Affine and degenerate affine BMW algebras: The center

Zajj Daugherty, Arun Ram, Rahbar Virk

Abstract

The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively.Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their de-generate versions are quotients. In this paper the theory is unified by treating the orthogonaland symplectic cases simultaneously; we make an exact parallel between the degenerate affineand affine cases via a new algebra which takes the role of the affine braid group for the de-generate setting. A main result of this paper is an identification of the centers of the affineand degenerate affine BMW algebras in terms of rings of symmetric functions which satisfya “cancellation property” or “wheel condition” (in the degenerate case, a reformulation ofa result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as thecohomology and K-theory of isotropic Grassmanians and symplectic loop Grassmanians. Wealso establish new intertwiner-like identities which, when projected to the center, producethe recursions for central elements given previously by Nazarov for degenerate affine BMWalgebras, and by Beliakova-Blanchet for affine BMW algebras.

1 Introduction

The degenerate affine BMW algebras Wk and the affine BMW algebras Wk arise naturally inthe context of Schur-Weyl duality and the application of Schur functors to modules in categoryO for orthogonal and symplectic Lie algebras and quantum groups (using the Schur functors of[39], [1], and [27]). The degenerate algebras Wk were introduced in [26] and the affine versionsWk appeared in [27], following foundational work of [16]-[18]. The representation theory ofWk and Wk contains the representation theory of any quotient: in particular, the degeneratecyclotomic BMW algebrasWr,k, the cyclotomic BMW algebras Wr,k, the degenerate affine Heckealgebras Hk, the affine Hecke algebras Hk, the degenerate cyclotomic Hecke algebras Hr,k, andthe cyclotomic Hecke algebras Hr,k as quotients. In [30, 33, 34, 2, 7] and other works, therepresentation theory of the affine BMW algebra is derived by cellular algebra techniques. Asindicated in [27], the Schur-Weyl duality also provides a path to the representation theory ofthe affine BMW algebras as an image of the representation theory of category O for orthogonaland symplectic Lie algebras and their quantum groups in the same way that the affine Heckealgebras arise in Schur-Weyl duality with the enveloping algebra of gln and its Drinfeld-Jimboquantum group.

In the literature, the algebras Wk and Wk have often been treated separately. One of thegoals of this paper is to unify the theory. To do this we have begun by adjusting the definitionsof the algebras carefully to make the presentations match, relation by relation. In the sameway that the affine BMW algebra is a quotient of the group algebra of the affine braid group,we have defined a new algebra, the degenerate affine braid algebra which has the degenerate

AMS 2010 subject classifications: 17B37 (17B10 20C08)

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affine BMW algebra and the degenerate affine Hecke algebras as quotients. We have done thiscarefully, to ensure that the Schur-Weyl duality framework is completely analogous for both thedegenerate affine and the affine cases. We have also added a parameter ε (which takes values±1) so that both the orthogonal and symplectic cases can be treated simultaneously. Our newpresentations of the algebras Wk and Wk are given in section 2.

In section 3 we consider some remarkable recursions for generating central elements in thealgebras Wk and Wk. These recursions were given by Nazarov [26] in the degenerate case, andthen extended to the affine BMW algebra by Beliakova-Blanchet [4]. Another proof in the affinecyclotomic case appears in [34, Lemma 4.21] and, in the degenerate case, in [2, Lemma 4.15]. Inall of these proofs, the recursion is obtained by a rather mysterious and tedious computation. Weshow that there is an “intertwiner” like identity in the full algebra which, when “projected to thecenter” produces the Nazarov recursions. Our approach provides new insight into where theserecursions are coming from. Moreover, the proof is exactly analogous in both the degenerateand the affine cases, and includes the parameter ε, so that both the orthogonal and symplecticcases are treated simultaneously.

In section 4 we identify the center of the degenerate and affine BMW algebras. In thedegenerate case this has been done in [26]. Nazarov stated that the center of the degenerateaffine BMW algebra is the subring of the ring of symmetric functions generated by the oddpower sums. We identify the ring in a different way, as the subring of symmetric functionswith the Q-cancellation property, in the language of Pragacz [28]. This is a fascinating ring.Pragacz identifies it as the cohomology ring of orthogonal and symplectic Grassmannians; thesame ring appears again as the cohomology of the loop Grassmannian for the symplectic groupin [23, 21]; and references for the relationship of this ring to the projective representation theoryof the symmetric group, the BKP hierarchy of differential equations, representations of Liesuperalgebras, and twisted Gelfand pairs are found in [24, Ch. II §8]. For the affine BMWalgebra, the Q-cancellation property can be generalized well to provide a suitable description ofthe center. From our perspective, one would expect that the ring which appears as the centerof the affine BMW algebra should also appear as the K-theory of the orthogonal and symplecticGrassmannians and as the K-theory of the loop Grassmannian for the symplectic group, but weare not aware that these identifications have yet been made in the literature.

The recent paper [30] classifies the irreducible representations of Wk by multisegments, andand the recent paper [6] adds to this program of study by setting up commuting actions be-tween the algebras Wk and Wk and the enveloping algebras of orthogonal and symplectic Liealgebras and their quantum groups, showing how the central elements which arise in the Nazarovrecursions coincide with central elements studied in Baumann [3], and providing an approachto admissibility conditions by providing “universal admissible parameters” in an appropriateground ring (arising naturally, from Schur-Weyl duality, as the center of the enveloping alge-bra, or quantum group). We would also like to mention forthcoming work of M. Ehrig and C.Stroppel which studies these algebras in the context of categorification.

Acknowledgements: Significant work on this paper was done while the authors were in res-idence at the Mathematical Sciences Research Institute (MSRI) in 2008, and the writing wascompleted when A. Ram was in residence at the Hausdorff Insitute for Mathematics (HIM) in2011. We thank MSRI and HIM for hospitality, support, and a wonderful working environmentduring these stays. This research has been partially supported by the National Science Founda-tion (DMS-0353038) and the Australian Research Council (DP-0986774). We thank S. Fominfor providing the reference [28] and Fred Goodman for providing the reference [4], many infor-mative discussions, much help in processing the theory and for correcting many of our errors.We thank J. Enyang for his helpful comments on the manuscript. Finally, many thanks to the

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referee for a critical reading and the discovery of an omission in our original definition of thedegenerate affine braid algebra.

2 Affine and degenerate affine BMW algebras

In this section, we define the affine Birman-Murakami-Wenzl (BMW) algebra Wk and its degen-erate version Wk. We have adjusted the definitions to unify the theory. In particular, in section2.2, we define a new algebra, the degenerate affine braid algebra Bk, which has the degenerateaffine BMW algebras Wk and the degenerate affine Hecke algebras Hk as quotients. The moti-vation for the definition of Bk is that the affine BMW algebras Wk and the affine Hecke algebrasHk are quotients of the group algebra of affine braid group CBk.

The definition of the degenerate affine braid algebra Bk also makes the Schur-Weyl dualityframework completely analogous in both the affine and degenerate affine cases. Both Bk andCBk are designed to act on tensor space of the form M ⊗V ⊗k. In the degenerate affine case thisis an action commuting with a complex semisimple Lie algebra g, and in the affine case this isan action commuting with the Drinfeld-Jimbo quantum group Uqg. The degenerate affine andaffine BMW algebras arise when g is son or spn and V is the first fundamental representationand the degenerate affine and affine Hecke algebras arise when g is gln or sln and V is the firstfundamental representation. In the case when M is the trivial representation and g is son, the“Jucys-Murphy” elements y1, . . . , yk in Bk become the “Jucys-Murphy” elements for the Braueralgebras used in [26] and, in the case that g = gln, these become the classical Jucys-Murphyelements in the group algebra of the symmetric group. The Schur-Weyl duality actions areexplained in [6].

