the simple modules of the affine and degenerate affineBMW algebras over an algebraically closed field.The main results of Ariki, Mathas and Rui [3] regarding degenerate cyclotomic BMW algebrasdepend on the hypothesis that 2 is invertible in the ground ring. We point out in that this hypothesiscan be eliminated; see Section 3.Finally, we characterize those parameter sets for affine BMW algebras over an algebraically closedfield that permit the algebras to have non–trivial cyclotomic quotients, or equivalently, finite dimen-sional modules M with e M = 0; see Theorem 7.9. The analogous result for degenerate affine BMWalgebras was proved in [3]; we have made a minor improvement by removing the restriction that thecharacteristic of the field should be different from 2; see Theorem 7.1.2. Preliminaries
Definition of degenerate affine and cyclotomic BMW algebras.
Fix a positive integer n and a commutative ring S with multiplicative identity. Let Ω = { ω a : a ≥ } be a sequence ofelements of S . Definition 2.1 ([17]) . The degenerate affine BMW algebra b N n,S = b N n,S (Ω) is the unital associative S –algebra with generators { s i , e i , y j : 1 ≤ i < n and 1 ≤ j ≤ n } and relations:(1) (Involutions) s i = 1, for 1 ≤ i < n .(2) (Affine braid relations)(a) s i s j = s j s i if | i − j | > s i s i +1 s i = s i +1 s i s i +1 , for 1 ≤ i < n − s i y j = y j s i if j = i, i + 1.(3) (Idempotent relations) e i = ω e i , for 1 ≤ i < n .(4) (Compression relations) e y a e = ω a e , for a > s i e j = e j s i , and e i e j = e j e i if | i − j | > e i y j = y j e i , if j = i, i + 1,(c) y i y j = y j y i , for 1 ≤ i, j ≤ n .(6) (Tangle relations)(a) e i s i = e i = s i e i , for 1 ≤ i ≤ n − s i e i +1 e i = s i +1 e i , and e i e i +1 s i = e i s i +1 , for 1 ≤ i ≤ n − e i +1 e i s i +1 = e i +1 s i , and s i +1 e i e i +1 = s i e i +1 , for 1 ≤ i ≤ n − e i +1 e i e i +1 = e i +1 , and e i e i +1 e i = e i , for 1 ≤ i ≤ n − s i y i − y i +1 s i = e i −
1, and y i s i − s i y i +1 = e i −
1, for 1 ≤ i < n .(8) (Anti–symmetry relations) e i ( y i + y i +1 ) = 0, and ( y i + y i +1 ) e i = 0, for 1 ≤ i < n . Definition 2.2 ([3]) . Fix an integer r ≥ u , . . . , u r in S . The degenerate cyclotomicBMW algebra N n,S,r = N n,S,r (Ω; u , . . . , u r ) is the quotient of the degenerate affine BMW algebra b N n,S (Ω) by the cyclotomic relation ( y − u ) . . . ( y − u r ) = 0 . Note that, due to the symmetry of the relations, b N n,S has a unique S –linear algebra involution ∗ (that is, an algebra anti-automorphism of order 2) such that e ∗ i = e i , s ∗ i = s i , and y ∗ i = y i for all i .The involution passes to cyclotomic quotients.2.2. Definition of affine and cyclotomic BMW algebras.
Fix an integer n ≥
0, and a commu-tative ring S with invertible elements ρ and q , and a sequence of elements Ω = ( ω a ) a ≥ , satisfying(2.1) ρ − − ρ = ( q − − q )( ω − . FREDERICK M. GOODMAN
Definition 2.3 ([14]) . The affine BMW algebra c W n,S = c W n,S ( ρ, q, Ω) is the unital associative S –algebra with generators y ± , g ± i and e i (1 ≤ i ≤ n −
1) and relations:(1) (Inverses) g i g − i = g − i g i = 1 and y y − = y − y = 1.(2) (Affine braid relations)(a) g i g i +1 g i = g i +1 g i g i +1 and g i g j = g j g i if | i − j | ≥ y g y g = g y g y and y g j = g j y if j ≥ e i = ω e i .(4) (Compression relations) For j ≥ e y j e = ω j e .(5) (Commutation relations)(a) g i e j = e j g i and e i e j = e j e i if | i − j | ≥ y e j = e j y if j ≥ g i e i = e i g i = ρ − e i and e i g i ± e i = ρe i .(b) e i e i ± e i = e i ,(c) g i g i ± e i = e i ± e i and e i g i ± g i = e i e i ± .(7) (Kauffman skein relation) g i − g − i = ( q − q − )(1 − e i ).(8) (Unwrapping relation) e y g y g = e = g y g y e . Definition 2.4 ([14]) . Fix an integer r ≥ u , . . . , u r in S , The cyclo-tomic BMW algebra W n,S,r = W n,S,r ( ρ, q, Ω; u , . . . , u r ) is the quotient of the affine BMW algebra c W n,S ( ρ, q, Ω) by the cyclotomic relation ( y − u ) . . . ( y − u r ) = 0 . As in the degenerate case, c W n,S has a unique S –linear algebra involution ∗ such that e ∗ i = e i and g ∗ i = g i , for all i , and y ∗ = y . The involution passes to cyclotomic quotients.2.3. Admissibility.Notation 2.5.
Let A n,S,r denote either the cyclotomic BMW algebra W n,S,r (with parameters ρ , q ,Ω = ( ω a ) a ≥ , and u , . . . , u r ) or the degenerate cyclotomic BMW algebra N n,S,r (with parametersΩ = ( ω a ) a ≥ and u , . . . , u r ) over a commutative ring S . Let(2.2) p ( u ) = ( u − u ) · · · ( u − u r ) = r X j =0 a j u j . The coefficients a j for j < r are signed elementary symmetric functions in u , . . . , u r , namely a j = ( − r − j ε r − j ( u , . . . , u r ), and a r = 1. Lemma 2.6.
The left ideal A ,S,r e in A ,S,r is equal to the S –span of { e , y e , . . . , y r − e } .Proof. For both the cyclotomic and degenerate cyclotomic BMW algebras, it is easy to check using therelations that the S –span of { e , y e , . . . , y r − e } is invariant under multiplication by the generatorson the left. (cid:3) Lemma 2.7.
Assume that e is not a torsion element over S in A ,S,r . Then the elements ω j , j ≥ satisfy the following recursion relation: (2.3) r X j =0 a j ω j + ℓ = 0 , for all ℓ ≥ . YCLOTOMIC BMW ALGEBRAS 5
Proof.
Multiply the cyclotomic condition: P rj =0 a j y j = 0 by y ℓ , and then multiply from both sidesby e . Use the compression and idempotent relations to obtain P rj =0 a j ω j + ℓ e = 0. Since e is nota torsion element over S , the result follows. (cid:3) Definition 2.8.
Consider the cyclotomic or degenerate cyclotomic BMW algebras over a commuta-tive ring S with suitable parameters. We say that the parameters are admissible if { e , y e , . . . , y r − e } is linearly independent over S in A ,S,r .For both the cyclotomic and degenerate cyclotomic BMW algebras, admissibility as defined abovetranslates into explicit conditions on the parameters. We review this for the two classes of algebrasseparately in the following two sections.3. Admissibility for degenerate cyclotomic BMW algebras
Consider the degenerate cyclotomic BMW algebras N n,S,r with parameters Ω = ( ω a ) a ≥ and u , . . . , u r over a commutative ring S . Define a , . . . , a r − by (2.2). Lemma 3.1 ([7], Lemma 4.1) . Suppose that { e , y e , . . . , y r − e } is linearly independent over S in N ,S,r . Then the parameters satisfy the following relations: (3.1) r − j − X µ =0 ω µ a µ + j +1 = − δ ( r − j is odd ) a j + δ ( j is even ) a j +1 , for ≤ j ≤ r − . We are going to show that admissibility (i.e. linear independence of { e , y e , . . . , y r − e } ) isequivalent to the parameters satisfying conditions (2.3) and (3.1). Lemma 3.2 ([7], Lemma 4.4) . There exist universal polynomials H a ( u , . . . , u r ) for a ≥ , symmet-ric in u , . . . u r , with integer coefficients, such that whenever S is a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r satisfying (2.3) and (3.1), one has (3.2) ω a = H a ( u , . . . , u r ) for a ≥ . Conversely, if ω a = H a ( u , . . . , u r ) for a ≥ , then the parameters satisfy (2.3) and (3.1).Proof. The system of relations (3.1) is a unitriangular linear system of equation for the variables ω , . . . , ω r − . In fact, if we list the equations in reverse order then the matrix of coefficients is a r − a r − a r − a a . . . a r − . Solving the system for ω , . . . , ω r − gives these quantities as polynomial functions of a , . . . , a r − ,thus symmetric polynomials in u , . . . , u r . The recursion relations P rj =0 a j ω j + m = 0, for all m ≥ a ≥ r . The converse is obvious, since the ω a given by (3.2) are the solutions of theequations (2.3) and (3.1). (cid:3) FREDERICK M. GOODMAN
The admissibility condition of Ariki, Mathas, and Rui.
