AFFINE AND DEGENERATE AFFINE BMW ALGEBRAS: THE CENTER

Transcription

1 Daugherty, Z., Ram, A. Vrk, R. Osaka J. Math. 5 (204), AFFINE AND DEGENERATE AFFINE BMW ALGEBRAS: THE CENTER ZAJJ DAUGHERTY, ARUN RAM RAHBAR VIRK (Receved October 2, 20, revsed August 7, 202) Abstract The degenerate affne affne BMW algebras arse naturally n the context of Schur Weyl dualty for orthogonal symplectc Le algebras quantum groups, respectvely. Cyclotomc BMW algebras, affne Hecke algebras, cyclotomc Hecke algebras, ther degenerate versons are quotents. In ths paper the theory s unfed by treatng the orthogonal symplectc cases smultaneously; we make an exact parallel between the degenerate affne affne cases va a new algebra whch takes the role of the affne brad group for the degenerate settng. A man result of ths paper s an dentfcaton of the centers of the affne degenerate affne BMW algebras n terms of rngs of symmetrc functons whch satsfy a cancellaton property or wheel condton (n the degenerate case, a reformulaton of a result of Nazarov). Mraculously, these same rngs also arse n Schubert calculus, as the cohomology K-theory of sotropc Grassmannans symplectc loop Grassmannans. We also establsh new ntertwner-lke denttes whch, when projected to the center, produce the recursons for central elements gven prevously by Nazarov for degenerate affne BMW algebras, by Belakova Blanchet for affne BMW algebras.. Introducton The degenerate affne BMW algebras W k the affne BMW algebras W k arse naturally n the context of Schur Weyl dualty the applcaton of Schur functors to modules n category O for orthogonal symplectc Le algebras quantum groups (usng the Schur functors of [4], [], [28]). The degenerate algebras W k were ntroduced n [27] the affne versons W k appeared n [28], followng foundatonal work of [7] [9]. The representaton theory of W k W k contans the representaton theory of any quotent: n partcular, the degenerate cyclotomc BMW algebras W r,k, the cyclotomc BMW algebras W r,k, the degenerate affne Hecke algebras H k, the affne Hecke algebras H k, the degenerate cyclotomc Hecke algebras H r,k, the cyclotomc Hecke algebras H r,k as quotents. In [3, 34, 35, 2, 8] other works, the representaton theory of the affne BMW algebra s derved by cellular algebra technques. As ndcated n [28], the Schur Weyl dualty also provdes a path to the representaton theory of the affne BMW algebras as an mage of the representaton theory of category O for orthogonal symplectc Le algebras ther quantum groups 200 Mathematcs Subject Classfcaton. 7B37.

2 258 Z. DAUGHERTY, A. RAM AND R. VIRK n the same way that the affne Hecke algebras arse n Schur Weyl dualty wth the envelopng algebra of gl n ts Drnfeld Jmbo quantum group. In the lterature, the algebras W k W k have often been treated separately. One of the goals of ths paper s to unfy the theory. To do ths we have begun by adjustng the defntons of the algebras carefully to make the presentatons match, relaton by relaton. In the same way that the affne BMW algebra s a quotent of the group algebra of the affne brad group, we have defned a new algebra, the degenerate affne brad algebra whch has the degenerate affne BMW algebra the degenerate affne Hecke algebras as quotents. We have done ths carefully, to ensure that the Schur Weyl dualty framework s completely analogous for both the degenerate affne the affne cases. We have also added a parameter (whch takes values ) so that both the orthogonal symplectc cases can be treated smultaneously. Our new presentatons of the algebras W k W k are gven n Secton 2. In Secton 3 we consder some remarkable recursons for generatng central elements n the algebras W k W k. These recursons were gven by Nazarov [27] n the degenerate case, then extended to the affne BMW algebra by Belakova Blanchet [4]. Another proof n the affne cyclotomc case appears n [35, Lemma 4.2], n the degenerate case, n [2, Lemma 4.5]. In all of these proofs, the recurson s obtaned by a rather mysterous tedous computaton. We show that there s an ntertwner lke dentty n the full algebra whch, when projected to the center produces the Nazarov recursons. Our approach provdes new nsght nto where these recursons are comng from. Moreover, the proof s exactly analogous n both the degenerate the affne cases, ncludes the parameter, so that both the orthogonal symplectc cases are treated smultaneously. In Secton 4 we dentfy the center of the degenerate affne BMW algebras. In the degenerate case ths has been done n [27]. Nazarov stated that the center of the degenerate affne BMW algebra s the subrng of the rng of symmetrc functons generated by the odd power sums. We dentfy the rng n a dfferent way, as the subrng of symmetrc functons wth the -cancellaton property, n the language of Pragacz [29]. Ths s a fascnatng rng. Pragacz dentfes t as the cohomology rng of orthogonal symplectc Grassmannans; the same rng appears agan as the cohomology of the loop Grassmannan for the symplectc group n [24, 22]; references for the relatonshp of ths rng to the projectve representaton theory of the symmetrc group, the BKP herarchy of dfferental equatons, representatons of Le superalgebras, twsted Gelf pars are found n [25, Chapter II 8]. For the affne BMW algebra, the -cancellaton property can be generalzed well to provde a sutable descrpton of the center. From our perspectve, one would expect that the rng whch appears as the center of the affne BMW algebra should also appear as the K-theory of the orthogonal symplectc Grassmannans as the K-theory of the loop Grassmannan for the symplectc group, but we are not aware that these dentfcatons have yet been made n the lterature.