2.1 The affine BMW algebra Wk

The affine braid group Bk is the group given by generators T1, T2, . . . , Tk−1 and Xε1 , withrelations

TiTj = TjTi, if j 6= i± 1, (2.1)

TiTi+1Ti = Ti+1TiTi+1, for i = 1, 2, . . . , k − 2, (2.2)

Xε1T1Xε1T1 = T1X

ε1T1Xε1 , (2.3)

Xε1Ti = TiXε1 , for i = 2, 3, . . . , k − 1. (2.4)

Let C be a commutative ring and let CBk be the group algebra of the affine braid group.Fix constants

q, z ∈ C and Z(`)0 ∈ C, for ` ∈ Z,

with q and z invertible. Let Yi = zXεi so that

Y1 = zXε1 , Yi = Ti−1Yi−1Ti−1, and YiYj = YjYi, for 1 ≤ i, j ≤ k. (2.5)

In the affine braid groupTiYiYi+1 = YiYi+1Ti. (2.6)

Assume that q − q−1 is invertible in C and define Ei in the group algebra of the affine braidgroup by

TiYi = Yi+1Ti − (q − q−1)Yi+1(1− Ei). (2.7)

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The affine BMW algebra Wk is the quotient of the group algebra CBk of the affine braid groupBk by the relations

EiT±1i = T±1i Ei = z∓1Ei, EiT

±1i−1Ei = EiT

±1i+1Ei = z±1Ei, (2.8)

E1Y`1E1 = Z

(`)0 E1, EiYiYi+1 = Ei = YiYi+1Ei. (2.9)

The affine Hecke algebra Hk is the affine BMW algebra Wk with the additional relations

Ei = 0, for i = 1, . . . , k − 1. (2.10)

Fix b1, . . . , br ∈ C. The cyclotomic BMW algebra Wr,k(b1, . . . , br) is the affine BMW algebraWk with the additional relation

(Y1 − b1) · · · (Y1 − br) = 0. (2.11)

The cyclotomic Hecke algebra Hr,k(b1, . . . , br) is the affine Hecke algebra Hk with the additionalrelation (2.11).

Since the composite map C[Y ±11 , . . . , Y ±1k ] → CBk → Wk → Hk is injective and the lasttwo maps are surjections, it follows that the Laurent polynomial ring C[Y ±11 , . . . , Y ±1k ] is asubalgebra of CBk and Wk.

Proposition 2.1. The affine BMW algebra Wk coincides with the one defined in [27].

Proof. In [27] the affine BMW algebra is defined as the quotient of the group algebra of theaffine braid group by the relations in (2.8) which are [27, (6.3b) and (6.3c)], the first relation in(2.9) which is [27, (6.3d)], the second relation in (2.9) for i = 1 which is [27, (6.3e)], and thesecond relation in (2.15) below where, in [27], the element Ei is defined by the first equation in(2.13) below.

Working in Wk, since Y −1i+1(TiYi)Yi+1 = Y −1i+1YiYi+1Ti = YiTi, conjugating (2.7) by Y −1i+1 gives

YiTi = TiYi+1 − (q − q−1)(1− Ei)Yi+1. (2.12)

Left multiplying (2.7) by Y −1i+1 and using the second identity in (2.5) shows that (2.7) is equivalent

to Ti − T−1i = (q − q−1)(1− Ei), so that

Ei = 1−Ti − T−1i

q − q−1and TiTi+1EiT

−1i+1T

−1i = Ei+1. (2.13)

Thus the Ei in Wk coincides with the Ei used in [27].Multiply the second relation in (2.13) on the left and the right by Ei, and then use the

relations in (2.8) to get

EiEi+1Ei = EiTiTi+1EiT−1i+1T

−1i Ei = EiTi+1EiT

−1i+1Ei = zEiT

−1i+1Ei = Ei,

so that

EiEi±1Ei = Ei, and E2i =

(1 +

z − z−1

q − q−1

)Ei (2.14)

is obtained by multiplying the first equation in (2.13) by Ei and using (2.8). As one can constructrepresentations on which E1 acts non-trivially, the first relation in (2.9) implies

Z(0)0 = 1 +

z − z−1

q − q−1and (Ti − z−1)(Ti + q−1)(Ti − q) = 0, (2.15)

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since (Ti − z−1)(Ti + q−1)(Ti − q)T−1i = (Ti − z−1)(T 2i − (q − q−1)Ti − 1)T−1i = (Ti − z−1)(Ti −

T−1i − (q − q−1)) = (Ti − z−1)(q − q−1)(−Ei) = −(z−1 − z−1)(q − q−1)Ei = 0. This shows thatthe relation [27, (6.3a)] follows from the relations in Wk.

To complete the proof let us show that the relations in Wk follow from [27, (6.3a-e)]. Giventhe equivalence of the definitions of Ei as established in (2.13), and the coincidences of therelations [27, (6.3b-e)] with the relations in (2.8) and (2.9) it only remains to show that thesecond set of relations in (2.9), for i > 1 follow from [27, (6.3a-e)]. But this is established by[27, (6.12)] and the identity

EiYiYi+1 =

(1−

Ti − T−1i

q − q−1

)YiYi+1 = YiYi+1

(1−

Ti − T−1i

q − q−1

)= YiYi+1Ei,

which follows from (2.6).

The relations

Ei+1Ei = Ei+1TiTi+1, EiEi+1 = T−1i+1T−1i Ei+1, (2.16)

TiEi+1Ei = T−1i+1Ei, and Ei+1EiTi+1 = Ei+1T−1i , (2.17)

are consequences of (2.8), and the second relation in (2.13).

2.2 The degenerate affine braid algebra BkLet C be a commutative ring, and let Sk denote the symmetric group on {1, . . . , k}. Fori ∈ {1, . . . , k}, write si for the transposition in Sk that switches i and i + 1. The degenerateaffine braid algebra is the algebra Bk over C generated by

tu (u ∈ Sk), κ0, κ1, and y1, . . . , yk, (2.18)

with relations

tutv = tuv, yiyj = yjyi, κ0κ1 = κ1κ0, κ0yi = yiκ0, κ1yi = yiκ1, (2.19)

κ0tsi = tsiκ0, κ1ts1κ1ts1 = ts1κ1ts1κ1, and κ1tsj = tsjκ1, for j 6= 1, (2.20)

tsi(yi + yi+1) = (yi + yi+1)tsi , and yjtsi = tsiyj , for j 6= i, i+ 1, (2.21)

κ1ts1y1ts1 = ts1y1ts1κ1, (2.22)

and

tsitsi+1γi,i+1tsi+1tsi = γi+1,i+2, where γi,i+1 = yi+1 − tsiyitsi for i = 1, . . . , k − 2. (2.23)

In the degenerate affine braid algebra Bk let c0 = κ0 and

cj = κ0 + 2(y1 + . . .+ yj), so that yj = 12(cj − cj−1), for j = 1, . . . , k. (2.24)

Then c0, . . . , ck commute with each other, commute with κ1, and the relations (2.21) are equiv-alent to

tsicj = cjtsi , for j 6= i. (2.25)

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Theorem 2.2. The degenerate affine braid algebra Bk has another presentation by generators

tu, for u ∈ Sk, κ0, . . . , κk and γi,j , for 0 ≤ i, j ≤ k with i 6= j, (2.26)

and relationstutv = tuv, twκitw−1 = κw(i), twγi,jtw−1 = γw(i),w(j), (2.27)

κiκj = κjκi, κiγ`,m = γ`,mκi, (2.28)

γi,j = γj,i, γp,rγ`,m = γ`,mγp,r, and γi,j(γi,r + γj,r) = (γi,r + γj,r)γi,j , (2.29)

for p 6= ` and p 6= m and r 6= ` and r 6= m and i 6= j, i 6= r and j 6= r.

The commutation relations between the κi and the γi,j can be rewritten in the form

[κr, γ`,m] = 0, [γi,j , γ`,m] = 0, and [γi,j , γi,m] = [γi,m, γj,m], (2.30)

for all r and all i 6= ` and i 6= m and j 6= ` and j 6= m.