Ariki, Mathas and Rui useda different approach to admissibility for degenerate cyclotomic BMW algebras in their fundamentalwork [3]. Let u , . . . , u r and t be algebraically independent indeterminants over Z . Define symmetricpolynomials q a ( u ) in u , . . . , u r by r Y i =1 u i t − u i t = X a ≥ q a ( u ) t a . The polynomials q a are known as Schur q –functions . Define(3.3) η ± a ( u ) = q a +1 ( u ) ± ( − r − q a ( u ) + 12 δ a, . for a ≥
0. Then (cf. [3], Lemma 3.8)(3.4) X a ≥ η ± a ( u ) t − a = ( 12 − t ) + ( t ± ( − r − r Y i =1 t + u i t − u i , as one sees by expanding the series, using the definition of the Schur q –functions. Ostensibly, η ± a ( u ) ∈ Z [1 / , u , . . . , u r ], but actually: Lemma 3.3. (1) q ( u ) = 1 . (2) For a ≥ , q a ≡ p a ( u ) mod 4 Z [ u , . . . , u r ] , where p a denotes the a –th power sum symmetricfunction. (3) η ± a ( u ) ∈ Z [ u , . . . , u r ] .Proof. Part (1) is obvious. Using the identity:1 + vt − vt = 1 + 2 vt (1 − vt ) − = 1 + 2 vt + 2 v t + 2 v t + . . . , one sees that the coefficient of t a in r Y i =1 u i t − u i t is 2 P i u ai plus a sum of terms divisible by 4. Thisgives (2), and (3) follows as well. (cid:3) Example 3.4.
Consider a ring S of characteristic 2 and u , . . . , u r ∈ S . Then q a ( u , . . . , u r ) = 0 for a ≥
1, but q a ( u , . . . , u r ) = P i u ai ; that is, we consider q a in Z [ u , . . . , u r ], and then evaluate at( u , . . . , u r ) ∈ S r . Furthermore, η +0 ( u , . . . , u r ) = δ ( r is odd) and η + a ( u , . . . , u r ) = p a ( u , . . . , u r ) , for a ≥ Definition 3.5 ([3]) . Let S be a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r .Say that the parameters are ( u , . . . , u r )–admissible, or that Ω is ( u , . . . , u r )–admissible, if for all a ≥ ω a = η + a ( u , . . . , u r ) . Lemma 3.6. (cf. [7], Lemma 5.1)(1) η + a ( u , . . . , u r ) = H a ( u , . . . , u r ) , where H a are the polynomials in Lemma 3.2. (2) Let S be a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r . The parametersare ( u , . . . , u r ) –admissible if and only if they satisfy (2.3) and (3.1). YCLOTOMIC BMW ALGEBRAS 7
Proof.
Part (1) is proved in [7], Section 5, by showing that the sequence ( η + a ( u )) a ≥ satisfies (2.3)and (3.1); that is,(3.6) r X j =0 a j η + j + ℓ ( u ) = 0 , for all ℓ ≥ , and(3.7) r − j − X µ =0 η + µ ( u ) a µ + j +1 = − δ ( r − j is odd) a j + δ ( j is even) a j +1 , for 0 ≤ j ≤ r −
1, where a j = ( − r − j ε r − j ( u , . . . , u r ). Part (2) follows from part (1) together withDefinition 3.5 and Lemma 3.2. (cid:3) Recovering the results of Ariki, Mathas, and Rui.
The main results of [3] regardingdegenerate cyclotomic BMW algebras are stated for ground rings S in which 2 is invertible. Theprimary reason for this restriction on the ground ring was that it seemed to be needed in order touse the quantities η + a ( u , . . . u r ), which play a central role in [3], via Definition 3.5. Using Lemma3.3, the restriction on the ground ring can be eliminated. We proceed to outline how the proofs haveto be adjusted.Let us define a universal ring with ( u , . . . , u r )–admissible parameters. Let u , . . . , u r be indeter-minants over Z . Let Z = Z [ u , . . . , u r ]; define a j = ( − j ε r − j ( u , . . . , u r ) for 0 ≤ j ≤ r , where ε k is the k –th elementary symmetric function, and define ω a for a ≥ ω a = H a ( u , . . . , u r ) = η + a ( u , . . . , u r ) for a ≥ . The parameters Ω = ( ω a ) a ≥ and u , . . . u r are ( u , . . . u r )–admissible by definition. (This is thesame construction as in [3], page 105, except that we don’t need to adjoin 1 / S is any commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r , such that Ω is ( u , . . . , u r )–admissible then there is a unique algebra homomorphism from Z to S taking u j u j . Since Ω is( u , . . . , u r )–admissible, it follows that ω a ω a for all a ≥
0. For all n ≥
0, we have(3.9) N n,S,r (Ω; u , . . . , u r ) ∼ = N n, Z ,r ( Ω ; u , . . . u r ) ⊗ Z S. See [11], Remark 3.4 for a justification.Let S be any commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r (with no conditionsimposed on the parameters). We recall a construction of a spanning set in N n = N n,S,r (Ω; u , . . . , u r )from [3]. Remark that there is a homomorphism from the Brauer algebra B n ( ω ) with parameter ω to N n,S,r taking s i s i and e i e i ; this follows from the presentation of the Brauer algebra citedin [3], Proposition 2.7. For a Brauer diagram γ , we will also write γ for the image of γ in N n,S,r .The “ r –regular monomials” in N n,S,r are defined to be the elements(3.10) y p γy q , where γ is a Brauer diagram, y p = y p · · · y np n , and y q = y q · · · y nq n ; moreover, p i and q i arenon-negative integers, in the interval 0 , , . . . , r −
1, and p i = 0 unless the i -th vertex at the bottomof γ is the left endpoint of a horizontal strand, and q i = 0 unless the i -th vertex at the top of γ iseither the left endpoint of a horizontal strand, or the top endpoint of a vertical strand. Note thatthere are at most n strictly positive exponents p i or q i , and the number of r –regular monomials is r n (2 n − FREDERICK M. GOODMAN
Proposition 3.7 ([3], Proposition 2.15) . Let S be any commutative ring with parameters Ω =( ω a ) a ≥ and u , . . . , u r . For all n ≥ , N n,S,r (Ω; u , . . . , u r ) is spanned over S by the set of r –regularmonomials. Furthermore, the ideal N n e n − N n is spanned by those r –regular monomials y p γy q suchthat γ has at least two horizontal strands. Remark 3.8.
It may appear from the presentation in [3] that this result depends on the invertibilityof 2 in the ground ring and on an additional condition on the parameters (called “admissibility” in [3],see Definition 2.10 in that paper). However, in fact, the result does not depend on any additionalassumptions. From Theorem 2.12 in [3], one only needs the statement that the degenerate affineBMW algebra is spanned by regular monomials, and the argument for this part of Theorem 2.12 isvalid over an arbitrary ring. The argument given for Proposition 2.15 itself in [3] is also valid overan arbitrary ring.