3 AFFINE BMW ALGEBRAS: THE CENTER 259 The recent paper [3] classfes the rreducble representatons of W k by multsegments, the recent paper [6] adds to ths program of study by settng up commutng actons between the algebras W k W k the envelopng algebras of orthogonal symplectc Le algebras ther quantum groups, showng how the central elements whch arse n the Nazarov recursons concde wth central elements studed n Baumann [3], provdng an approach to admssblty condtons by provdng unversal admssble parameters n an approprate ground rng (arsng naturally, from Schur Weyl dualty, as the center of the envelopng algebra, or quantum group). We would also lke to menton the recent paper of A. Sartor [36] whch establshes smlar results n the case of the degenerate affne walled Brauer algebra the recent work of M. Ehrg C. Stroppel [7] whch studes these algebras n the context of categorfcaton. 2. Affne degenerate affne BMW algebras In ths secton, we defne the affne Brman Murakam Wenzl (BMW) algebra W k ts degenerate verson W k. We have adjusted the defntons to unfy the theory. In partcular, n Secton 2.2, we defne a new algebra, the degenerate affne brad algebra B k, whch has the degenerate affne BMW algebras W k the degenerate affne Hecke algebras H k as quotents. The motvaton for the defnton of B k s that the affne BMW algebras W k the affne Hecke algebras H k are quotents of the group algebra of affne brad group C B k. The defnton of the degenerate affne brad algebra B k also makes the Schur Weyl dualty framework completely analogous n both the affne degenerate affne cases. Both B k C B k are desgned to act on tensor space of the form M Å V Åk. In the degenerate affne case ths s an acton commutng wth a complex semsmple Le algebra g, n the affne case ths s an acton commutng wth the Drnfeld Jmbo quantum group U q g. The degenerate affne affne BMW algebras arse when g s so n or sp n V s the frst fundamental representaton the degenerate affne affne Hecke algebras arse when g s gl n or sl n V s the frst fundamental representaton. In the case when M s the trval representaton g s so n, the Jucys Murphy elements y,, y k n B k become the Jucys Murphy elements for the Brauer algebras used n [27], n the case that g gl n, these become the classcal Jucys Murphy elements n the group algebra of the symmetrc group. The Schur Weyl dualty actons are explaned n [6]. 2.. The affne BMW algebra W k. The affne brad group B k s the group gven by generators T, T 2,, T k X, wth relatons (2.) (2.2) (2.3) (2.4) T T j T j T, f j, T T T T T T, for, 2,, k 2, X T X T T X T X, X T T X, for 2, 3,, k.

4 260 Z. DAUGHERTY, A. RAM AND R. VIRK Let C be a commutatve rng let C B k be the group algebra of the affne brad group. Fx constants q, z ¾ C Z (l) 0 ¾ C, for l ¾, wth q z nvertble. Let Y z X so that (2.5) Y z X, Y T Y T, Y Y j Y j Y, for, j k. In the affne brad group (2.6) T Y Y Y Y T. Assume that q q s nvertble n C defne E n the group algebra of the affne brad group by (2.7) T Y Y T (q q )Y ( E ). The affne BMW algebra W k s the quotent of the group algebra C B k of the affne brad group B k by the relatons (2.8) (2.9) E T T E z E, E T E E T E z E, E Y l E Z (l) 0 E, E Y Y E Y Y E. The affne Hecke algebra H k s the affne BMW algebra W k wth the addtonal relatons (2.0) E 0, for,, k. Fx b,, b r ¾ C. The cyclotomc BMW algebra W r,k (b,, b r ) s the affne BMW algebra W k wth the addtonal relaton (2.) (Y b ) (Y b r ) 0. The cyclotomc Hecke algebra H r,k (b,, b r ) s the affne Hecke algebra H k wth the addtonal relaton (2.). Snce the composte map C[Y,, Y k ] C B k W k H k s njectve the last two maps are surjectons, t follows that the Laurent polynomal rng C[Y,, Y k ] s a subalgebra of C B k W k. Proposton 2.. The affne BMW algebra W k concdes wth the one defned n [28]. Proof. In [28] the affne BMW algebra s defned as the quotent of the group algebra of the affne brad group by the relatons n (2.8) whch are [28, (6.3b)

5 AFFINE BMW ALGEBRAS: THE CENTER 26 (6.3c)], the frst relaton n (2.9) whch s [28, (6.3d)], the second relaton n (2.9) for whch s [28, (6.3e)], the second relaton n (2.5) below where, n [28], the element E s defned by the frst equaton n (2.3) below. Workng n W k, snce Y (T Y )Y Y Y Y T Y T, conjugatng (2.7) by Y gves (2.2) Y T T Y (q q )( E )Y. Left multplyng (2.7) by Y usng the second dentty n (2.5) shows that (2.7) s equvalent to T T (q q )( E ), so that (2.3) E T T T q q T E T T E. Thus the E n W k concdes wth the E used n [28]. Multply the second relaton n (2.3) on the left the rght by E, then use the relatons n (2.8) to get E E E E T T E T T E E T E T E ze T E E, so that (2.4) E E E E, E 2 z z E q q s obtaned by multplyng the frst equaton n (2.3) by E usng (2.8). As one can construct representatons on whch E acts non-trvally, the frst relaton n (2.9) mples (2.5) Z (0) 0 z z q q (T z )(T q )(T q) 0, snce (T z )(T q )(T q)t (T z )(T 2 (q q )T )T (T z )(T T (q q )) (T z )(q q )( E ) (z z )(q q )E 0. Ths shows that the relaton [28, (6.3a)] follows from the relatons n W k. To complete the proof let us show that the relatons n W k follow from [28, (6.3a-e)]. Gven the equvalence of the defntons of E as establshed n (2.3), the concdences of the relatons [28, (6.3b-e)] wth the relatons n (2.8) (2.9) t only remans to show that the second set of relatons n (2.9), for follow from [28, (6.3a-e)]. But ths s establshed by [28, (6.2)] the dentty E Y Y T T Y q q Y Y Y T T Y q q Y E, whch follows from (2.6).

6 262 Z. DAUGHERTY, A. RAM AND R. VIRK The relatons (2.6) (2.7) E E E T T, E E T T E, T E E T E, E E T E T, are consequences of (2.8), the second relaton n (2.3) The degenerate affne brad algebra B k. Let C be a commutatve rng, let S k denote the symmetrc group on {,, k}. For ¾ {,, k}, wrte s for the transposton n S k that swtches. The degenerate affne brad algebra s the algebra B k over C generated by (2.8) t u (u ¾ S k ), 0,, y,, y k, wth relatons (2.9) (2.20) (2.2) (2.22) t u t Ú t uú, y y j y j y, 0 0, 0 y y 0, y y, 0 t s t s 0, t s t s t s t s, t s j t s j, for j, t s (y y ) (y y )t s, y j t s t s y j, for j,, t s y t s t s y t s, (2.23) t s t s, t s t s,2, where, y t s y t s for,, k 2. In the degenerate affne brad algebra B k let c 0 0 (2.24) c j 0 2(y y j ), so that y j 2 (c j c j ), for j,, k. Then c 0,, c k commute wth each other, commute wth, the relatons (2.2) are equvalent to (2.25) t s c j c j t s, for j. Theorem 2.2. The degenerate affne brad algebra B k has another presentaton by generators (2.26) t u, for u ¾ S k, 0,, k, j, for 0, j k wth j,