Proof. The generators in (2.26) are written in terms of the generators in (2.18) by the formulas

κ0 = κ0, κ1 = κ1, tw = tw, (2.31)

γ0,1 = y1 − 12κ1, and γj,j+1 = yj+1 − tsjyjtsj , for j = 1, . . . , k − 1, (2.32)

andκm = tuκ1tu−1 , γ0,m = tuγ0,1tu−1 and γi,j = tvγ1,2tv−1 , (2.33)

for u, v ∈ Sk such that u(1) = m, v(1) = i and v(2) = j.The generators in (2.18) are written in terms of the generators in (2.26) by the formulas

κ0 = κ0, κ1 = κ1, tw = tw, and yj = 12κj +

∑0≤`<j

γ`,j . (2.34)

Let us show that relations in (2.19-2.22) follow from the relations in (2.27-2.29).

(a) The relation tutv = tuv in (2.19) is the first relation in (2.27).

(b) The relation yiyj = yjyi in (2.19): Assume that i < j. Using the relations in (2.28) and(2.29),

[yi, yj ] =[12κi +

∑`<i

γ`,i,12κj +

∑m<j

γm,j]

=[∑`<i

γ`,i,∑m<j

γm,j]

=∑`<i

[γ`,i,

∑m<j

γm,j]

=∑`<i

[γ`,i, (γ`,j + γi,j) +

∑m<j

m 6=`,m 6=i

γm,j]

= 0.

(c) The relation κ0κ1 = κ1κ0 in (2.19) is part of the first relation in (2.28), and the relationsκ0yi = yiκ0 and κ1yi = yiκ1 in (2.19) follow from the relations κiκj = κjκi and κiγ`,m =γ`,mκi in (2.28).

(d) The relations κ0tsi = tsiκ0 and κ1tsj = tsjκ1 for j 6= 1 from (2.20) follow from the relationtwκit

−1w = κw(i) in (2.27), and the relation κ1ts1κ1ts1 = ts1κ1ts1κ1 from (2.20) follows from

κ1κ2 = κ2κ1, which is part of the first relation in (2.28).

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(e) The relations in (2.21) and (2.23) all follow from the relations twκitw−1 = κw(i) andtwγi,jtw−1 = γw(i),w(j) in (2.27).

(f) By second relation in (2.28) and the (already established) second relation in (2.20)

[κ1, ts1y1ts1 ] = [κ1, ts1(y1 − 12κ1)ts1 + 1

2 ts1κ1ts1 ] = [κ1, γ02 + 12 ts1κ1ts1 ] = 0,

which establishes (2.22).

To complete the proof let us show that the relations of (2.27-2.29) follow from the relations in(2.19-2.22).

(a) The relation tutv = tuv in (2.27) is the first relation in (2.19).

(b) The relations twκitw−1 = κw(i) in (2.27) follow from the first and last relations in (2.20)(and force the definition of κm in (2.33)).

(c) Since γ0,1 = y1 − 12κ1, the relations twγ0,jtw−1 = γ0,w(j) in (2.28) follow from the last

relation in each of (2.20) and (2.21) (and force the definition of γ0,m in (2.33)).

(d) Since γ1,2 = y2 − ts1y1ts1 , the first relation in (2.21) gives that γ2,1 = γ1,2 since

ts1γ1,2ts1 − γ1,2 = (ts1y2ts1 − y1)− y2 + ts1y1ts1 = ts1(y1 + y2)ts1 − (y1 + y2) = 0. (2.35)

The relations twγ1,2tw−1 = γw(1),w(2) in (2.27) then follow from (2.35) and the last relationin (2.21) (and force the definitions γi,j = tvγ1,2tv−1 in (2.33)).

(e) The third relation in (2.19) is κ0κ1 = κ1κ0 and the second relation in (2.20) gives κ1κ2 =κ2κ1. The relations κiκj = κjκi in (2.28) then follow from the second set of relations in(2.27).

(f) The second relation in (2.20) gives [κ1, κ2] = 0. Multiplying (2.22) on the left and rightby ts1 gives [y1, κ2] = [y1, ts1κ1ts1 ] = 0. Using these and the relations in (2.19),

[κ1, γ0,2 + γ1,2] = [κ1, (y2 − 12κ2 − γ1,2) + γ1,2] = −[κ1,

12κ2] = 0, (2.36)

and[γ0,1, γ0,2 + γ1,2] = [y1 − 1

2κ1, y2 −12κ2] = 1

4 [κ1, κ2] = 0, (2.37)

so that

[γ0,1, κ2] = [γ0,1, 2y2 − 2(γ0,2 + γ1,2)] = [γ0,1, 2y2] = [y1 − 12κ1, 2y2] = −[κ1, y2] = 0.

Conjugating the last relation by ts1 gives

[κ1, γ0,2] = 0, and thus [κ1, γ1,2] = 0,

by (2.36). By the third and fourth relations in (2.19),

[κ0, γ0,1] = [κ0, y1 − 12κ1] = 0, and [κ1, γ0,1] = [κ1, y1 − 1

2κ1] = 0.

By the relations in (2.20) and (2.19),

[κ0, γ1,2] = [κ0, y2 − ts1y1ts1 ] = 0 and [κ1, γ2,3] = [κ1, y3 − ts2y2ts2 ] = 0.

Putting these together with the (already established) relations in (2.27) provides the secondset of relations in (2.28).

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(g) From the commutativity of the yi and the second relation in (2.21)

γ1,2γ3,4 = (y2 − ts1y1ts1)(y4 − ts3y3ts3) = (y4 − ts3y3ts3)(y2 − ts1y1ts1) = γ3,4γ1,2.

By the last relation in (2.19) and the last relation in (2.20),

[γ0,1, γ2,3] = [y1 − 12κ1, y3 − ts2y2ts2 ] = 0.

Together with the (already established) relations in (2.27), we obtain the first set of rela-tions in (2.29).

(h) Conjugating (2.37) by ts2ts1ts2 gives [γ0,2, γ0,3 + γ2,3] = 0, and this and the (alreadyestablished) relations in (2.28) and the first set of relations in (2.29) provide

0 = [y2, y3] = [12κ2 + γ0,2 + γ1,2,12κ3 + γ0,3 + γ1,3 + γ2,3]

= [γ0,2 + γ1,2, γ0,3 + γ1,3 + γ2,3] = [γ1,2, γ0,3 + γ1,3 + γ2,3] = [γ1,2, γ1,3 + γ2,3].

Note also that

[γ1,2, γ1,0 + γ2,0] = [γ1,2, γ0,1 + γ0,2] = −[γ0,1, γ1,2] + [γ1,2, γ0,2]

= [γ0,1, γ0,2] + [γ1,2, γ0,2] = ts1 [γ0,2 + γ1,2, γ0,1]ts1 = 0,

by (two applications of) (2.37). The last set of relations in (2.29) now follow from the lastset of relations in (2.27).

By the first formula in (2.24) and the last formula in (2.34),

cj =

j∑i=0

κi + 2∑

0≤`<m≤jγ`,m. (2.38)

2.3 The degenerate affine BMW algebra Wk

Let C be a commutative ring and let Bk be the degenerate affine braid algebra over C as definedin Section 2.2. Define ei in the degenerate affine braid algebra by

tsiyi = yi+1tsi − (1− ei), for i = 1, 2, . . . , k − 1, (2.39)

so that, with γi,i+1 as in (2.23),γi,i+1tsi = 1− ei. (2.40)

Fix constantsε = ±1 and z

(`)0 ∈ C, for ` ∈ Z≥0.

The degenerate affine Birman-Wenzl-Murakami (BMW) algebra Wk (with parameters ε and z(`)0 )

is the quotient of the degenerate affine braid algebra Bk by the relations

eitsi = tsiei = εei, eitsi−1ei = eitsi+1ei = εei, (2.41)

e1y`1e1 = z

(`)0 e1, ei(yi + yi+1) = 0 = (yi + yi+1)ei. (2.42)

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The degenerate affine Hecke algebra Hk is the quotient of Wk by the relations

ei = 0, for i = 1, . . . , k − 1. (2.43)

Fix b1, . . . , br ∈ C. The degenerate cyclotomic BMW algebra Wr,k(b1, . . . , br) is the degenerateaffine BMW algebra with the additional relation

(y1 − b1) · · · (y1 − br) = 0. (2.44)

The degenerate cyclotomic Hecke algebra Hr,k(b1, . . . , br) is the degenerate affine Hecke algebraHk with the additional relation (2.44).