Theorem 3.9 ([3]) . Let F = Q ( u , . . . , u r ) denote the field of fractions of Z . Then the algebra N n,F,r ( Ω ; u , . . . , u r ) is split–semisimple of dimension r n (2 n − . This theorem is proved by explicit construction of sufficiently many irreducible representations.
Corollary 3.10. (cf. [3], Theorem 5.5)
Let S be a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r . Assume that Ω is ( u , . . . , u r ) –admissible. Then for all n ≥ , N n,S,r (Ω; u , . . . , u r ) is a free S –module with basis the set of r –regular monomials.Proof. Because of (3.9), it suffices to show that N n, Z ,r ( Ω ; u , . . . , u r ) is a free Z –module with basisthe set M of r –regular monomials. By Proposition 3.7, M is a spanning set, and M ⊗ { m ⊗ m ∈ M} is a spanning set in N n, Z ,r ⊗ Z F = N n,F,r . But by Theorem 3.9, the latter algebra over F has dimension r n (2 n − M ⊗ F . It follows that M islinearly independent over Z . (cid:3) The following theorem concerns cellularity of degenerate cyclotomic BMW algebras. The definitionof cellularity is given in Section 6.
Theorem 3.11. (cf. [3], Theorem 7.17)
Let S be a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r . Assume that Ω is ( u , . . . , u r ) –admissible. Then N n,S,r (Ω; u , . . . , u r ) is a cellularalgebra of rank r n (2 n − .Proof. Because of (3.9), it suffice to prove this when S = Z . For this special case, one can followthe proof in [3], Theorem 7.17, substituting Corollary 3.10 for [3], Theorem 5.5. For an alternativetreatment of cellularity, see [9], Section 6.5. (cid:3) Equivalence of admissibility conditions.
The following theorem establishes the equivalenceof the various admissibility criteria for degenerate cyclotomic BMW algebras.
Theorem 3.12. (cf. [7], Theorem 5.2)
Let S be a commutative ring with parameters Ω = ( ω a ) a ≥ and u , . . . , u r . Consider the degenerate cyclotomic BMW algebra N = N ,S,r (Ω; u , . . . , u r ) . Thefollowing are equivalent: (1) The parameters are admissible, i.e. { e , y e , . . . , y r − e } is linearly independent over S in N . (2) { y a e y b , s y a y b , y a y b : 0 ≤ a, b ≤ r − } is an S –basis of N . (3) The parameters satisfy (2.3) and (3.1). (4)
The parameters are ( u , . . . , u r ) –admissible.Proof. (1) = ⇒ (3) results from Lemmas 2.7 and 3.1. We have (3) ⇐⇒ (4) by Lemma 3.6. Theimplication (4) = ⇒ (2) is a special case of Corollary 3.10. Finally, the implication (2) = ⇒ (1) istrivial. (cid:3) YCLOTOMIC BMW ALGEBRAS 9 Admissibility for cyclotomic BMW algebras
Fix an integral domain S with parameters ρ , q , Ω = ( ω a ) a ≥ and u , . . . , u r ; assume that ρ and q are invertible and that equation (2.1) holds. Consider the cyclotomic BMW algebras W n,S,r = W n,S,r ( ρ, q, Ω; u , . . . , u r ).4.1. Admissibility conditions of Wilcox and Yu.
Explicit relations on the parameters that areequivalent to admissibility (i.e. linear independence of { e , y e , . . . , y r − e } ) have been found byWilcox and Yu [23, 22, 25]. The form of these relations depends on whether q = 1 is satisfied in S . Note that the conditions q = 1 (in the non–degenerate case) and char( S ) = 2 (in the degeneratecase) should be regarded as analogous. Theorem 4.1 (Wilcox & Yu, [23]) . Let S be an integral domain with parameters ρ , q , Ω = ( ω a ) a ≥ ,and u , . . . , u r satisfying Equation (2.1) and ( q − q − ) = 0 . The following conditions are equivalent: (1) { e , y e , . . . , y r − e } ⊆ W ,S,r is linearly independent over S . (2) The parameters satisfy the recursion relation (2.3) and the following relations: (4.1) ( q − q − ) h r − ℓ X j =1 a j + ℓ ω j i = − ρ ( a ℓ − a r − ℓ /a )+ ( q − q − ) h ⌊ ( ℓ + r ) / ⌋ X j =max( ℓ +1 , ⌈ r/ ⌉ ) a j − ℓ − min( ℓ, ⌈ r/ ⌉− X j = ⌈ ℓ/ ⌉ a j − ℓ i for ≤ ℓ ≤ r − , and (4.2) ρ = ± a if r is odd, and ρ ∈ { q − a , − qa } if r is even. Note that Equations (2.1), (4.1), and (4.2) determine ω , . . . , ω r − and ρ in terms of q , u , . . . , u r while the recursion relation (2.3) determines ω a for a ≥ r .In [22] and [25] Wilcox and Yu derive explicit relations on the parameters that are equivalent tolinear independence of { e , y e , . . . , y r − e } also in the case that q − q − = 0; their new conditionsreduce to those of Theorem 4.1 in the case q − q − = 0.4.2. The admissibility criterion of Rui and Xu.
Rui and Xu [21], following [3], take a differentapproach to admissibility for cyclotomic BMW algebras when q − q − = 0. Let u , . . . , u r , ρ , q , and t be algebraically independent indeterminants over Z . Define(4.3) G ( t ) = G ( u , . . . , u r ; t ) = r Y ℓ =1 t − u ℓ t u ℓ − . Let(4.4) Z ( t ) = Z ( t ; u , . . . , u r , ρ , q ) = − ρ − + ( q − q − ) t t − A ( t ) G ( t − ) , where(4.5) A ( t ) = ( ρ − ( Q j u j ) + ( q − q − ) t/ ( t −
1) if r is odd, and ρ − ( Q j u j ) − ( q − q − ) t / ( t −
1) if r is even . Definition 4.2 (Rui and Xu, [21]) . Let S be an integral domain with parameters ρ , q , Ω = ( ω a ) a ≥ and u , . . . , u r satisfying (2.1) and q − q − = 0. One says that the parameters are ( u , . . . , u r ) –admissible , or that Ω is ( u , . . . , u r ) –admissible , if(4.6) ( q − q − ) X a ≥ ω a t − a = Z ( t ; u , . . . , u r , ρ, q ) , where Z is defined in equations (4.4) and (4.5). Remark 4.3.
Let S be an integral domain with ( u , . . . , u r )–admissible parameters, as in Definition4.2. With p = Q j u j , we have(4.7) ρ = ± p if r is odd, and ρ ∈ { q − p, − qp } if r is even.The condition (4.7) on ρ was included in the definition of u –admissibility in [21], but it actuallyfollows from (2.1) and (4.6), as explained in [6], Remark 3.10.4.3. Equivalence of admissibility conditions.
The following theorem establishes the equivalenceof the various admissibility conditions for cyclotomic BMW algebras, in case the ground ring isintegral and q − q − = 0. Theorem 4.4 ([6], Theorem 4.4) . Let S be an integral domain with parameters ρ , q , Ω = ( ω a ) a ≥ ,and u , . . . , u r satisfying Equation (2.1) and ( q − q − ) = 0 . The following are equivalent: (1) { e , y e , . . . , y r − e } ⊆ W ,S,r is linearly independent over S . (2) The parameters satisfy the recursion relation (2.3) and the conditions (4.1) and (4.2) ofWilcox and Yu. (3) Ω is ( u , . . . , u r ) –admissible. Semi–admissibility
Let A n,S,r = A n,S,r ( u , . . . , u r ) denote either the cyclotomic BMW algebra W n,S,r , with param-eters ρ , q , Ω = ( ω a ) a ≥ and u , . . . , u r , or the degenerate cyclotomic BMW algebra N n,S,r , withparameters Ω = ( ω a ) a ≥ and u , . . . , u r , over an integral domain S .From here on, we impose the following standing assumption: Assumption 5.1.