7 AFFINE BMW ALGEBRAS: THE CENTER 263 relatons (2.27) (2.28) (2.29) t u t Ú t uú, t Û t Û Û(), t Û, j t Û Û(),Û( j), j j, l,m l,m,, j j,, p,r l,m l,m p,r,, j (,r j,r ) (,r j,r ), j, for p l p m r l r m j, r j r. The commutaton relatons between the the, j can be rewrtten n the form (2.30) [ r, l,m ] 0, [, j, l,m ] 0, [, j,,m ] [,m, j,m ], for all r all l m j l j m. Proof of Theorem 2.2. The generators n (2.26) are wrtten n terms of the generators n (2.8) by the formulas (2.3) (2.32) 0 0,, t Û t Û, 0, y 2, j, j y j t s j y j t s j, for j,, k, (2.33) m t u t u, 0,m t u 0, t u, j t Ú,2 t Ú, for u, Ú ¾ S k such that u() m, Ú() Ú(2) j. The generators n (2.8) are wrtten n terms of the generators n (2.26) by the formulas (2.34) 0 0,, t Û t Û, y j 2 j l, j. 0l j Let us show that relatons n (2.9) (2.22) follow from the relatons n (2.27) (2.29). (a) The relaton t u t Ú t uú n (2.9) s the frst relaton n (2.27). (b) The relaton y y j y j y n (2.9): Assume that j. Usng the relatons n (2.28) (2.29), [y, y j ] m, j m, j 2 l l l,, 2 j m j ¾ l,, m, j m j l l l,, m j l,, ( l, j, j ) m j ml, m m, j 0.

8 264 Z. DAUGHERTY, A. RAM AND R. VIRK (c) The relaton 0 0 n (2.9) s part of the frst relaton n (2.28), the relatons 0 y y 0 y y n (2.9) follow from the relatons j j l,m l,m n (2.28). (d) The relatons 0 t s t s 0 t s j t s j for j from (2.20) follow from the relaton t Û t Û Û() n (2.27), the relaton t s t s t s t s from (2.20) follows from 2 2, whch s part of the frst relaton n (2.28). (e) The relatons n (2.2) (2.23) all follow from the relatons t Û t Û Û() t Û, j t Û Û(),Û( j) n (2.27). (f) By second relaton n (2.28) the (already establshed) second relaton n (2.20), t s y 2 t s 2 t s t s [, t s y t s ], 02 2 t s t s 0, whch establshes (2.22). To complete the proof let us show that the relatons of (2.27) (2.29) follow from the relatons n (2.9) (2.22). (a) The relaton t u t Ú t uú n (2.27) s the frst relaton n (2.9). (b) The relatons t Û t Û Û() n (2.27) follow from the frst last relatons n (2.20) ( force the defnton of m n (2.33)). (c) Snce 0, y (2), the relatons t Û 0, j t Û 0,Û( j) n (2.28) follow from the last relaton n each of (2.20) (2.2) ( force the defnton of 0,m n (2.33)). (d) Snce,2 y 2 t s y t s, the frst relaton n (2.2) gves that 2,,2 snce (2.35) t s,2 t s,2 (t s y 2 t s y ) y 2 t s y t s t s (y y 2 )t s (y y 2 ) 0. The relatons t Û,2 t Û Û(),Û(2) n (2.27) then follow from (2.35) the last relaton n (2.2) ( force the defntons, j t Ú,2 t Ú n (2.33)). (e) The thrd relaton n (2.9) s 0 0 the second relaton n (2.20) gves 2 2. The relatons j j n (2.28) then follow from the second set of relatons n (2.27). (f) The second relaton n (2.20) gves [, 2 ] 0. Multplyng (2.22) on the left rght by t s gves [y, 2 ] [y, t s t s ] 0. Usng these the relatons n (2.9), (2.36) [, 0,2,2 ], (2.37) [ 0,, 0,2,2 ] y 2 2 2,2,2, 2 2 y 2, y [, 2 ] 0, 0,

9 AFFINE BMW ALGEBRAS: THE CENTER 265 so that [ 0,, 2 ] [ 0,, 2y 2 2( 0,2,2 )] [ 0,, 2y 2 ] y 2, 2y 2 [, y 2 ] 0. Conjugatng the last relaton by t s gves [, 0,2 ] 0, thus [,,2 ] 0, by (2.36). By the thrd fourth relatons n (2.9), [ 0, 0, ] 0, y 2 0, [, 0, ] By the relatons n (2.20) (2.9),, y 2 0. [ 0,,2 ] [ 0, y 2 t s y t s ] 0 [, 2,3 ] [, y 3 t s2 y 2 t s2 ] 0. Puttng these together wth the (already establshed) relatons n (2.27) provdes the second set of relatons n (2.28). (g) From the commutatvty of the y the second relaton n (2.2),2 3,4 (y 2 t s y t s )(y 4 t s3 y 3 t s3 ) (y 4 t s3 y 3 t s3 )(y 2 t s y t s ) 3,4,2. By the last relaton n (2.9) the last relaton n (2.20), [ 0,, 2,3 ] y 2, y 3 t s2 y 2 t s2 0. Together wth the (already establshed) relatons n (2.27), we obtan the frst set of relatons n (2.29). (h) Conjugatng (2.37) by t s2 t s t s2 gves [ 0,2, 0,3 2,3 ] 0, ths the (already establshed) relatons n (2.28) the frst set of relatons n (2.29) provde 0 [y 2, y 3 ] 2 2 0,2,2, 2 3 0,3,3 2,3 [ 0,2,2, 0,3,3 2,3 ] [,2, 0,3,3 2,3 ] [,2,,3 2,3 ]. Note also that [,2,,0 2,0 ] [,2, 0, 0,2 ] [ 0,,,2 ] [,2, 0,2 ] [ 0,, 0,2 ] [,2, 0,2 ] t s [ 0,2,2, 0, ]t s 0, by (two applcatons of) (2.37). The last set of relatons n (2.29) now follow from the last set of relatons n (2.27).