Since the composite map C[y1, . . . , yk] → Bk → Wk → Hk is injective (see [19, Theorem3.2.2]) and the last two maps are surjections, it follows that the polynomial ring C[y1, . . . , yk] isa subalgebra of Bk and Wk.

Proposition 2.3. Let C = C, κ0, κ1 ∈ C and ε = 1. Then the degenerate affine BMW algebraWk coincides with the one defined in [26].

Proof. In [26], the degenerate affine BMW algebra is defined with the first two relations in (2.19)and the second set of relations in (2.21), which are [26, (4.1)] and the first relations in [26, (1.2)and (1.3)], the relations in (2.39) which are [26, (4.2)], the relations in (2.42) which are [26,(4.3) and (4.4)], the first relations in (2.41) which is the third set of relations in [26, (1.2)], therelations in (2.48) below which are the last two relations in [26, (1.3)] and the second relationin [26, (1.2)], the relations in (2.50) below which are [26, (1.4)], and the relations

eitsj = tsjei and eiej = ejei, for |j − i| > 1, (2.45)

which are the second and third relations in [26, (1.5)].Working in Wk and conjugating (2.39) by tsi and using the first relation in (2.41) gives

yitsi = tsiyi+1 − (1− ei). (2.46)

Then, by (2.40) and (2.23),

γi,i+1 = tsi − εei, and ei+1 = tsitsi+1eitsi+1tsi . (2.47)

Multiply the second relation in (2.47) on the left and the right by ei, and then use the relationsin (2.41) to get

eiei+1ei = eitsitsi+1eitsi+1tsiei = eitsi+1eitsi+1ei = ε eitsi+1ei = ei,

so thateiei±1ei = ei. Note that e2i = z

(0)0 ei (2.48)

is, for i = 1, a special case of the first identity in (2.42) and then, for general i, follows from thesecond identity in (2.47). The relations

ei+1ei = ei+1tsitsi+1 , eiei+1 = tsi+1tsiei+1, (2.49)

tsiei+1ei = tsi+1ei, and ei+1eitsi+1 = ei+1tsi (2.50)

result from (2.41) and the second relation in (2.47). The relations in (2.45) follow from (2.39) thefirst two relations in (2.19) and the last relations in (2.21). Thus the relations in the definitionof the degenerate affine BMW algebra in [26] follow from the defining relations of Wk.

To complete the proof we must show that the first relations in (2.21), the relations in (2.23),and the second relations in (2.41) follow from the defining relations used in [26]. Because of theassumption that κ0, κ1 ∈ C the other relations in (2.19)-(2.23) are automatic.

9

(a) Multiplying the first relation in (2.50) on the left by ei and using the first relations in(2.41) and the first relations in (2.48) provides part of the second relations in (2.41) andthe other part is obtained similarly by multiplying the second relations in (2.50) on theright by ei+1.

(b) Conjugating (2.39) by tsi produces (2.46) and then adding (2.39) and (2.46) produces thefirst relations in (2.21).

(c) Using (2.50),

ei+1 = tsitsi(ei+1tsi)tsi = tsitsiei+1eitsi+1tsi = tsitsi+1eitsi+1tsi

which, with (2.39), gives the relations in (2.23).

3 Identities in affine and degenerate affine BMW algebras

In [26], Nazarov defined some naturally occurring central elements in the degenerate affineBMW algebra Wk and proved a remarkable recursion for them. This recursion was generalizedto analogous central elements in the affine BMW algebra Wk by Beliakova-Blanchet [4]. Inboth cases, the recursion was accomplished with an involved computation. In this section,we provide a new proof of the Nazarov and Beliakov-Blanchet recursions by lifting them outof the center, to intertwiner-like identities in Wk and Wk (Propositions 3.1 and 3.3). Theseintertwiner-like identities for the degenerate affine and affine BMW algebras are reminiscent ofthe intertwiner identities for the degenerate affine and affine Hecke algebras found, for example,in [20, Prop. 2.5(c)] and [29, Prop. 2.14(c)], respectively. The central element recursions of [26]and [4] are then obtained by multiplying the intertwiner-like identities by the projectors ek andEk, respectively. We shall not include our new proofs of Proposition 3.1 and Theorem 3.2 heresince, given our parallel setup of the degenerate affine and the affine BMW algebras in Section2, the proof is exactly parallel to the proofs of Proposition 3.3 and Theorem 3.4.

3.1 The degenerate affine case

Let Wk be the degenerate affine BMW algebra as defined in (2.41- 2.42) and let 1 ≤ i < k − 1.Let u be a variable and let

u+i =1

u− yiand u−i =

1

u+ yi. (3.1)

Proposition 3.1. In the degenerate affine BMW algebra Wi+1,(ei

1

1− yi+1− tsi −

1

2u− (yi + yi+1)

)(ei

1

1− yi+ tsi −

1

2u− (yi + yi+1)

)=−(2u− (yi + yi+1) + 1)(2u− (yi + yi+1)− 1)

(2u− (yi + yi+1))2, (3.2)

and(u+i+1 + tsi − ei

1

2u− (yi + yi+1)

)− u+i

(u+i+1 + tsi − ei

1

2u− (yi + yi+1)

)u+i (3.3)

=

(tsiu

+i tsi + tsi − ei

1

2u− (yi + yi+1)

)− u+i+1

(eiu

+i ei + εei − ei

1

2u− (yi + yi+1)

)u+i+1.

10

The identities (3.4) and (3.5) of the following theorem are [26, Lemma 2.5], and [26, Lemma3.8], respectively.

Theorem 3.2. [26] Let Wk be the degenerate affine BMW algebra as defined in (2.41-2.42) and

let 1 ≤ i < k − 1. Let z0(u) =∑

`∈Z≥0z(`)0 u−`. Then(

eiu−i − ε−

12u

) (eiu

+i + ε− 1

2u

)ei = −

(ε+ 1

2u

) (ε− 1

2u

)ei, and (3.4)(

ei+1u+i+1 + ε− 1

2u

)ei+1 =

(z0(u)

u+ ε− 1

2u

) i∏j=1

(u+ yj − 1)(u+ yj + 1)(u− yj)2

(u+ yj)2(u− yj + 1)(u− yj − 1)ei+1. (3.5)

3.2 The affine case

Let Wk be the affine BMW algebra as defined in (2.8-2.9) and let 1 ≤ i < k − 1. Let u be avariable,

U+i =

Yiu− Yi

, and note that U+i U

+i+1 =

YiYi+1

u2 − YiYi+1(U+

i + U+i+1 + 1). (3.6)

By the definition of Ei in (2.7),

(u− Yi+1)Ti = Ti(u− Yi)− (q − q−1)Yi+1(1− Ei),

and, by (2.12),(u− Yi)Ti = Ti(u− Yi+1) + (q − q−1)(1− Ei)Yi+1,

so that

Ti1

u− Yi=

1

u− Yi+1Ti − (q − q−1) Yi+1

u− Yi+1(1− Ei)

1

u− Yi, and (3.7)

Ti1

u− Yi+1=

1

u− YiTi + (q − q−1) 1

u− Yi(1− Ei)

Yi+1

u− Yi+1. (3.8)

The relations

TiU+i = U+

i+1T−1i − (q − q−1)U+

i+1(1− Ei)U+i

= U+i+1

(T−1i − (q − q−1)(1− Ei)U+

i

), and

(3.9)

T−1i U+i+1 = U+

i Ti − (q − q−1)U+i EiU

+i+1 + (q − q−1)U+

i+1U+i

= U+i

(Ti + (q − q−1)(1− Ei)U+

i+1

) (3.10)

are obtained by multiplying (3.7) and (3.8) on the right (resp. left) by Yi and using the relationTiYi = Yi+1T

−1i .