The ground ring S is an integral domain, and the left ideal A ,S,r e ⊆ A ,S,r istorsion free as an S –module. This assumption holds, in particular, whenever S is a field.Under Assumption 5.1, exactly one of the following three possibilities must hold:(1) e = 0 in A ,S,r . In this case, e n − = 0 in A n,S,r for all n ≥
2. The (degenerate) cyclotomicBMW algebras reduce to (degenerate) cyclotomic Hecke algebras.(2) The parameters are admissible, i.e. { e , y e , . . . , y r − e } is linearly independent over S in A ,S,r . This case has been studied in the literature and is well understood.(3) There is a d with 0 < d < r such that { e , y e , . . . , y d − e } is linearly independent over S in A ,S,r , but { e , y e , . . . , y d e } is linearly dependent. This case remains to be investigated. Definition 5.2.
Consider the cyclotomic or degenerate cyclotomic BMW algebras A n,S,r over anintegral domain S with suitable parameters. Let 0 < d < r . We say that the parameters are d –semi–admissible if { e , y e , . . . , y d − e } is linearly independent over S in A ,S,r , but { e , y e , . . . , y d e } is linearly dependent. YCLOTOMIC BMW ALGEBRAS 11
Suppose d –semi–admissibility of the parameters. Then there is a polynomial of p ( u ) ∈ S [ u ] ofdegree d such that p ( y ) e = 0 but r ( y ) e = 0 for any non–zero polynomial r ( u ) ∈ S [ u ] of degreeless than d . Let F denote the field of fractions of S , and write p ( u ) = ( u − u ) · · · ( u − u r ) ∈ S [ u ].Since p ( y ) = 0, it follows that p ( u ) divides p ( u ) in F [ u ]. Because of unique factorization in F [ u ], wehave (after renumbering the roots u i of p ( u )) p ( u ) = α ( u − u ) · · · ( u − u d ) for some non–zero α in F .In fact α ∈ S , since it is the leading coefficient of p ( u ). Then we have α ( y − u ) · · · ( y − u d ) e = 0.Because we assumed A ,S,r e is torsion–free over S , we can conclude that ( y − u ) · · · ( y − u d ) e = 0.Thus without loss of generality, we can take p ( u ) = ( u − u ) · · · ( u − u d ). Assumption 5.3.
For the remainder of this section, we assume the parameters of A n,S,r are d –semi–admissible for some d with < d < r . Assume without loss of generality that p ( y ) e = 0 ,where p ( u ) = ( u − u ) · · · ( u − u d ) = P dj =0 b j u j . Lemma 5.4.
There is a surjective homomorphism θ : A n,S,r ( u , . . . , u r ) → A n,S,d ( u , . . . , u d ) takinggenerators to generators. Moreover, θ maps the ideal generated by e n − in A n,S,r ( u , . . . , u r ) ontothe ideal generated by e n − in A n,S,d ( u , . . . , u d ) .Proof. The existence of the surjective homomorphism θ is evident because the generators of A n,S,d ( u , . . . , u d ) satisfy the defining relations of A n,S,r ( u , . . . , u r ).In general, if A and B are algebras and ϕ : A → B is a surjective algebra homomorphism, thenfor any e ∈ A , we have ϕ ( AeA ) = Bϕ ( e ) B . In particular, θ maps the ideal generated by e n − in A n,Sr ( u , . . . , u r ) onto the ideal generated by e n − in A n,S,d ( u , . . . , u e ). (cid:3) Lemma 5.5. (1)
The sequence
Ω = ( ω a ) a ≥ satisfies the recurrence relation P dj =0 b j ω j + ℓ = 0 for all ℓ ≥ . (2) The parameters
Ω = ( ω a ) a ≥ and u , . . . u d in the degenerate case (respectively, ρ, q, Ω =( ω a ) a ≥ , and u , . . . u d in the non-degenerate case) are admissible. That is, the set { e , y e , . . . , y d − e } is linearly independent over S in A ,S,d ( u , . . . , u d ) .Proof. For part (1), multiply the equation p ( y ) e = 0 by e y ℓ on the left, and employ the idempo-tent and compression relations to get P dj =0 b j ω j + ℓ e = 0. The conclusion follows since e is not atorsion element over S .We should pause to see why something needs to be proved for part (2). We have assumed that { e , y e , . . . , y d − e } ⊆ A ,S,r ( u , . . . , u r ) is linearly independent, and we have to prove that { e , y e , . . . , y d − e } ⊆ A ,S,d ( u , . . . , u d ) is linearly independent. The latter set is the image ofthe former under the algebra homomorphism θ : A ,S,r ( u , . . . , u r ) → A ,S,d ( u , . . . , u d ) . Consider the degenerate case. Apply the proof of (1) = ⇒ (3) in Theorem 3.12 to the linearlyindependent set { e , y e , . . . , y d − e } ⊆ N ,S,r ( u , . . . , u r ). This yields the analogue of condition(3.1) with r replaced by d and a j by b j , namely(5.1) d − j − X µ =0 ω µ b µ + j +1 = − δ ( d − j is odd) b j + δ ( j is even) b j +1 , for 0 ≤ j ≤ d −
1. Part (1) of this lemma together with the implication (3) = ⇒ (1) in Theorem3.12, applied now to N ,S,d (Ω; u , . . . , u d ), gives the conclusion (2).For the non-degenerate case, one uses the theorem of Wilcox and Yu (Theorem 4.1 in the case q − q − = 0, or [22] in general) in the same manner. (cid:3) A spanning set for W n,S,r . In this section, write W n for W n,S,r ( ρ, q, Ω; u , . . . , u r ).Define elements y ′ j and y ′′ j for j ≥ y ′ = y ′′ = y ,y ′ j = g j − y ′ j − g − j − and y ′′ j = g − j − y ′′ j − g j − for j ≥ . Since the elements y ′ j and y ′′ j are all conjugate, we have p ( y ′ j ) = ( y ′ j − u ) · · · ( y ′ j − u r ) = 0 , for all j ,and similarly for the elements y ′′ j . Lemma 5.6.