10 266 Z. DAUGHERTY, A. RAM AND R. VIRK By the frst formula n (2.24) the last formula n (2.34), (2.38) c j j 0 2 0lm j 2.3. The degenerate affne BMW algebra W k. Let C be a commutatve rng let B k be the degenerate affne brad algebra over C as defned n Secton 2.2. Defne e n the degenerate affne brad algebra by l,m. (2.39) t s y y t s ( e ), for, 2,, k, so that, wth, as n (2.23), (2.40), t s e. Fx constants z (l) 0 ¾ C, for l ¾ 0. The degenerate affne Brman Wenzl Murakam (BMW) algebra W k (wth parameters z (l) 0 ) s the quotent of the degenerate affne brad algebra B k by the relatons (2.4) (2.42) e t s t s e e, e t s e e t s e e, e y l e z (l) 0 e, e (y y ) 0 (y y )e. The degenerate affne Hecke algebra H k s the quotent of W k by the relatons (2.43) e 0, for,, k. Fx b,, b r ¾ C. The degenerate cyclotomc BMW algebra W r,k (b,, b r ) s the degenerate affne BMW algebra wth the addtonal relaton (2.44) (y b ) (y b r ) 0. The degenerate cyclotomc Hecke algebra H r,k (b,,b r ) s the degenerate affne Hecke algebra H k wth the addtonal relaton (2.44). Snce the composte map C[y,, y k ] B k W k H k s njectve (see [20, Theorem 3.2.2]) the last two maps are surjectons, t follows that the polynomal rng C[y,, y k ] s a subalgebra of B k W k. Proposton 2.3. Let C, 0, ¾. Then the degenerate affne BMW algebra W k concdes wth the one defned n [27].

11 AFFINE BMW ALGEBRAS: THE CENTER 267 Proof. In [27], the degenerate affne BMW algebra s defned wth the frst two relatons n (2.9) the second set of relatons n (2.2), whch are [27, (4.)] the frst relatons n [27, (.2) (.3)], the relatons n (2.39) whch are [27, (4.2)], the relatons n (2.42) whch are [27, (4.3) (4.4)], the frst relatons n (2.4) whch s the thrd set of relatons n [27, (.2)], the relatons n (2.48) below whch are the last two relatons n [27, (.3)] the second relaton n [27, (.2)], the relatons n (2.50) below whch are [27, (.4)], the relatons (2.45) e t s j t s j e e e j e j e, for j, whch are the second thrd relatons n [27, (.5)]. Workng n W k conjugatng (2.39) by t s usng the frst relaton n (2.4) gves (2.46) y t s t s y ( e ). Then, by (2.40) (2.23), (2.47), t s e, e t s t s e t s t s. Multply the second relaton n (2.47) on the left the rght by e, then use the relatons n (2.4) to get so that e e e e t s t s e t s t s e e t s e t s e e t s e e, (2.48) e e e e. Note that e 2 z (0) 0 e s, for, a specal case of the frst dentty n (2.42) then, for general, follows from the second dentty n (2.47). The relatons (2.49) (2.50) e e e t s t s, e e t s t s e, t s e e t s e, e e t s e t s result from (2.4) the second relaton n (2.47). The relatons n (2.45) follow from (2.39) the frst two relatons n (2.9) the last relatons n (2.2). Thus the relatons n the defnton of the degenerate affne BMW algebra n [27] follow from the defnng relatons of W k. To complete the proof we must show that the frst relatons n (2.2), the relatons n (2.23), the second relatons n (2.4) follow from the defnng relatons used n [27]. Because of the assumpton that 0, ¾ the other relatons n (2.9) (2.23) are automatc.

12 268 Z. DAUGHERTY, A. RAM AND R. VIRK (a) Multplyng the frst relaton n (2.50) on the left by e usng the frst relatons n (2.4) the frst relatons n (2.48) provdes part of the second relatons n (2.4) the other part s obtaned smlarly by multplyng the second relatons n (2.50) on the rght by e. (b) Conjugatng (2.39) by t s produces (2.46) then addng (2.39) (2.46) produces the frst relatons n (2.2). (c) Usng (2.50), e t s t s (e t s )t s t s t s e e t s t s t s t s e t s t s whch, wth (2.39), gves the relatons n (2.23). 3. Identtes n affne degenerate affne BMW algebras In [27], Nazarov defned some naturally occurrng central elements n the degenerate affne BMW algebra W k proved a remarkable recurson for them. Ths recurson was generalzed to analogous central elements n the affne BMW algebra W k by Belakova Blanchet [4]. In both cases, the recurson was accomplshed wth an nvolved computaton. In ths secton, we provde a new proof of the Nazarov Belakova Blanchet recursons by lftng them out of the center, to ntertwner-lke denttes n W k W k (Propostons ). These ntertwner-lke denttes for the degenerate affne affne BMW algebras are remnscent of the ntertwner denttes for the degenerate affne affne Hecke algebras found, for example, n [2, Proposton 2.5 (c)] [30, Proposton 2.4 (c)], respectvely. The central element recursons of [27] [4] are then obtaned by multplyng the ntertwner-lke denttes by the projectors e k E k, respectvely. We shall not nclude our new proofs of Proposton 3. Theorem 3.2 here snce, gven our parallel setup of the degenerate affne the affne BMW algebras n Secton 2, the proof s exactly parallel to the proofs of Proposton 3.3 Theorem The degenerate affne case. Let W k be the degenerate affne BMW algebra as defned n (2.4) (2.42) let k. Let u be a varable let (3.) u u y u u y. Proposton 3.. In the degenerate affne BMW algebra W, e t s e t s y 2u (y y ) y 2u (y y ) (3.2) (2u (y y ) )(2u (y y ) ), (2u (y y )) 2

13 AFFINE BMW ALGEBRAS: THE CENTER 269 (3.3) u t s e 2u (y y ) t s u t s t s e 2u (y y ) u u e u e e e 2u (y y ) u t s e u. 2u (y y ) u The denttes (3.4) (3.5) of the followng theorem are [27, Lemma 2.5], [27, Lemma 3.8], respectvely. Theorem 3.2 ([27]). (2.4) (2.42) let k. Let z 0 (u) È l¾ 0 z (l) 0 u l. Then (3.4) Let W k be the degenerate affne BMW algebra as defned n e u e u e 2u 2u 2u 2u e, (3.5) e u 2u z0 (u) u 2u e j (u y j )(u y j )(u y j ) 2 (u y j ) 2 (u y j )(u y j ) e The affne case. Let W k be the affne BMW algebra as defned n (2.8) (2.9) let k. Let u be a varable, (3.6) U Y, note that U u Y By the defnton of E n (2.7),, by (2.2), U Y Y (U u 2 U Y Y ). (u Y )T T (u Y ) (q q )Y ( E ), so that (u Y )T T (u Y ) (q q )( E )Y, (3.7) T u Y T (q q Y ) ( E ), u Y u Y u Y