Taking the coefficient of u−(`+1) on each side of (3.7) and (3.8) gives

TiY`i = Y `

i+1Ti − (q − q−1)(Y `i+1(1− Ei) + Y `−1

i+1 (1− Ei)Yi + · · ·+ Yi+1(1− Ei)Y `−1i ), (3.11)

TiY`i+1 = Y `

i Ti + (q − q−1)(Y `−1i (1− Ei)Yi+1 + Y `−2

i (1− Ei)Y 2i+1 + · · ·+ (1− Ei)Y `

i+1), (3.12)

respectively, for ` ∈ Z≥0. Therefore,

TiY−`i = Y −`i+1Ti + (q − q−1)

(Y−(`−1)i+1 (1− Ei)Y −1i + · · ·+ (1− Ei)Y −`i

), (3.13)

TiY−`i+1 = Y −`i Ti − (q − q−1)

(Y −`i (1− Ei) + · · ·+ Y −1i (1− Ei)Y −(`−1)i+1

). (3.14)

11

Proposition 3.3. Let Q = q − q−1. Then, in the affine BMW algebra Wi+1,(Ei

Yi+1

u− Yi+1− TiQ− YiYi+1

u2 − YiYi+1

)(Ei

Yiu− Yi

+T−1i

Q− YiYi+1

u2 − YiYi+1

)=−(u2 − q2YiYi+1)(u

2 − q−2YiYi+1)

Q2(u2 − YiYi+1)2, and (3.15)

(U+i+1 +

TiQ− Ei

YiYi+1

u2 − YiYi+1

)−Q2(U+

i + 1)

(U+i+1 +

TiQ− Ei

YiYi+1

u2 − YiYi+1

)U+i (3.16)

=

(TiU

+i T−1i +

TiQ− Ei

YiYi+1

u2 − YiYi+1

)−Q2U+

i+1

(EiU

+i Ei + z

EiQ− Ei

YiYi+1

u2 − YiYi+1

)(U+

i+1 + 1).

Proof. Putting (3.6) into (3.9) says that if

A =TiQ

+YiYi+1

u2 − YiYi+1and B = EiU

+i +

T−1i

Q− YiYi+1

u2 − YiYi+1

then

AU+i = U+

i+1B −YiYi+1

u2 − YiYi+1. Next, AEi = EiA

follows from (2.8) and (2.9). So(Ei

Yi+1

u− Yi+1− TiQ− YiYi+1

u2 − YiYi+1

)(Ei

Yiu− Yi

+T−1i

Q− YiYi+1

u2 − YiYi+1

)= Ei(U

+i+1B)−AB = Ei

(AU+

i +YiYi+1

u2 − YiYi+1

)−AB = A(EiU

+i −B) + Ei

YiYi+1

u2 − YiYi+1

= −(TiQ

+YiYi+1

u2 − YiYi+1

)(T−1i

Q− YiYi+1

u2 − YiYi+1

)+ Ei

YiYi+1

u2 − YiYi+1,

and, by (2.13), multiplying out the right hand side gives (3.15).Rewrite T−1i U+

i+1 = U+i T−1i +QU+

i (1− Ei)(U+i+1 + 1) as

T−1i U+i+1 −Q(U+

i+1 + 1)U+i = U+

i T−1i −QU+

i Ei(U+i+1 + 1),

and multiply on the left by Ti to get

U+i+1 −QTi(U

+i+1 + 1)U+

i = TiU+i T−1i −QTiU+

i Ei(U+i+1 + 1). (3.17)

Then, since Ti = T−1i +Q(1− Ei), equations (3.10) and (3.9) imply

Ti(U+i+1 + 1) = Q(U+

i + 1)

(TiQ

+ (1− Ei)U+i+1

)and TiU

+i = QU+

i+1

(T−1i

Q− (1− Ei)U+

i

),

and so (3.17) is

U+i+1 −Q

2(U+i + 1)

(TiQ

+ (1− Ei)U+i+1

)U+i (3.18)

= TiU+i T−1i −Q2U+

i+1

(T−1i

Q− (1− Ei)U+

i

)Ei(U

+i+1 + 1).

12

Using (3.6) and

addingTiQ− Ei

YiYi+1

u2 − YiYi+1−Q2 YiYi+1

u2 − YiYi+1(U+

i + 1)Ei(U+i+1 + 1) to each side

of (3.18) gives

U+i+1 +

TiQ− Ei

YiYi+1

u2 − YiYi+1−Q2(U+

i + 1)

(U+i+1 +

TiQ− Ei

YiYi+1

u2 − YiYi+1

)U+i

= TiU+i T−1i +

TiQ− Ei

YiYi+1

u2 − YiYi+1−Q2U+

i+1

(EiU

+i +

T−1i

Q− YiYi+1

u2 − YiYi+1

)Ei(U

+i+1 + 1)

= TiU+i T−1i +

TiQ− Ei

YiYi+1

u2 − YiYi+1−Q2U+

i+1

(EiU

+i Ei + z

EiQ− Ei

YiYi+1

u2 − YiYi+1

)(U+

i+1 + 1),

completing the proof of (3.16).

Let Z+0 and Z−0 be the generating functions

Z+0 =

∑`∈Z≥0

Z(`)0 u−` and Z−0 =

∑`∈Z≤0

Z(`)0 u−`.

If

U−i =Y −1i

u− Y −1i

then EiU+i+1 = EiU

−i and U+

i+1Ei = U−i Ei, (3.19)

by the second identity in (2.9). The first identity in (2.9) is equivalent to

E1U+1 E1 = (Z+

0 − Z(0)0 )E1.

In the following theorem, the identity (3.20) is equivalent to [12, Lemma 2.8, parts (2) and (3)]or [13, Lemma 2.6(4)] and the identity (3.21) is equivalent to the identity found in [4, Lemma7.4].

Theorem 3.4. [4, 12, 13] Let Wk be the affine BMW algebra as defined in (2.8-2.9) and let1 ≤ i < k − 1. Then(

EiU−i −

z−1

q − q−1− 1

u2 − 1

)(EiU

+i +

z

q − q−1− 1

u2 − 1

)Ei =

−(u2 − q2)(u2 − q−2)(u2 − 1)2(q − q−1)2

Ei,

(3.20)

and(Ei+1U

+i+1 +

z

q − q−1− 1

u2 − 1

)Ei+1

=

(Z+0 +

z−1

q − q−1− u2

u2 − 1

) i∏j=1

(u− Yj)2(u− q−2Y −1j )(u− q2Y −1j )

(u− Y −1j )2(u− q2Yj)(u− q−2Yj)

Ei+1. (3.21)

Proof. Multiply (3.15) on the right by Ei and use Z(0)i−1 = 1 + (z − z−1)/(q − q−1) to get (3.20).

Multiplying (3.16) on the left and right by Ei+1 and using the relations in (2.8), (2.9), (2.14),and

Ei+1TiU+i T−1i Ei+1 = Ei+1TiTi+1U

+i T−1i+1T

−1i Ei+1 = Ei+1EiU

+i EiEi+1,

13

gives (Ei+1U

+i+1 +

z

Q− 1

u2 − 1

)Ei+1

(1−Q2(U+

i + 1)U+i

)= Ei+1

(EiU

+i +

z

Q− 1

u2 − 1

)EiEi+1

−Q2U−i Ei+1

(EiU

+i +

z

Q− 1

u2 − 1

)EiEi+1(U

−i + 1)

= (1−Q2LU−iRU−i +1)

(Ei+1

(EiU

+i +

z

Q− 1

u2 − 1

)EiEi+1

)where LU−i

is the operator of left multiplication by U−i and RU−i +1 is the operator of right

multiplication by U−i + 1. Then, by induction,(Ei+1U

+i+1 +

z

Q− 1

u2 − 1

)Ei+1

i∏j=1

(1−Q2U+

j (U+j + 1)

)

=

i∏j=1

(1−Q2LU−jRU−j +1)

(Ei+1Ei . . . E2(E1U+1 +

z

Q− 1

u2 − 1)E1E2 . . . EiEi+1

)

=i∏

j=1

(1−Q2LU−jRU−j +1)

(Ei+1Ei . . . E2(Z

+0 − Z

(0)0 +

z

Q− 1

u2 − 1)E1E2 . . . EiEi+1

)

= (Z+0 − Z

(0)0 +

z

Q− 1

u2 − 1)

i∏j=1

(1−Q2LU−jRU−j +1)

(Ei+1Ei . . . E2E1E2 . . . EiEi+1)

= (Z+0 − Z

(0)0 +

z

Q− 1

u2 − 1)

i∏j=1

(1−Q2U−j (U−j + 1))Ei+1.