In any affine or cyclotomic BMW algebra, e i and g i commute with y ′ j and y ′′ j if j
6∈ { i, i + 1 } .Proof. We will prove the commutation relations for the element y ′ j ; the proof for the elements y ′′ j isessentially the same.For i ≥ e i and g i commute with y and with g ± , . . . , g ± i − , hence with y ′ j for j < i . One seesfrom the defining relations that(5.2) g i g i +1 e i g − i +1 g − i = e i +1 and g i − g i +1 − e i g i +1 g i = e i +1 . for all i . Using this, and the already established commutation relation [ e i +1 , y ′ i ] = 0, we have(5.3) e i y ′ i +2 = e i ( g i +1 g i ) y ′ i ( g − i g − i +1 )= ( g i +1 g i ) e i +1 y ′ i ( g − i g − i +1 )= ( g i +1 g i ) y ′ i e i +1 ( g − i g − i +1 )= ( g i +1 g i ) y ′ i ( g − i g − i +1 ) e i = y ′ i +2 e i . Similarly, using the braid relations and the commutation relation [ g i +1 , y ′ i ] = 0, we obtain that[ g i , y ′ i +2 ] = 0. If j ≥ i + 3, we have(5.4) y ′ j = ( g j − · · · g i +2 ) y ′ i +2 ( g − i +2 · · · g − j − ) , and we see that g i and e i commute with y ′ j because they commute with all the factors on the righthand side of (5.4). (cid:3) Lemma 5.7. In W n , we have p ( y ′ j ) e i = e i p ( y ′ j ) = 0 for all j = i + 1 . The same statement holdswith y ′ j replaced by y ′′ j .Proof. We will verify explicitly that p ( y ′ j ) e i = 0 for j = i + 1. An identical argument shows thesame with y ′ j replaced by y ′′ j , and the statement e i p ( y ′ j ) = e i p ( y ′′ j ) = 0 for j = i + 1 follows as wellby applying the involution ∗ .We first show that p ( y ′ j ) e j = 0 for all j , by induction. This is given for j = 1. If p ( y ′ j ) e j = 0holds for some particular value of j , then0 = g j g j +1 p ( y ′ j ) e j g − j +1 g − j = p ( y ′ j +1 ) e j +1 , and our assertion follows.Next we check that p ( y ′ j ) e i = 0 for all j ≤ i , by induction on i − j . We have already checked thecase j = i . If this holds for some particular j ≤ i , then p ( y ′ j ) e i +1 = p ( y ′ j ) e i +1 e i e i +1 = e i +1 p ( y ′ j ) e i e i +1 = 0 . It remains to check that p ( y ′ j ) e i = 0 for i ≤ j −
2. We have p ( y ′ j ) e i = g j − · · · ( g i +1 g i ) p ( y ′ i )( g i − g i +1 − ) · · · g j − − e i = g j − · · · ( g i +1 g i )[ p ( y ′ i ) e i +1 ]( g i − g i +1 − ) · · · g j − − = 0 , YCLOTOMIC BMW ALGEBRAS 13 since p ( y ′ i ) e i +1 = 0 by the previous part of the proof. (cid:3) We now describe a certain basis of the affine BMW algebra c W n = c W n,S ( ρ, q, Ω) that was introducedin [5], Section 3.2. Given a permutation π ∈ S n , with reduced expression π = s i s i · · · s i ℓ , let g π = g i g i · · · g i ℓ in c W n ; in fact, g π is independent of the choice of the reduced expression of π ,see [5], Section 2.4. Fix an integer f with 0 ≤ f ≤ n , and let γ be a Brauer diagram with 2 f horizontal strands and s = n − f vertical strands. Then γ has a unique factorization in the Braueralgebra of the form(5.5) γ = α ( e e · · · e f − ) πβ − , where π is a permutation of { f + 1 , . . . , n − , n } and α and β are in a certain subset D f,n of S n described in [5], Section 3.2. Consider a sequence of n integers( a , b , c ) = ( a , a , . . . , a f − , b , b , . . . , b f − , c f +1 , . . . , c n ) . Corresponding to γ and the sequence ( a , b , c ), we let T γ, a , b , c be the following element of c W n , T γ, a , b , c = g α y ′′ a ( e e · · · e f − ) g π y ′′ c y ′ b ( g β ) ∗ , where y ′′ a = y ′′ a y ′′ a · · · y ′′ f − a f − ,y ′ b = y ′ f − b f − · · · y ′ b y ′ b , and y ′′ c = y ′′ nc n · · · y ′′ f +2 c f +2 y ′′ f +1 c f +1 If γ has no horizontal strands (i.e. γ is a permutation diagram), the elements T γ, a , b , c still makesense, but then f = 0, α and β are trivial, γ = π , and a and b are empty sequences. We have T γ, a , b , c = T γ, c = g γ y ′′ c . It is shown in [5], Section 3.2 that the set of T γ, a , b , c , as γ ranges over Brauer diagrams and ( a , b , c )ranges over n –tuples of integers forms an S –basis of c W n , and, moreover, the subset corresponding toBrauer diagrams with 2 f > c W n e n − c W n .Let b ′ ( n ) denote the number of Brauer diagrams on n strands with at least one horizontal strand, b ′ ( n ) = (2 n − − n !. Lemma 5.8.
The ideal W n e n − W n is spanned by a set of d n b ′ ( n ) elements. The algebra W n isspanned by a set of d n b ′ ( n ) + r n n ! elements.Proof. We also write T γ, a , b , c for the image of that element in the cyclotomic BMW algebra W n . Theset of all T γ, a , b , c spans W n , while those with γ a Brauer diagram with 2 f > W n e n − W n .If γ is a permutation diagram, then we can write any element T γ, a , b , c = T γ, c as a linear combinationof elements T γ, c ′ , with 0 ≤ c ′ i ≤ r , using the relations p ( y ′′ j ) = ( y ′′ j − u ) · · · ( y ′′ j − u r ) = 0 . In the following, take f > γ be a Brauer diagram with 2 f horizontal strands. We claimthat any element T γ, a , b , c can be written as a linear combination of elements T γ, a ′ , b ′ , c ′ where a ′ i , b ′ i ,and c ′ i lie in the interval 0 , , . . . , d −
1. Using the commutation relations of Lemma 5.6, we can write T γ, a , b , c = g α ( y ′′ a e )( y ′′ a e ) · · · ( y ′′ f − a f − e f − ) g π y ′′ c y ′ b ( g β ) ∗ Now, using Lemma 5.7, we can write any such element as a linear combination elements T γ, a ′ , b , c with the a ′ i in the desired interval. Using the commutation relations again, we can also write T γ, a ′ , b , c = g α y ′′ a ′ g π y ′′ c ( e f − y ′ f − b f − ) · · · ( e y ′ b )( e y ′ b )( g β ) ∗ , and using Lemma 5.7, we can write any such element as a linear combination of elements T γ, a ′ , b ′ , c with the b ′ i in the desired interval. Finally, e f − commutes with g π and with all y ′′ f + j . Using e f − p ( y ′′ f + j ) = 0, we can reduce any T γ, a ′ , b ′ , c to a linear combination elements T γ, a ′ , b ′ , c ′ with the c ′ i in the desired interval.It follows that W n is spanned by elements T γ, c , where γ is a permutation diagram and 0 ≤ c i ≤ r ,and by elements T γ, a , b , c where γ is a Brauer diagram with at least 2 horizontal strands and 0 ≤ a i , b i , c i ≤ d . Moreover, the latter set spans W n e n − W n . (cid:3) A spanning set for N n,S,r . In this section, write N n for N n,S,r (Ω; u , . . . , u r ).Consider first the free non-commutative polynomial algebra in the generators { s i , e i , y j : 1 ≤ i
The ideal N n e n − N n is spanned by a set of d n b ′ ( n ) elements. The algebra N n isspanned by a set of d n b ′ ( n ) + r n n ! elements.Proof. It is enough to work instead in the associated graded algebra G . We have that G is spannedby the elements T γ, c , where γ is a permutation diagram and 0 ≤ c i ≤ r − i , and by theelements T γ, a , b , c where γ is a Brauer diagram with at least 2 horizontal strands. The argument ofLemma 5.8, with Lemma 5.7 replaced by Lemma 5.9, shows that any T γ, a , b , c , where γ has horizontalstrands, can be written as a linear combination of elements T γ, a ′ , b ′ , c ′ , with 0 ≤ a i , b i , c i ≤ d − G e n − G . (cid:3) Freeness of A n,S,r . Let us recall from Lemma 5.4 that there is a surjective algebra homomor-phism θ : A n,S,r ( u , . . . , u r ) → A n,S,d ( u , . . . , u d ) and that θ maps the ideal generated by e n − in A n,S,r ( u , . . . , u r ) onto the ideal generated by e n − in A n,S,d ( u , . . . , u d ). Proposition 5.11. θ induces an isomorphism from the ideal generated by e n − in A n,S,r ( u , . . . , u r ) onto the ideal generated by e n − in A n,S,d ( u , . . . , u d ) .Proof. Write h e n − i r for the ideal generated by e n − in A n,S,r ( u , . . . , u r ) and h e n − i d for the idealgenerated by e n − in A n,S,d ( u , . . . , u d ).The parameters of A n,S,d ( u , . . . , u d ) are admissible, by Lemma 5.5. Hence, we know that A n,S,d ( u , . . . , u d ) is a free S module of rank d n (2 n − h e n − i d is free of rank d n ((2 n − − n !) = d n b ′ ( n ) , where b ′ ( n ) denotes the number of Brauer diagrams on n strands with at least one horizontal strand.We know that h e n − i r has a spanning set of the same cardinality by Lemmas 5.8 and 5.10. Therefore, θ : h e n − i r → h e n − i d is an isomorphism. (In fact, if B is spanning set of h e n − i r of cardinality d n b ′ ( n ), then θ ( B ) spans h e n − i d . Since S is an integral domain and h e n − i d is free over S with abasis of the same cardinality, it follows that θ ( B ) is a basis of h e n − i d . Therefore, B is a basis of h e n − i r , and θ is an isomorphism.) (cid:3) Theorem 5.12.