14 270 Z. DAUGHERTY, A. RAM AND R. VIRK Y (3.8) T T (q q ) ( E ). u Y u Y u Y u Y The relatons (3.9) (3.0) T U T U U U U T (q q )U ( E )U U (T (q q )( E )U ), T (q q )U E U (q q )U U (T (q q )( E )U ) are obtaned by multplyng (3.7) (3.8) on the rght (resp. left) by Y usng the relaton T Y Y T. Takng the coeffcent of u (l) on each sde of (3.7) (3.8) gves (3.) (3.2) T Y l Y l T (q q )(Y l ( E ) Y l ( E )Y Y ( E )Y l ), T Y l Y l T (q q )(Y l ( E )Y Y l 2 ( E )Y 2 ( E )Y l ), respectvely, for l ¾ 0. Therefore, (3.3) (3.4) T Y l T Y l (l ) Y l T (q q )(Y ( E )Y Y l T (q q )(Y l ( E ) Y ( E )Y ( E )Y l ), (l ) ). Proposton 3.3. Let q q. Then, n the affne BMW algebra W, Y T Y Y Y E E u Y (3.5) u 2 T Y Y Y Y u Y u 2 Y Y (u2 q 2 Y Y )(u 2 q 2 Y Y ), 2 (u 2 Y Y ) 2 (3.6) U T E Y Y u 2 Y Y 2 (U ) U T E Y Y U u 2 Y Y T U T T E Y Y u 2 Y Y 2 U E U E z E Y Y E (U u 2 Y Y ).

15 AFFINE BMW ALGEBRAS: THE CENTER 27 Proof. Puttng (3.6) nto (3.9) says that f A T Y Y u 2 Y Y then B E U T AU U B Y Y u 2 Y Y Y Y u 2 Y Y. Next, AE E A follows from (2.8) (2.9). So E Y u Y T E (U B) AB E Y Y u 2 Y Y A(E U B) E Y Y u 2 Y Y T Y Y T u 2 Y Y Y E T u Y AU Y Y u 2 Y Y Y Y u 2 Y Y, by (2.3), multplyng out the rght h sde gves (3.5). Rewrte T U U T U ( E )(U ) as T U (U )U U T multply on the left by T to get (3.7) U T (U )U T U T Then, snce T T U T U AB Y Y u 2 Y Y Y Y E, u 2 Y Y E (U ), E (U ). ( E ), equatons (3.0) (3.9) mply T (U ) (U ) T ( E )U T U T U ( E )U,

16 272 Z. DAUGHERTY, A. RAM AND R. VIRK so (3.7) s (3.8) Usng (3.6) addng U 2 (U ) T T U T 2 U ( E )U T U ( E )U E (U ). T E Y Y 2 Y Y (U u 2 Y Y u 2 )E (U Y Y ) to each sde of (3.8) gves U T E T U T T 2 U T U T 2 U Y Y u 2 Y Y E Y Y u 2 Y Y E U T T E Y Y u 2 Y Y E U E z E completng the proof of (3.6). 2 (U ) U T Y Y E u 2 (U Y Y ) E E Y Y (U u 2 Y Y ), Y Y u 2 Y Y U If Let Z 0 Z 0 be the generatng functons 0 u l Z 0 Z 0 l¾ 0 Z (l) l¾ 0 Z (l) 0 u l. (3.9) U Y u Y then E U E U U E U E, by the second dentty n (2.9). The frst dentty n (2.9) s equvalent to E U E (Z 0 Z (0) 0 )E. In the followng theorem, the dentty (3.20) s equvalent to [3, Lemma 2.8, parts (2) (3)] or [4, Lemma 2.6(4)] the dentty (3.2) s equvalent to the dentty found n [4, Lemma 7.4].

17 AFFINE BMW ALGEBRAS: THE CENTER 273 Theorem 3.4 ([4, 3, 4]). Let W k be the affne BMW algebra as defned n (2.8) (2.9) let k. Then (3.20) E U z q q u 2 (u2 q 2 )(u 2 q 2 ) (u 2 ) 2 (q q ) 2 E, E U z E q q u 2 (3.2) E U z q q Z 0 z u2 q q u 2 u 2 E j (u Y j ) 2 (u q 2 Y j )(u q 2 Y j ) (u Y j ) 2 (u q 2 Y j )(u q 2 Y j ) E. Proof. Multply (3.5) on the rght by E use Z (0) (z z )(q q ) to ge (3.20). Multplyng (3.6) on the left rght by E usng the relatons n (2.8), (2.9), (2.4), E T U T E E T T U T T E E E U E E, gves E U z E E U z E u 2 ( 2 (U )U ) u 2 2 U E E U z E ( 2 L U R U ) E E u 2 E U z E E (U ) E E u 2 where L U s the operator of left multplcaton by U. Then, by nducton, rght multplcaton by U E U z j E u 2 ( 2 U j (U j )) ( 2 L U j R U j ) j E E E 2 E U z R U s the operator of E u 2 E 2 E E

18 274 Z. DAUGHERTY, A. RAM AND R. VIRK j ( 2 L U j R U j ) E E E 2 Z 0 Z (0) 0 z Z 0 Z (0) 0 z Z 0 Z (0) 0 z So (3.2) follows from u 2 u 2 j j ( 2 L U j R U j ) (q q ) 2 U j (U j ) (q q ) 2 U j (U j ) ( 2 U j (U j ))E. (q q ) 2 Y j (u Y j )(Y j (u Y j ) ) (q q ) 2 Y j (u Y j )(Y j (u Y j ) ) ((u Y j ) 2 (q q ) 2 Y j u)((u Y j ) 2 ) ((u Y j ) 2 (q q ) 2 Y j u)((u Y j ) 2 ) (u q 2 Y j )(u q 2 Y j )(u Y j ) 2 (u q 2 Y j )(u q 2 Y j )(u Y j ) 2 Z (0) 0 (z z )(q q ). E u 2 E 2 E E (E E E 2 E E 2 E E ) 4. The center of the affne degenerate affne BMW algebras In ths secton, we dentfy the center of W k W k. Both centers arse as algebras of symmetrc functons wth a cancellaton property (n the language of [29]) or wheel condton (n the language of [9]). In the degenerate case, Z(W k ) s the rng of symmetrc functons n y,, y k wth the -cancellaton property of Pragacz. By [29, Theorem 2. ()], ths s the same rng as the rng generated by the odd power sums, whch s the way that Nazarov [27] dentfed Z(W k ). The cancellaton property n the case of W k s analogous, exhbtng the center of the affne BMW algebra Z(W k ) as a subalgebra of the rng of symmetrc Laurent polynomals. At the end of ths secton, n an attempt to make the theory for the affne BMW algebra completely analogous to that for the degenerate affne BMW algebra, we have formulated an alternate descrpton of Z(W k ) as a rng generated by negatve power sums. 4.. Bases of W k W k. The Brauer algebra, dependng on a parameter x, s gven by generators e,, e k s,, s k relatons as gven n [27, (.2) (.5)] (where our e s denoted Æs our x s denoted N). The Brauer algebra