So (3.21) follows from

1− (q − q−1)2U−j (U−j + 1)

1− (q − q−1)2U+j (U+

j + 1)=

1− (q − q−1)2 Y −1j

u−Y −1j

(Y −1j

u−Y −1j

+ 1

)1− (q − q−1)2 Yj

u−Yj

(Yj

u−Yj + 1)

=((u− Y −1j )2 − (q − q−1)2Y −1j u) 1

(u−Y −1j )2

((u− Yj)2 − (q − q−1)2Yju) 1(u−Yj)2

=(u− q−2Y −1j )(u− q2Y −1j )(u− Yj)2

(u− q−2Yj)(u− q2Yj)(u− Y −1j )2

and Z(0)0 = 1 + (z − z−1)/(q − q−1).

4 The center of the affine and degenerate affine BMW algebras

In this section, we identify the center ofWk and Wk. Both centers arise as algebras of symmetricfunctions with a “cancellation property” (in the language of [28]) or “wheel condition” (in thelanguage of [8]). In the degenerate case, Z(Wk) is the ring of symmetric functions in y1, . . . , yk

14

with the Q-cancellation property of Pragacz. By [28, Theorem 2.11(Q)], this is the same ring asthe ring generated by the odd power sums, which is the way that Nazarov [26] identified Z(Wk).

The cancellation property in the case of Wk is analogous, exhibiting the center of the affineBMW algebra Z(Wk) as a subalgebra of the ring of symmetric Laurent polynomials. At theend of this section, in an attempt to make the theory for the affine BMW algebra completelyanalogous to that for the degenerate affine BMW algebra, we have formulated an alternatedescription of Z(Wk) as a ring generated by “negative” power sums.

4.1 Bases of Wk and Wk

The Brauer algebra, depending on a parameter x, is given by generators e1, . . . , ek−1 ands1, . . . , sk−1 and relations as given in [26, (1.2)-(1.5)] (where our ei is denoted si and our xis denoted N). The Brauer algebra also has a diagrammatic presentation (see [5]) with basis

Dk = { diagrams on k dots }, (4.1)

where a (Brauer) diagram on k dots is a graph with k dots in the top row, k dots in thebottom row and k edges pairing the dots. We label the vertices of the top row, left to right,with 1, 2, . . . , k and the vertices in the bottom row, left to right, with 1′, 2′, . . . , k′ so that, forexample,

d =••••••••••••••...................................................................................................................

...........................................

........................................................

............................................................................ ......................................

.....

..................................

.....................................................................................................................................................................

..............................................................= (13)(21′)(45)(66′)(74′)(2′7′)(3′5′) (4.2)

is a Brauer diagram on 7 dots. Setting

x = z(0)0 and si = εtsi

realizes the Brauer algebra as a subalgebra of the degenerate affine BMW algebra Wk. TheBrauer algebra is also the quotient ofWk by y1 = 0 and, hence, can be viewed as the degeneratecyclotomic BMW algebra W1,k(0).

Theorem 4.1. [26, 2] Let Wk be the degenerate affine BMW algebra and let Wr,k(b1, . . . , br) bethe degenerate cyclotomic BMW algebra as defined in (2.41)-(2.42) and (2.44), respectively. Forn1, . . . , nk ∈ Z≥0 and a diagram d on k dots let

dn1,...,nk = yn1i1· · · yn`

i`dy

n`+1

i`+1· · · ynk

ik,

where, in the lexicographic ordering of the edges (i1, j1), . . . , (ik, jk) of d, i1, . . . , i` are in the toprow of d and i`+1, . . . , ik are in the bottom row of d. Let Dk be the set of diagrams on k dots,as in (4.1).

(a) If κ0, κ1 ∈ C and(z0(−u)−

(12 + ε u

)) (z0(u)−

(12 − ε u

))=(12 − ε u

)(12 + ε u

)(4.3)

then {dn1,...,nk | d ∈ Dk, n1, . . . , nk ∈ Z≥0} is a C-basis of Wk.

(b) If κ0, κ1 ∈ C, (4.3) holds, and

(z0(u) + u− 1

2

)= (u− 1

2(−1)r)

(r∏i=1

u+ biu− bi

)(4.4)

then {dn1,...,nk | d ∈ Dk, 0 ≤ n1, . . . , nk ≤ r − 1} is a C-basis of Wr,k(b1, . . . , br).

15

Part (a) of Theorem 4.1 is [26, Theorem 4.6] (see also [2, Theorem 2.12]) and part (b) is [2,Prop. 2.15 and Theorem 5.5].

Theorem 4.2. [13, 37] Let Wk be the affine BMW algebra and let Wr,k(b1, . . . , br) be the cyclo-tomic BMW algebra as defined in Section 2.1. Let d ∈ Dk be a Brauer diagram, where Dk is asin (4.1). Choose a minimal length expression of d as a product of e1, . . . , ek−1, s1, . . . , sk−1,

d = a1 · · · a`, ai ∈ {e1, . . . , ek−1, s1, . . . , sk−1},

such that the number of si in this product is the number of crossings in d. For each ai which isin {s1, . . . , sk−1} fix a choice of sign εj = ±1 and set

Td = A1 · · ·A`, where Aj =

{Ei, if aj = ei,

Tεji , if aj = si.

For n1, . . . , nk ∈ Z letTn1,...,nkd = Y n1

i1· · ·Y n`

i`TdY

n`+1

i`+1· · ·Y nk

ik,

where, in the lexicographic ordering of the edges (i1, j1), . . . , (ik, jk) of d, i1, . . . , i` are in the toprow of d and i`+1, . . . , ik are in the bottom row of d.

(a) If(Z−0 −

z

q − q−1− u2

u2 − 1

)(Z+0 +

z−1

q − q−1− u2

u2 − 1

)=−(u2 − q2)(u2 − q−2)(u2 − 1)2(q − q−1)2

(4.5)

then {Tn1,...,nkd | d ∈ Dk, n1, . . . , nk ∈ Z} is a C-basis of Wk.

(b) If (4.5) holds and

Z+0 +

z−1

q − q−1− u2

u2 − 1=

(z

q − q−1+

uz

u2 − 1

) r∏j=1

u− b−1ju− bj

(4.6)

then {Tn1,...,nkd | d ∈ Dk, 0 ≤ n1, . . . , nk ≤ r − 1} is a C-basis of Wr,k(b1, . . . , br).

Part (a) of Theorem 4.2 is [13, Theorem 2.25] and part (b) is [13, Theorem 5.5] and [37,Theorem 8.1]. We refer to these references for proof, remarking only that one key point inshowing that {Tn1,...,nk

d | d ∈ Dk, n1, . . . , nk ∈ Z} spans Wk is that if (i, j) is a top-to-bottomedge in d then

YiTd = TdYj + (terms with fewer crossings), (4.7)

and, if (i, j) is a top-to-top edge in d then

YiTd = Y −1j Td + (terms with fewer crossings). (4.8)

4.2 The center of Wk

The degenerate affine BMW algebra is the algebraWk over C defined in Section 2.3 and the poly-nomial ring C[y1, . . . , yk] is a subalgebra of Wk. The symmetric group Sk acts on C[y1, . . . , yk]by permuting the variables. A classical fact (see, for example, [19, Theorem 3.3.1]) is that thecenter of the degenerate affine Hecke algebra Hk is the ring of symmetric functions

Z(Hk) = C[y1, . . . , yk]Sk = {f ∈ C[y1, . . . , yk] | wf = f , for w ∈ Sk}.