For all n ≥ , A n,S,r is a free S –module of rank d n b ′ ( n ) + r n n ! , and A n,S,r imbedsin A n +1 ,S,r .Proof. The ideal h e n − i r is free of rank d n b ′ ( n ), by Proposition 5.11, and the quotient A n,S,r / h e n − i r is isomorphic to the cyclotomic Hecke algebra or degenerate cyclotomic Hecke algebra, which is freeof rank r n n !. Therefore, A n,S,r is free of rank d n b ′ ( n ) + r n n !.We have given spanning sets of the same cardinality in Lemmas 5.8 and 5.10, and hence thosespanning sets are actually S –bases. It is straightforward to check that the homomorphism from A n,S,r to A n +1 ,S,r taking generators to generators maps the given basis of A n,S,r injectively into thebasis of A n +1 ,S,r . Therefore the map is injective. (cid:3) Cellularity
The following is a slight weakening of the original definition of cellularity from Graham andLehrer [13].
Definition 6.1 ([13]) . Let R be an integral domain and A a unital R –algebra. A cell datum for A consists of an algebra involution ∗ of A ; a partially ordered set (Λ , ≥ ) and for each λ ∈ Λ a set T ( λ );and a subset C = { c λs,t : λ ∈ Λ and s, t ∈ T ( λ ) } ⊆ A ; with the following properties:(1) C is an R –basis of A .(2) For each λ ∈ Λ, let ˘ A λ be the span of the c µs,t with µ > λ . Given λ ∈ Λ, s ∈ T ( λ ), and a ∈ A ,there exist coefficients r sv ( a ) ∈ R such that for all t ∈ T ( λ ): ac λs,t ≡ X v r sv ( a ) c λv,t mod ˘ A λ . (3) ( c λs,t ) ∗ ≡ c λt,s mod ˘ A λ for all λ ∈ Λ and, s, t ∈ T ( λ ). A is said to be a cellular algebra if it has a cell datum.For brevity, we will write that ( C , Λ) is a cellular basis of A . In the original definition in [13] it isrequired that ( c λs,t ) ∗ = c λt,s . All the conclusions of [13] remain valid with the weaker definition, and,in fact, the two definitions are equivalent if 2 is invertible in R . The main advantage of the weakerdefinition is that it allows a graceful treatment of extensions. Definition 6.2.
Let A be an algebra with involution and let J be a ∗ –invariant ideal. Say that J isa cellular ideal if it satisfies the axioms for a cellular algebra (except for being unital) with cellularbasis { c λs,t : λ ∈ Λ J and s, t ∈ T ( λ ) } ⊆ J and we have, as in point (2) of the definition of cellularity, ac λs,t ≡ X v r sv ( a ) c λv,t mod ˘ J λ not only for a ∈ J but also for a ∈ A . Lemma 6.3. (On extensions of cellular algebras.) If J is a cellular ideal in A , and H = A/J iscellular (with respect to the involution induced from the involution on A ), then A is cellular.Proof. Let (Λ J , ≥ ) be the partially ordered set in the cell datum for J and C J the cellular basis.Let (Λ H , ≥ ) be the partially ordered set in the cell datum for H and { ¯ h µu,v } the cellular basis. LetΛ = Λ J ∪ Λ H , with partial order agreeing with the original partial orders on Λ J and on Λ H and with λ > µ if λ ∈ Λ J and µ ∈ Λ H . A cellular basis of A is C J ∪ { h µs,t } , where h µs,t is any lift of ¯ h µs,t . (cid:3) Theorem 6.4.
Consider the sequence A n,S,r of cyclotomic or degenerate cyclotomic BMW algebrasover an integral domain S . Suppose that Assumption 5.1 holds. Then (1) A n,S,r imbeds in A n +1 ,S,r for all n ≥ . (2) A n,S,r is a cellular algebra.Proof. In the case e = 0 in A ,S,r , the cyclotomic or degenerate cyclotomic BMW algebras reduceto cyclotomic or degenerate cyclotomic Hecke algebras; in this case the results are known. If theparameters are admissible, these results are obtained in the papers cited in the introduction.It remains to verify the results in the semi–admissible case. We already have shown in the semi–admissible case that A n,S,r is a free S module, and that A n,S,r imbeds in A n +1 ,S,r . Adopt the notationand conventions of Section 5. We know that A n,S,d ( u , . . . , u d ) has admissible parameters by Lemma YCLOTOMIC BMW ALGEBRAS 17 h e n − i d isa cellular ideal in A n,S,d ( u , . . . , u d ). It follows that h e n − i r is a cellular ideal in A n,S,r , with cellularbasis { θ − ( c λ s , t ) } , where { c λ s , t } is a cellular basis of h e n − i d . The crucial point regarding the expansionof aθ − ( c λ s , t ) in terms of basis elements, for a ∈ A n,S,r follows because aθ − ( c λ s , t ) = θ − ( θ ( a ) c λ s , t ).Since A n,S,r / h e n − i r is isomorphic to the cyclotomic Hecke algebra, or degenerate cyclotomicHecke algebra, which is cellular, it follows from Lemma 6.3 that A n,S,r is cellular. (cid:3) Corollary 6.5.
Any cyclotomic or degenerate cyclotomic BMW algebra over a field is cellular.Proof.
In case the ground ring is a field, Assumption 5.1 holds automatically. (cid:3)
Corollary 6.6.
Let F be an algebraically closed field and consider an affine (resp. degenerate affine)BMW algebra b A n,F over F . Let M be a simple finite dimensional b A n,F –module. If e M = 0 , then M factors through a cyclotomic (resp. degenerate cyclotomic) Hecke algebra. If e M = 0 , then M factors through cyclotomic (resp. degenerate cyclotomic) BMW algebra with admissible parameters.Proof. In the degenerate case, this result is contained in [3], Theorem 7.19 and Proposition 3.11 (butwith the hypothesis that the characteristic of the field is = 2.)Because the field is algebraically closed, the minimal polynomial of y on M factors over F . Hence M factors through some cyclotomic (resp. degenerate cyclotomic) BMW algebra. If e M = 0, then M factors through the corresponding cyclotomic (resp. degenerate cyclotomic) Hecke algebra. If e M = 0, then the parameters of the cyclotomic (resp. degenerate cyclotomic) BMW algebra mustbe either admissible or semi–admissible.Let us assume a cyclotomic (resp. degenerate cyclotomic) BMW algebra A n,F,r = A nF,r ( u , . . . , u r )with d –semi–admissible parameters ( d < r ). Then M is the simple head of a cell module ∆ λ , andsince e M = 0, the cell module belongs to the ideal h e n − i r . But the cell modules belonging to h e n − i r factor through θ : A nF,r ( u , . . . , u r ) → A nF,d ( u , . . . , u d ), and the latter algebra has admis-sible parameters. (cid:3) The following proposition depends only on the material in this paper up through Lemma 5.5.
Proposition 6.7.
Let F be an algebraically closed field and consider an affine (resp. degenerateaffine) BMW algebra b A n,F over F . The following are equivalent: (1) There exist r > and u , . . . , u r ∈ F such that the parameters of b A n,F together with u , . . . , u r are admissible. (2) b A n,F admits a finite dimensional module on which e is non–zero.Proof. If (1) holds, then A n,F,r ( u , . . . , u r ) is a finite dimensional b A n,F module on which e = 0.If (2) holds, let u , . . . , u r be the roots of the minimal polynomial of y acting on M . The module M factors through the cyclotomic algebra A n,F,r ( u , . . . , u r ). Since e M = 0, it follows that e = 0in A n,F,r ( u , . . . , u r ) and hence also in A ,F,r ( u , . . . , u r ). Since F is a field, Assumption 5.1 holdsfor A ,F,r ( u , . . . , u r ). Therefore, the parameters of A ,F,r ( u , . . . , u r ) are either admissible or d –semi–admissible for some d with 0 < d < r . In the latter case, after renumbering the roots u i , A ,F,d ( u , . . . , u d ) has admissible parameters, by Lemma 5.5. Thus (1) holds. (cid:3) Rationality of parameters for affine algebras
Rationality of parameters for degenerate affine BMW algebras.
Ariki, Mathas, andRui call the parameter set Ω of a degenerate affine (or cyclotomic) BMW algebra rational if thegenerating function P a ≥ ω a t − a is a rational function. They prove the following theorem, under theadditional hypothesis that the characteristic of the field is different from 2. Theorem 7.1.