19 AFFINE BMW ALGEBRAS: THE CENTER 275 also has a dagrammatc presentaton (see [5]) wth bass (4.) D k {dagrams on k dots}, where a (Brauer) dagram on k dots s a graph wth k dots n the top row, k dots n the bottom row k edges parng the dots. We label the vertces of the top row, left to rght, wth, 2,, k the vertces n the bottom row, left to rght, wth ¼, 2 ¼,, k ¼ so that, for example, (4.2) d (3)(2 ¼ )(45)(66 ¼ )(74 ¼ )(2 ¼ 7 ¼ )(3 ¼ 5 ¼ ) s a Brauer dagram on 7 dots. Settng x z (0) 0 s t s realzes the Brauer algebra as a subalgebra of the degenerate affne BMW algebra W k. The Brauer algebra s also the quotent of W k by y 0, hence, can be vewed as the degenerate cyclotomc BMW algebra W,k (0). Theorem 4. ([27, 2]). Let W k be the degenerate affne BMW algebra let W r,k (b,, b r ) be the degenerate cyclotomc BMW algebra as defned n (2.4) (2.42) (2.43), respectvely. For n,, n k ¾ 0 a dagram d on k dots let d n,,n k y n y n l l dy n l l y n k k, where, n the lexcographc orderng of the edges (, j ),, ( k, j k ) of d,,, l are n the top row of d l,, k are n the bottom row of d. Let D k be the set of dagrams on k dots, as n (4.). (a) If 0, ¾ C (4.3) z 0 ( u) 2 u z 0 (u) u u u then {d n,,n k d ¾ Dk, n,, n k ¾ 0 } s a C-bass of W k. (b) If 0, ¾ C, (4.3) holds, (4.4) z 0 (u) u 2 u 2 ( )r r u b u b then {d n,,n k d ¾ Dk, 0 n,, n k r } s a C-bass of W r,k (b,, b r ). Part (a) of Theorem 4. s [27, Theorem 4.6] (see also [2, Theorem 2.2]) part (b) s [2, Proposton 2.5 Theorem 5.5].

20 276 Z. DAUGHERTY, A. RAM AND R. VIRK Theorem 4.2 ([4, 39]). Let W k be the affne BMW algebra let W r,k (b,, b r ) be the cyclotomc BMW algebra as defned n Secton 2.. Let d ¾ D k be a Brauer dagram, where D k s as n (4.). Choose a mnmal length expresson of d as a product of e,, e k, s,, s k, d a a l, a ¾ {e,, e k, s,, s k }, such that the number of s n ths product s the number of crossngs n d. For each a whch s n {s,, s k } fx a choce of sgn j set E, f a T d A A l, where A j j e, T j, f a j s. For n,, n k ¾ let T n,,n k d Y n Y n l l T d Y n l l Y n k k, where, n the lexcographc orderng of the edges (, j ),, ( k, j k ) of d,,, l are n the top row of d l,, k are n the bottom row of d. (a) If (4.5) Z 0 z q q u2 u 2 (u2 q 2 )(u 2 q 2 ) (u 2 ) 2 (q q ) 2 Z 0 z q q then {T n,,n k d d ¾ D k, n,, n k ¾ } s a C-bass of W k. (b) If (4.5) holds (4.6) Z 0 z u2 q q u 2 z q q u2 u 2 uz r u 2 j u b j u b j then {T n,,n k d d ¾ D k, 0 n,, n k r } s a C-bass of W r,k (b,, b r ). Part (a) of Theorem 4.2 s [4, Theorem 2.25] part (b) s [4, Theorem 5.5] [39, Theorem 8.]. We refer to these references for proof, remarkng only that one key pont n showng that {T n,,n k d d ¾ D k, n,, n k ¾ } spans W k s that f (, j) s a top-to-bottom edge n d then (4.7) Y T d T d Y j (terms wth fewer crossngs),, f (, j) s a top-to-top edge n d then (4.8) Y T d Y j T d (terms wth fewer crossngs).

21 AFFINE BMW ALGEBRAS: THE CENTER The center of W k. The degenerate affne BMW algebra s the algebra W k over C defned n Secton 2.3 the polynomal rng C[y,, y k ] s a subalgebra of W k. The symmetrc group S k acts on C[y,, y k ] by permutng the varables. A classcal fact (see, for example, [20, Theorem 3.3.]) s that the center of the degenerate affne Hecke algebra H k s the rng of symmetrc functons Z(H k ) C[y,, y k ] S k { f ¾ C[y,, y k ] Û f f, for Û ¾ S k }. Theorem 4.3 gves an analogous characterzaton of the center of the degenerate affne BMW algebra. We shall not nclude the proof here snce, gven our parallel setup of the degenerate affne BMW algebras the affne BMW algebras n Secton 2, the proof s exactly parallel to the proof of Theorem 4.4. Theorem 4.3. The center of the degenerate affne BMW algebra W k s R k { f ¾ C[y,, y k ] S k f (y, y, y 3,, y k ) f (0, 0, y 3,, y k )}. The power sum symmetrc functons p are gven by p y y 2 y k, for ¾ 0. The Hall Lttlewood polynomals (see [25, Chapter III (2.)]) are gven by P (yá t) P (y,, y k Á t) Ú (t) Û¾S k Û y y k k jk x tx j, x x j where Ú (t) s a normalzng constant (a polynomal n t) so that the coeffcent of y y k k n P (yá t) s equal to. The Schur -functons (see [25, Chapter III (8.7)]) are 0, f s not strct, 2 l() P (yá ), f s strct, where l() s the number of (nonzero) parts of the partton s strct f all ts (nonzero) parts are dstnct. Let R k be as n Theorem 4.3. Then (see [27, Corollary 4.0], [29, Theorem 2. ()] [25, Chapter III 8]) (4.9) R k C[p, p 3, p 5, ] Cspan-{ s strct}. More generally, let r ¾ 0 let be a prmtve rth root of unty. Defne R r,k { f ¾ [ ][y,, y k ] S k f (y, y,, r y, y r,, y k ) f (0, 0,, 0, y r,, y k )}.