16

Theorem 4.3 gives an analogous characterization of the center of the degenerate affine BMWalgebra. We shall not include the proof here since, given our parallel setup of the degenerateaffine BMW algebras and the affine BMW algebras in Section 2, the proof is exactly parallel tothe proof of Theorem 4.4.

Theorem 4.3. The center of the degenerate affine BMW algebra Wk is

Rk = {f ∈ C[y1, . . . , yk]Sk | f(y1,−y1, y3, . . . , yk) = f(0, 0, y3, . . . , yk)}.

The power sum symmetric functions pi are given by

pi = yi1 + yi2 + · · ·+ yik, for i ∈ Z>0.

The Hall-Littlewood polynomials (see [24, Ch. III (2.1)]) are given by

Pλ(y; t) = Pλ(y1, . . . , yk; t) =1

vλ(t)

∑w∈Sk

w

yλ11 · · · yλkk ∏1≤i<j≤k

xi − txjxi − xj

,

where vλ(t) is a normalizing constant (a polynomial in t) so that the coefficient of yλ11 · · · yλkk in

Pλ(y; t) is equal to 1. The Schur Q-functions (see [24, Ch. III (8.7)]) are

Qλ =

{0, if λ is not strict,

2`(λ)Pλ(y;−1), if λ is strict,

where `(λ) is the number of (nonzero) parts of λ and the partition λ is strict if all its (nonzero)parts are distinct. Let Rk be as in Theorem 4.3. Then (see [26, Cor. 4.10], [28, Theorem2.11(Q)] and [24, Ch. III §8])

Rk = C[p1, p3, p5, . . .] = Cspan-{Qλ | λ is strict}. (4.9)

More generally, let r ∈ Z>0 and let ζ be a primitive rth root of unity. Define

Rr,k = {f ∈ Z[ζ][y1, . . . , yk]Sk | f(y1, ζy1, . . . , ζ

r−1y1, yr+1, . . . , yk) = f(0, 0, . . . , 0, yr+1, . . . , yk)}.

ThenRr,k ⊗Z[ζ] Q(ζ) = Q(ζ)[pi | i 6= 0 mod r], (4.10)

andRr,k has Z[ζ]-basis {Pλ(y; ζ) | mi(λ) < r and λ1 ≤ k}, (4.11)

where mi(λ) is the number parts of size i in λ. The ring Rr,k is studied in [25], [22], [24, Ch.III Ex. 5.7 and Ex. 7.7], [35], [8], and others. The proofs of (4.10) and (4.11) follow from [24,Ch. III Ex. 7.7], [35, Lemma 2.2 and following remarks] and the arguments in the proofs of [8,Lemma 3.2 and Proposition 3.5].

4.3 The center of Wk

The affine BMW algebra is the algebra Wk over C defined in Section 2.1 and the ring of Lau-rent polynomials C[Y ±11 , . . . , Y ±1k ] is a subalgebra of Wk. The symmetric group Sk acts onC[Y ±11 , . . . , Y ±1k ] by permuting the variables. A classical fact (see, for example, [15, Proposition2.1]) is that the center of the affine Hecke algebra Hk is the ring of symmetric functions,

Z(Hk) = C[Y ±11 , . . . , Y ±1k ]Sk = {f ∈ C[Y ±11 , . . . , Y ±1k ] | wf = f , for w ∈ Sk}.

Theorem 4.4 is a characterization of the center of the affine BMW algebra.

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Theorem 4.4. The center of the affine BMW algebra Wk is

Rk = {f ∈ C[Y ±11 , . . . , Y ±1k ]Sk | f(Y1, Y−11 , Y3, . . . , Yk) = f(1, 1, Y3, . . . , Yk)}.

Proof. Step 1: f ∈Wk commutes with all Yi ⇔ f ∈ C[Y ±11 , . . . , Y ±1k ]:Assume f ∈Wk and write

f =∑

cn1,...,nkd Tn1,...,nk

d ,

in terms of the basis in Theorem 4.2. Let d ∈ Dk with the maximal number of crossings suchthat cn1,...,nk

d 6= 0 and, using the notation after (4.2), suppose there is an edge (i, j) of d suchthat j 6= i′. Then, by (4.7) and (4.8),

the coefficient of YiTn1,...,nkd in Yif is cn1,...,nk

d

andthe coefficient of YiT

n1,...,nkd in fYi is 0.

If Yif = fYi it follows that there is no such edge, and so d = 1 (and therefore Td = 1). Thusf ∈ C[Y ±11 , . . . , Y ±1k ]. Conversely, if f ∈ C[Y ±11 , . . . , Y ±1k ], then Yif = fYi.Step 2: f ∈ C[Y ±11 , . . . , Y ±1k ] commutes with all Ti ⇔ f ∈ Rk:Assume f ∈ C[Y ±11 , . . . , Y ±1k ] and write

f =∑a,b∈Z

Y a1 Y

b2 fa,b, where fa,b ∈ C[Y ±13 , . . . , Y ±1k ].

Then f(1, 1, Y3, . . . , Yk) =∑a,b∈Z

fa,b and

f(Y1, Y−11 , Y3, . . . , Yk) =

∑a,b∈Z

Y a−b1 fa,b =

∑`∈Z

Y `1

(∑b∈Z

f`+b,b

). (4.12)

By direct computation using (3.12) and (3.14),

T1Ya1 Y

b2 = Y a

1 Ya2 T1Y

b−a2 = s1(Y

a1 Y

b2 )T1 + (q − q−1)Y

a1 Y

b2 − s1(Y a

1 Yb2 )

1− Y1Y −12

+ Eb−a,

where

E` =

−(q − q−1)∑r=1

Y `−r1 E1Y

−r1 , if ` > 0,

(q − q−1)−∑r=1

Y `+r−11 E1Y

r−11 , if ` < 0,

0, if ` = 0.

It follows that

T1f = (s1f)T1 + (q − q−1) f − s1f1− Y1Y −12

+∑`∈Z6=0

E`

(∑b∈Z

f`+b,b

). (4.13)

Thus, if f(Y1, Y−11 , Y3, . . . , Yk) = f(1, 1, Y3, . . . , Yk) then, by (4.12),∑

b∈Zf`+b,b = 0, for ` 6= 0. (4.14)

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Hence, if f ∈ C[Y ±11 , . . . , Y ±1k ]Sk and f(Y1, Y−11 , Y3, . . . , Yk) = f(1, 1, Y3, . . . , Yk) then s1f = f

and (4.14) holds so that, by (4.13), T1f = fT1. Similarly, f commutes with all Ti.Conversely, if f ∈ C[Y ±11 , . . . , Y ±1k ] and Tif = fTi then

sif = f and∑b∈Z

f`+b,b = 0, for ` 6= 0,

so that f ∈ C[Y ±11 , . . . , Y ±1k ]Sk and f(Y1, Y−11 , Y3, . . . , Yk) = f(1, 1, Y3, . . . , Yk).

It follows from (2.7) that Rk = Z(Wk).

The symmetric group Sk acts on Zk by permuting the factors. The ring

C[Y ±11 , . . . , Y ±1k ]Sk has basis {mλ | λ ∈ Zk with λ1 ≥ λ2 ≥ · · · ≥ λk},

wheremλ =

∑µ∈Skλ

Y µ11 · · ·Y

µkk .

The elementary symmetric functions are

er = m(1r,0k−r) and e−r = m(0k−r,(−1)r), for r = 0, 1, . . . , k,

and the power sum symmetric functions are

pr = m(r,0k−1) and p−r = m(0k−1,−r), for r ∈ Z>0.

The Newton identities (see [24, Ch. I (2.11′)]) say

`e` =∑r=1

(−1)r−1pre`−r and `e−` =∑r=1

(−1)r−1p−re−(`−r), (4.15)

where the second equation is obtained from the first by replacing Yi with Y −1i . For ` ∈ Z andλ = (λ1, . . . , λk) ∈ Zk,

e`kmλ = mλ+(`k), where λ+ (`k) = (λ1 + `, . . . , λk + `).