Consider the degenerate affine BMW algebra b N n , n ≥ , over an algebraically closedfield F , with parameters Ω = ( ω a ) a ≥ . Suppose that e = 0 in b N n . The following are equivalent. (1)
The generating function P a ≥ ω a t − a is a rational function in F ( t ) . (2) Ω satisfies a linear homogeneous recursion; i.e. there exist r > , N ≥ and a , a , . . . , a r − ∈ F such that ω r + ℓ + P r − j =0 a j ω j + ℓ = 0 , for all ℓ ≥ N . (3) There exist r > and a , a , . . . , a r − ∈ F such that ω r + ℓ + P r − j =0 a j ω j + ℓ = 0 , for all ℓ ≥ . (4) There exist r > and u , . . . , u r ∈ F such that the parameters Ω and u , . . . , u r are admis-sible. (5) b N n admits a finite dimensional module on which e is non–zero.Proof. (1) ⇐⇒ (2) ⇐ = (3) is easy, and (3) ⇐ = (4) holds by Lemma 2.7. Proposition 6.7 gives (4) ⇐⇒ (5). The implication (1) = ⇒ (4) is proved in [3], Proposition 3.11, under the assumption thatthe characteristic of the field is not equal to 2. So it remains only to prove this implication for a fieldof characteristic 2. This will be done with the aid of two lemmas. (cid:3) Lemma 7.2.
Consider the degenerate affine BMW algebra b N ,S over a ring S , with parameters Ω = ( ω a ) a ≥ . Suppose that e is not a torsion element over S . Then: (1) 2 ω a +1 = − ω a + P a +1 b =1 ( − b − ω b − ω a +1 − b for a ≥ . (2) If the characteristic of S is , then ω a = ω a for a ≥ .Proof. Part (1) is [3], Corollary 2.4. If the characteristic is 2, then the equation in part (1) simplifiesto ω a = ω a . (cid:3) The proof of the following lemma was suggested by Kevin Buzzard, via mathoverflow.net . Lemma 7.3.
Let F be an algebraically closed field of characteristic . Suppose that Ω = ( ω a ) a ≥ satisfies a linear homogeneous recursion, as in Theorem 7.1 (2) and ω a = ω a for a ≥ . Then thereexist distinct u , . . . , u d ∈ F such that ω a = P di =1 u ai for all a ≥ , and ω ∈ { , } .Proof. Our assumptions include ω = ω . Thus ω ∈ { , } . Let v , . . . , v m be the distinct rootsof the characteristic polynomial of the linear recursion relation satisfied by Ω. Then there existpolynomials h , . . . , h m such that ω a = P mi =1 h i ( a ) v ai for a ≥ N . Let α i be the constant term of h i for each i . Since char( F ) = 2, we have h i (2 a ) = α i for all a . For a ≥ N ,(7.1) X i α i v ai = ω a = ω a = X i α i v ai . Because the characteristic of F is 2, each element has a unique 2 k –th root for all k ≥
1; in particularall the v i are distinct, so Equation (7.1) implies that α i = α i for all i , i.e. α i ∈ { , } . Let u , . . . , u d be the list of those v j such that α j = 1. Then we have ω a = P i u ai for a ≥ N . For an arbitrary a ≥
1, chose k such that 2 k − a ≥ N . Then ω a is the unique 2 k –th root of ω k a = P i u k ai , namely ω a = P i u ai . (cid:3) Conclusion of the proof of Theorem 7.1.
Let us prove (1) = ⇒ (4) when the characteristic of thefield is 2. Since the ground ring is a field and e = 0, we have ω a = ω a for a ≥
0, by Lemma 7.2.Hence, by Lemma 7.3, there exist u , . . . , u d ∈ F such that ω a = p a ( u , . . . , u d ) = p a ( u , . . . , u d , YCLOTOMIC BMW ALGEBRAS 19 for a ≥ ω ∈ { , } . Using Example 3.4 and Definition 3.5, Ω is either ( u , . . . , u d )–admissible or( u , . . . , u d , (cid:3) Corollary 7.4 (Rui and Si [19]) . Assume char( F ) = 2 . The conditions of Theorem 7.1 are equivalentto the existence of a simple finite dimensional module on which e is non–zero, as long as Ω is notthe zero sequence or n = 2 .Proof. By the results of [19], a degenerate cyclotomic BMW algebra N n,F,r (Ω; u , . . . , u r ) with admis-sible parameters has a simple module on which e is non–zero, as long as Ω is not the zero sequenceor n = 2. (Rui and Si assumed char( F ) = 2, and I have not checked whether their results remainvalid in characteristic 2.) (cid:3) Rationality of parameters for affine BMW algebras.
We are going to obtain a resultanalogous to Theorem 7.1 for the affine BMW algebras.
Lemma 7.5.
Consider an affine BMW algebra c W n,S with parameters ρ , q , and Ω = ( ω a ) a ≥ . (1) There exist elements ω − a ∈ S such that e y − a e = ω − a e for a ≥ . (2) Suppose that e is not a torsion element over S . Then: (7.2) − ω a + ω − a + ρ ( q − q − ) a X i =1 ( ω a − i ω − i − ω a − i ) = 0 for a ≥ . (3) Suppose that S is an integral domain, that q − q − = 0 , and that e is not a torsion elementover S . Then: (7.3) X a ≥ ω a t − a − t t − ρ − q − q − X b ≥ ω − b t − b − t − − ρ − q − q − = t ( t − − q − q − ) . Proof.
Statement (1) is from [10], Corollary 3.13. Statement (2) is proved in [21], Lemma 2.17 and(in a different but equivalent form) in [10], Corollary 3.13 and [11], Lemma 2.6. The equation (7.3)appears as (2.30) in [21]. If S is integral and q − q − = 0, then (7.2) is equivalent to (7.3). To seethis, expand the left side of (7.3) and isolate the coefficient of t − n for each n ≥ (cid:3) Remark 7.6.
The equivalence of (7.2) and (7.3) seems to be implicit in [21]. The left side of (7.3)can also be written as: X a ≥ ω a t − a − t t − ρ − q − q − X b ≥ ω − b t − b − t t − − ρq − q − . Ram et. al. [4] have given an interesting non-inductive direct proof of (7.3).
Lemma 7.7.
Consider an cyclotomic BMW algebra W n,S,r with parameters ρ , q , and Ω = ( ω a ) a ≥ and u , . . . , u r . Let a i be given by equation (2.2). If e is not a torsion element over S , then P rj =0 a j ω j + ℓ = 0 for all ℓ ∈ Z .Proof. Same as the proof of Lemma 2.7. (cid:3)
Lemma 7.8.
Consider an affine BMW algebra c W n,F with parameters ρ , q , and Ω = ( ω a ) a ≥ over afield F . Suppose that there exist r > and a , a , . . . , a r − ∈ S such that ω a + r + P r − j =0 a j ω j + a = 0 for all a ∈ Z . Then w + ( t ) = P a ≥ ω a t − a and w − ( t ) = P b ≥ ω − b t − b are rational functions in F ( t ) and w − ( t ) = − w + ( t − ) . Moreover, w + (0) = 0 and w + ( ∞ ) = ω .Proof. Let p ( t ) = t r + P r − j =0 a j t j . Then one computes, using the recursion on ( ω a ) a ∈ Z , that p ( t ) w + ( t ) = q ( t ), where q is an explicit polynomial of degree ≤ r . Similarly, p ( t ) w − ( t − ) = q ( t ).Using the recursion again, one sees that q = − q . The coefficient of t r in q ( t ) is ω and the constantterm is zero; this gives w + (0) = 0 and w + ( ∞ ) = ω . (cid:3) Theorem 7.9.