22 278 Z. DAUGHERTY, A. RAM AND R. VIRK Then (4.0) R r,k Å [ ] É( ) É( )[p 0 mod r], (4.) R r,k has [ ]-bass {P (yá ) m () r k}, where m () s the number parts of sze n. The rng R r,k s studed n [26], [23], [25, Chapter III Example 5.7 Example 7.7], [37], [9], others. The proofs of (4.0) (4.) follow from [25, Chapter III Example 7.7], [37, Lemma 2.2 followng remarks] the arguments n the proofs of [9, Lemma 3.2 Proposton 3.5] The center of W k. The affne BMW algebra s the algebra W k over C defned n Secton 2. the rng of Laurent polynomals C[Y,, Y k ] s a subalgebra of W k. The symmetrc group S k acts on C[Y,, Y k ] by permutng the varables. A classcal fact (see, for example, [6, Proposton 2.]) s that the center of the affne Hecke algebra H k s the rng of symmetrc functons, Z(H k ) C[Y,, Y k ] S k { f ¾ C[Y,, Y k ] Û f f, for Û ¾ S k }. Theorem 4.4 s a characterzaton of the center of the affne BMW algebra. Theorem 4.4. The center of the affne BMW algebra W k s R k { f ¾ C[Y,, Y k ] S k f (Y, Y, Y 3,, Y k ) f (,, Y 3,, Y k )}. Proof. STEP : f ¾ W k commutes wth all Y f ¾ C[Y,, Y k ]: Assume f ¾ W k wrte f c n,,n k d T n,,n k d, n terms of the bass n Theorem 4.2. Let d ¾ D k wth the maxmal number of crossngs such that c n,,n k d 0, usng the notaton after (4.2), suppose there s an edge (, j) of d such that j ¼. Then, by (4.7) (4.8), the coeffcent of Y T n,,n k d n Y f s c n,,n k d the coeffcent of Y T n,,n k d n f Y s 0. If Y f f Y t follows that there s no such edge, so d ( therefore T d ). Thus f ¾ C[Y,, Y k ]. Conversely, f f ¾ C[Y,, Y k ], then Y f f Y.

23 AFFINE BMW ALGEBRAS: THE CENTER 279 STEP 2: f ¾ C[Y,, Y k ] commutes wth all T f ¾ R k : Assume f ¾ C[Y,, Y k ] wrte f a,b¾ Y a Y b 2 f a,b, where f a,b ¾ C[Y 3,, Y k ]. Then f (,, Y 3,, Y k ) È a,b¾ f a,b (4.2) f (Y, Y, Y 3,, Y k ) a,b¾ Y a b f a,b Y l l¾ b¾ f lb,b. By drect computaton usng (3.2) (3.4), T Y a Y 2 b Y a Y 2 a T Y2 b a s (Y a Y 2 b )T (q q ) Y ay 2 b s (Y ay 2 b) Y Y 2 E b a, where E l (q q ) (q q ) l r l r E Y r, f l 0, Y l r Y lr E Y r, f l 0, 0, f l 0. It follows that (4.3) T f (s f )T (q q f s f ) Y Y 2 l¾ 0 E l b¾ f lb,b. Thus, f f (Y, Y, Y 3,, Y k ) f (,, Y 3,, Y k ) then, by (4.2), (4.4) b¾ f lb,b 0, for l 0. Hence, f f ¾ C[Y,, Y k ] S k f (Y, Y, Y 3,, Y k ) f (,, Y 3,, Y k ) then s f f (4.4) holds so that, by (4.3), T f f T. Smlarly, f commutes wth all T. Conversely, f f ¾ C[Y,, Y k ] T f f T then s f f b¾ f lb,b 0, for l 0, so that f ¾ C[Y,, Y k ] S k f (Y, Y, Y 3,, Y k ) f (,, Y 3,, Y k ). It follows from (2.7) that R k Z(W k ).

24 280 Z. DAUGHERTY, A. RAM AND R. VIRK The symmetrc group S k acts on k by permutng the factors. The rng C[Y,, Y k ] S k has bass {m ¾ k wth 2 k }, where m ¾S k Y Y k k. The elementary symmetrc functons are e r m ( r,0 k r ) e r m (0 k r,( ) r ), for r 0,,, k, the power sum symmetrc functons are p r m (r,0 k ) p r m (0 k, r), for r ¾ 0. The Newton denttes (see [25, Chapter I (2. ¼ )]) say (4.5) le l l r ( ) r p r e l r le l l r ( ) r p re (l r), where the second equaton s obtaned from the frst by replacng Y wth Y. For l ¾ (,, k ) ¾ k, In partcular, e l k m m (l k ), where (l k ) ( l,, k l). (4.6) e r e k e k r, for r 0,, k. Defne (4.7) p p p p p p, for ¾ 0. The consequence of (4.6) (4.5) s that [Y,, Y k ] S k [e k, e,, e k ] [e k ][e, e 2,, e k2, e k e (k )2,, e k e 2, e k e ] [e k ][e, e 2,, e k2, e (k )2,, e 2, e ] [e k ][p, p 2,, p k2, p (k )2,, p 2, p ] [e k ][p, p 2,, p k2, p (k )2,, p 2, p ].

25 For ¾ k wth l 0 defne AFFINE BMW ALGEBRAS: THE CENTER 28 p p p l p p p l. Then [Y,, Y k ] S k has bass e l k p p l ¾, l() k, l() 2 k 2. In analogy wth (4.9) we expect that f R k s as n Theorem 4.4 then R k C[e k ][p, p 2, ]. ACKNOWLEDGEMENTS. Sgnfcant work on ths paper was done whle the authors were n resdence at the Mathematcal Scences Research Insttute (MSRI) n 2008, the wrtng was completed when A. Ram was n resdence at the Hausdorff Insttute for Mathematcs (HIM) n 20. We thank MSRI HIM for hosptalty, support, a wonderful workng envronment durng these stays. Ths research has been partally supported by the Natonal Scence Foundaton (DMS ) the Australan Research Councl (DP ). We thank S. Fomn for provdng the reference [29] Fred Goodman for provdng the reference [4], many nformatve dscussons, much help n processng the theory for correctng many of our errors. We thank J. Enyang for hs helpful comments on the manuscrpt. Fnally, many thanks to the referee for a crtcal readng the dscovery of an omsson n our orgnal defnton of the degenerate affne brad algebra. References [] T. Arakawa T. Suzuk: Dualty between sl n (C) the degenerate affne Hecke algebra, J. Algebra 209 (998), [2] S. Ark, A. Mathas H. Ru: Cyclotomc Nazarov Wenzl algebras, Nagoya Math. J. 82 (2006), [3] P. Baumann: On the center of quantzed envelopng algebras, J. Algebra 203 (998), [4] A. Belakova C. Blanchet: Sken constructon of dempotents n Brman Murakam Wenzl algebras, Math. Ann. 32 (200), [5] R. Brauer: On algebras whch are connected wth the semsmple contnuous groups, Ann. of Math. (2) 38 (937), [6] Z. Daugherty, A. Ram R. Vrk: Affne degenerate affne BMW algebras: actons on tensor space, Selecta Math. (N.S.) 9 (203), [7] M. Ehrg C. Stroppel: Nazarov-Wenzl algebras, codeal subalgebras categorfed skew Howe dualty, arxv: [8] J. Enyang: Specht modules semsmplcty crtera for Brauer Brman Murakam Wenzl algebras, J. Algebrac Combn. 26 (2007), [9] B. Fegn, M. Jmbo, T. Mwa, E. Mukhn Y. Takeyama: Symmetrc polynomals vanshng on the dagonals shfted by roots of unty, Int. Math. Res. Not. 2003,