In particular,e−r = e−1k ek−r, for r = 0, . . . , k. (4.16)

Definep+i = pi + p−i and p−i = pi − p−i, for i ∈ Z>0. (4.17)

The consequence of (4.16) and (4.15) is that

C[Y ±11 , . . . , Y ±1k ]Sk = C[e±1k , e1, . . . , ek−1]

= C[e±1k ][e1, e2, . . . , eb k2c, eke−b k−1

2c, . . . , eke−2, eke−1]

= C[e±1k ][e1, e2, . . . , eb k2c, e−b k−1

2c, . . . , e−2, e−1]

= C[e±1k ][p1, p2, . . . , pb k2c, p−b k−1

2c, . . . , p−2, p−1]

= C[e±1k ][p+1 , p+2 , . . . , p

+

b k2c, p−b k−1

2c, . . . , p−2 , p

−1 ].

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For ν ∈ Zk with ν1 ≥ · · · ≥ ν` > 0 define

p+ν = p+ν1 · · · p+ν`

and p−ν = p−ν1 · · · p−ν`.

Then

C[Y ±11 , . . . , Y ±1k ]Sk has basis {e`kp+λ p−µ | ` ∈ Z, `(λ) ≤ bk2c, `(µ) ≤ bk−12 c}. (4.18)

In analogy with (4.9) we expect that if Rk is as in Theorem 4.4 then

Rk = C[e±1k ][p−1 , p−2 , . . .]. (4.19)

References

[1] T. Arakawa and T. Suzuki: Duality between sln(C) and the degenerate affine Hecke algebraof type A, J. Algebra 209 (1998) 288–304.

[2] S. Ariki, A. Mathas and H. Rui: Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182(2006) 47–134.

[3] P. Baumann: On the center of quantized enveloping algebras, J. Algebra 203 (1998) 244–260.

[4] A. Beliakova and C. Blanchet: Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001) 347–373.

[5] R. Brauer: On algebras which are connected with the semisimple continuous groups, Ann.Math. 38 (1937) 857-872.

[6] Z. Daugherty, A. Ram and R. Virk: Affine and degenerate affine BMW algebras: Actionson tensor space, arXiv:1205.1852. To appear in Selecta Math..

[7] J. Enyang: Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras, J. Alg. Comb. 26 (2007) 291-341.

[8] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, and Y. Takeyama: Symmetric polynomialsvanishing on the diagonals shifted by roots of unity, Int. Math. Res. Notices 2003 1015–1034.

[9] F. M. Goodman: Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra321 (2009) 3299–3320.

[10] F. M. Goodman: Comparison of admissibility conditions for cyclotomic Birman-Wenzl-Murakami algebras, J. Pure and Applied Algebra 214 (2010) 2009–2016.

[11] F. M. Goodman: Admissibility conditions for degenerate cyclotomic BMW algebras,Comm. Algebra 39 (2011) 452461.

[12] F. M. Goodman and H. Hauschild: Affine Birman-Wenzl-Murakami algebras and tanglesin the solid torus, Fundamenta Mathematicae 190 (2006) 77–137.

[13] F. M. Goodman and H. Mosley: Cyclotomic Birman-Wenzl-Murakami algebras I: Freenessand realization as tangle algebras, J. Knot Theory Ramifications 18 (2009) 1089–1127.

20

[14] F. M. Goodman and H. Mosley: Cyclotomic Birman-Wenzl-Murakami algebras II: Admis-sibility relations and representation theory, Algebr. Represent. Theory 14 (2011) 139,

[15] I. Grojnowski and M. Vazirani: Strong multiplicity one theorems for affine Hecke algebrasof type A, Transformation Groups 6 (2001) 143-155.

[16] R. Haring-Oldenburg: The reduced Birman-Wenzl algebra of Coxeter type B, J. Algebra213 (1999) 437–466.

[17] R. Haring-Oldenburg: An Ariki-Koike like extension of the Birman-Murakami-Wenzl alge-bra, preprint 1998. arXiv:q-alg/9712030

[18] R. Haring-Oldenburg: Cyclotomic Birman-Murakami-Wenzl algebras, J. Pure Appl. Alg.161 (2001) 113–144.

[19] A. Kleshchev: Linear and projective representations of symmetric groups, Cambridge Tractsin Mathematics, 163. Cambridge University Press, Cambridge, 2005,

[20] C. Kriloff and A. Ram: Representations of graded Hecke algebras, Representation Theory6 (2002), 31-69.

[21] T. Lam: Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras, Bull.Lond. Math. Soc. 43 (2011) 328334.

[22] A. Lascoux, B. Leclerc, J.-Y. Thibon: Green polynomials and Hall-Littlewood functions atroots of unity, Europ. J. Combinatorics 15 (1994) 173–180.

[23] T. Lam, A. Schilling, and M. Shimozono: Schubert polynomials for the affine Grassmannianof the symplectic group, Math. Zeitschrift 264 (4) (2010) 765–811.

[24] I. G. Macdonald: Symmetric functions and Hall polynomials, Second edition, OxfordUniversity Press, 1995.

[25] A. O. Morris: On an algebra of symmetric functions, Quart. J. Math. 16 (1965) 53–64.

[26] M. Nazarov: Young’s Orthogonal Form for Brauer’s Centralizer Algebra, J. Algebra 182(1996) 664–693.

[27] R. Orellana and A. Ram: Affine braids, Markov traces and the category O, in Proceedingsof the International Colloquium on Algebraic Groups and Homogeneous Spaces Mumbai2004, V.B. Mehta ed., Tata Institute of Fundamental Research, Narosa Publishing House,Amer. Math. Soc. (2007) 423–473.

[28] P. Pragacz: Algebro-geometric applications of Schur S and Q polynomials, in Topics ininvariant theory (Paris, 1989/1990), Lecture Notes in Math. 1478, Springer, Berlin (1991),130–191.

[29] A. Ram: Affine Hecke algebras and generalized standard Young tableaux, J. Algebra 260(2003) 367-415.

[30] H. Rui, On the classification of finite dimensional irreducible modules for affine BMWalgebras, arXiv:1206.3771.

[31] H. Rui and M. Si: On the structure of cyclotomic Nazarov-Wenzl algebras, J. Pure Appl.Algebra 212 (2008) 2209–2235.

21

[32] H. Rui and M. Si: Gram determinants and semisimplicity criteria for Birman-Wenzlalgebras, J. Reine Angew. Math. 631 (2009) 153–179.

[33] H. Rui and J. Xu: On the semisimplicity of cyclotomic Brauer algebras, II, J. Algebra312 (2007) 995–1010.

[34] H. Rui and J. Xu: The representations of cyclotomic BMW algebras, J. Pure Appl. Algebra213 (2009) 2262–2288.

[35] B. Totaro: Towards a Schubert calculus for complex reflection groups, Math. Proc. Camb.Phil. Soc. 134 (2003) 83–93.

[36] S. Wilcox and S. Yu: The cyclotomic BMW algebra associated with the two string type Bbraid group, Comm. Algebra 39 (2011) 44284446.

[37] S. Wilcox and S. Yu: On the freeness of the cyclotomic BMW algebras: admissibility andan isomorphism with the cyclotomic Kauffman tangle algebras. arXiv:0911.5284.

[38] S. Yu: The cyclotomic Birman-Murakami-Wenzl algebras, Ph.D. Thesis, University of Syd-ney (2007). arXiv:0810.0069.

[39] A. Zelevinsky: Resolvents, dual pairs and character formulas, Funct. Anal. Appl. 21 (1987)152-154.

Zajj Daugherty:Department of Mathematics, Dartmouth College, Hanover, NH 03755 USA.zajj.b.daugherty@dartmouth.edu

Arun Ram:Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Aus-tralia.aram@unimelb.edu.au

Rahbar Virk:Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309USA.rahbar.virk@colorado.edu

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