Consider an affine BMW algebra c W n over an algebraically closed field F , withparameters ρ , q , and Ω = ( ω a ) a ≥ . Suppose that e = 0 in c W n . Consider the following statements: (1) w + ( t ) = P a ≥ ω a t − a and w − ( t ) = P b ≥ ω − b t − b are rational functions in F ( t ) and w − ( t ) = − w + ( t − ) . Moreover, w + ( t ) does not have a pole at or at ∞ . (2) There exist r > and a , a , . . . , a r − ∈ F such that ω r + ℓ + P r − j =0 a j ω j + ℓ = 0 , for all ℓ ∈ Z . (3) There exist r > and u , . . . , u r ∈ F such that the parameters ρ , q , Ω , and u , . . . , u r areadmissible. (4) c W n admits a finite dimensional module on which e is non–zero.The following implications hold: (1) ⇐ = (2) ⇐ = (3) ⇐⇒ (4) . If q − q − = 0 , then all the conditions are equivalent.Proof. The implication (1) ⇐ = (2) is from Lemma 7.8, and (2) ⇐ = (3) from Lemma 7.7. Theequivalence (3) ⇐⇒ (4) comes from Proposition 6.7.It remains to prove (1) = ⇒ (3) if q − q − = 0. Assume (1). Since the ground ring is a field and e is assumed to be non–zero, (7.3) holds. But by assumption, we have that w + ( t ) = P a ≥ ω a t − a and w − ( t ) = P b ≥ ω − b t − b are rational functions, and w − ( t ) = − w + ( t − ). Substituting in (7.3), andwriting h ( t ) = − t t − ρ − q − q − , we have(7.4) − (cid:2) w + ( t ) + h ( t ) (cid:3) (cid:2) w + ( t − ) + h ( t − ) (cid:3) = t ( t − − ( q − q − ) − . Define B ( t ) = ( q − q − ) − + tt − t + q )( t − q − )( q − q − )( t − . Note that(7.5) − B ( t ) B ( t − ) = t ( t − − ( q − q − ) − . We can write w + ( t ) in the form w + ( t ) = − h ( t ) + B ( t ) A t m Q sℓ =1 ( tu ℓ − Q rj =1 ( t − v j ) , YCLOTOMIC BMW ALGEBRAS 21 where m ∈ Z , A ∈ F , no u ℓ or v j is zero, and u ℓ = v − j for all j, ℓ . Then, taking into accountequations (7.4) and (7.5) we have(7.6) 1 = A Q sℓ =1 ( tu ℓ − t − u ℓ − Q rj =1 ( t − v j )( t − − v j ) = A ( − t ) r − s Q ℓ u ℓ Q j v j Q sℓ =1 ( t − u − ℓ )( t − u ℓ ) Q rj =1 ( t − v − j )( t − v j ) . Considering the restrictions placed on the u ℓ and v j , we must have r = s , A = 1, and the multisets { u , . . . , u s } and { v , . . . , v s } coincide. Thus(7.7) w + ( t ) = − h ( t ) + ( − α B ( t ) t m s Y j =1 tu j − t − u j , with α ∈ { , } and m ∈ Z . Because w + does not have a pole at 0 or ∞ , we have m = 0. Using thedefinition of h ( t ), we have finally(7.8) w + ( t ) = t t − − ρ − ( q − q − ) − + ( − α B ( t ) s Y j =1 tu j − t − u j . Moreover, using w + ( ∞ ) = ω , we obtain that( ω − q − q − ) = − ρ − + ( − α Q j u j , and (2.1) implies that ρ = ( − α Q j u j . Now there are four cases to consider, according to the parityof α and of s . Case 1, α = 0 and s is odd. Then ρ = Q j u j . Comparing the expression (7.8) for w + ( t ) withthe formulas (4.4) and (4.5) and Definition 4.2, we see that the parameters ρ , q , Ω are ( u , . . . , u s )–admissible. Case 2, α = 1 and s is odd. Then ρ = − Q j u j . Let v = ( u , . . . , u s , − , w + ( t ) = t t − − ρ − ( q − q − ) − − B ( t ) (cid:16)Q sj =1 u j (cid:17) s Y j =1 t − u − j t − u j = t t − − ρ − ( q − q − ) − + B ( t ) (cid:16)Q s +2 j =1 v j (cid:17) s +2 Y j =1 t − v − j t − v j , and ρ = − Q sj =1 u j = Q s +2 j =1 v j . Again, comparing with the formulas of Section 4.2, we see that theparameters ρ , q , Ω are ( u , . . . , u s , − , Case 3, α = 0 and s is even. Then ρ = Q j u j . By a similar calculation as in Case 2, one checksthat the parameters ρ , q , Ω are ( u , . . . , u s , Case 4, α = 1 and r is even. Then ρ = − Q j u j . By a similar calculation again, one checks thatthe parameters ρ , q , Ω are ( u , . . . , u s , − r > v , . . . , v r such that ρ , q , Ω and v , . . . , v r satisfythe Rui-Xu criterion for admissibility. Thus we have shown (1) = ⇒ (3) when q − q − = 0. (cid:3) Corollary 7.10 (Rui and Si [20]) . Assume q − q − = 0 . The conditions of Proposition 7.9 areequivalent to the existence of a simple finite dimensional module on which e is non–zero, as long as Ω is not the zero sequence or n = 2 .Proof. By the results of [20], a cyclotomic BMW algebra W n,S,r ( ρ, q, Ω; u , . . . , u r ) with admissibleparameters and q − q − = 0 has a simple module on which e is non–zero, as long as Ω is not thezero sequence or n = 2. (cid:3) Conjecture 7.11.
Theorem 7.9 remains valid when q − q − = 0.8. Construction of examples of semi–admissible parameters
Examples of cyclotomic (resp. degenerate cyclotomic) BMW algebras with semi–admissible pa-rameters can easily be constructed. For the sake of clarity, we carry this out for the degeneratecyclotomic BMW algebras only; non-degenerate cyclotomic BMW algebras with q = 1 can betreated in a similar way, using the admissibility criterion of Rui and Xu [21].Let S be an integral domain with 1 / ∈ S . Take 0 < d < r and u , . . . , u r ∈ S . Assume that u i = ± u j for any i, j and that u i = ± / i . Let p ( u ) = Q ≤ j ≤ r ( u − u j ) and p ( u ) = Q ≤ j ≤ d ( u − u j ). Define ω a for a ≥ u , . . . , u d )–admissibility criterion of [3],(8.1) ω a = q a +1 ( u , . . . , u d ) + 12 ( − d − q a ( u , . . . , u d ) + 12 δ a, . By [3], this is equivalent to(8.2) X a ≥ ω a u − a = 1 / − u + ( u − ( − d / d Y j =1 u + u j u − u j By the implication (4) = ⇒ (1) in Theorem 3.12 (which is from [3]), the parameters Ω = ( ω a ) a ≥ and u , . . . , u d are admissible; i.e. the set { e , y e , . . . , y d − e } is linearly independent over S in N ,S,d (Ω , u , . . . , u d ).Now consider N ,S,r (Ω; u , . . . , u r ). Since we have an algebra map θ : N ,S,r (Ω; u , . . . , u r ) → N ,S,d (Ω; u , . . . , u d ) , we have { e , y e , . . . , y d − e } is linearly independent over S in N ,S,r (Ω; u , . . . , u r ). Let r ′ bemaximal such that { e , y e , . . . , y r ′ − e } is linearly independent in N ,S,r (Ω; u , . . . , u r ). Thenby the argument following Definition 5.2, there is a subset { v , . . . , v r ′ } of { u , . . . , u r } such that p ( y ) e := Q r ′ j =1 ( y − v j ) e = 0, and h ( y ) e = 0 for any polynomial h of degree less than r ′ .Now by Lemma 5.5 and Theorem 3.12, the set of parameters Ω , v , . . . , v r ′ satisfies the ( v , . . . , v r ′ )–admissibility conditions. Hence we also have(8.3) X a ≥ ω a u − a = 1 / − u + ( u − ( − r ′ / r ′ Y j =1 u + v j u − v j . Comparing Equations (8.2) and (8.3), and taking into account the assumptions on { u , . . . , u r } ,we conclude that d = r ′ and { v , . . . , v d } = { u , . . . , u d } . Thus the parameters Ω , u , . . . , u r are d –semi–admissible. References
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Department of Mathematics, University of Iowa, Iowa City, Iowa
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