26 282 Z. DAUGHERTY, A. RAM AND R. VIRK [0] F.M. Goodman: Cellularty of cyclotomc Brman Wenzl Murakam algebras, J. Algebra 32 (2009), [] F.M. Goodman: Comparson of admssblty condtons for cyclotomc Brman Wenzl Murakam algebras, J. Pure Appl. Algebra 24 (200), [2] F.M. Goodman: Admssblty condtons for degenerate cyclotomc BMW algebras, Comm. Algebra 39 (20), [3] F.M. Goodman H. Hauschld: Affne Brman Wenzl Murakam algebras tangles n the sold torus, Fund. Math. 90 (2006), [4] F.M. Goodman H.H. Mosley: Cyclotomc Brman Wenzl Murakam algebras, I, Freeness realzaton as tangle algebras, J. Knot Theory Ramfcatons 8 (2009), [5] F.M. Goodman H.H. Mosley: Cyclotomc Brman Wenzl Murakam algebras, II, admssblty relatons freeness, Algebr. Represent. Theory 4 (20), 39. [6] I. Grojnowsk M. Vazran: Strong multplcty one theorems for affne Hecke algebras of type A, Transform. Groups 6 (200), [7] R. Härng-Oldenburg: The reduced Brman Wenzl algebra of Coxeter type B, J. Algebra 23 (999), [8] R. Härng-Oldenburg: An Ark Koke lke extenson of the Brman Murakam Wenzl algebra, preprnt (998), arxv:q-alg/ [9] R. Härng-Oldenburg: Cyclotomc Brman Murakam Wenzl algebras, J. Pure Appl. Algebra 6 (200), [20] A. Kleshchev: Lnear Projectve Representatons of Symmetrc Groups, Cambrdge Tracts n Mathematcs 63, Cambrdge Unv. Press, Cambrdge, [2] C. Krloff A. Ram: Representatons of graded Hecke algebras, Represent. Theory 6 (2002), [22] T. Lam: Affne Schubert classes, Schur postvty, combnatoral Hopf algebras, Bull. Lond. Math. Soc. 43 (20), [23] A. Lascoux, B. Leclerc J.-Y. Thbon: Green polynomals Hall Lttlewood functons at roots of unty, European J. Combn. 5 (994), [24] T. Lam, A. Schllng M. Shmozono: Schubert polynomals for the affne Grassmannan of the symplectc group, Math. Z. 264 (200), [25] I.G. Macdonald: Symmetrc Functons Hall Polynomals, second edton, Oxford Unv. Press, New York, 995. [26] A.O. Morrs: On an algebra of symmetrc functons, uart. J. Math. Oxford Ser. (2) 6 (965), [27] M. Nazarov: Young s orthogonal form for Brauer s centralzer algebra, J. Algebra 82 (996), [28] R. Orellana A. Ram: Affne brads, Markov traces the category O; n Algebrac Groups Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math, Tata Inst. Fund. Res., Mumba, 2007, [29] P. Pragacz: Algebro-geometrc applcatons of Schur S- -polynomals; n Topcs n Invarant Theory (Pars, 989/990), Lecture Notes n Math. 478, Sprnger, Berln, 99, [30] A. Ram: Affne Hecke algebras generalzed stard Young tableaux, J. Algebra 260 (2003), [3] H. Ru: On the classfcaton of fnte dmensonal rreducble modules for affne BMW algebras, arxv: [32] H. Ru M. S: On the structure of cyclotomc Nazarov Wenzl algebras, J. Pure Appl. Algebra 22 (2008), [33] H. Ru M. S: Gram determnants semsmplcty crtera for Brman Wenzl algebras, J. Rene Angew. Math. 63 (2009), [34] H. Ru J. Xu: On the semsmplcty of the cyclotomc Brauer algebras, II, J. Algebra 32 (2007), [35] H. Ru J. Xu: The representatons of cyclotomc BMW algebras, J. Pure Appl. Algebra 23 (2009), [36] A. Sartor: The degenerate affne walled Brauer algebra, arxv:

27 AFFINE BMW ALGEBRAS: THE CENTER 283 [37] B. Totaro: Towards a Schubert calculus for complex reflecton groups, Math. Proc. Cambrdge Phlos. Soc. 34 (2003), [38] S. Wlcox S. Yu: The cyclotomc BMW algebra assocated wth the two strng type B brad group, Comm. Algebra 39 (20), [39] S. Wlcox S. Yu: On the freeness of the cyclotomc BMW algebras: admssblty an somorphsm wth the cyclotomc Kauffman tangle algebras. arxv: [40] S. Yu: The cyclotomc Brman Murakam Wenzl algebras, Ph.D. Thess, Unversty of Sydney (2007). arxv: [4] A.V. Zelevnskĭ: Resolutons, dual pars character formulas, Funktsonal. Anal. Prlozhen. 2 (987), Zajj Daugherty Department of Mathematcs Dartmouth College Hanover, NH U.S.A. e-mal: zajj.b.daugherty@dartmouth.edu Arun Ram Department of Mathematcs Statstcs Unversty of Melbourne Parkvlle VIC 300 Australa e-mal: aram@unmelb.edu.au Rahbar Vrk Department of Mathematcs Unversty of Colorado Campus Box 395 Boulder, Colorado U.S.A. e-mal: rahbar.vrk@colorado.edu