JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 26, No 3 (2007)

John Enyang

Nagoya University Graduate School of Mathematics Chikusa-ku Nagoya 464-8602 Japan

A construction of bases for cell modules of the Birman-Murakami-Wenzl (or B-M-W) algebra B _n ( q, r) by lifting bases for cell modules of B _n - 1( q, r) is given. By iterating this procedure, we produce cellular bases for B-M-W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys-Murphy elements from the representation theory of the Iwahori-Hecke algebra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B-M-W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori-Hecke algebra of the symmetric group.

to the determinant of Gram matrix associated to Sλin Example 3.3 that Sλis absolutely irreducible and hence that B3( \hat q, \hat r) remains semisimple over κ(Theorems 4.3 and 4.4). link to page 51 link to page 51 failed to look up index:1.label J Algebr Comb (2007) 26: 291-341 9 Brauer algebras The foregoing construction for the B-M-W algebras applies with minor modification to the Brauer algebras over an arbitrary field. We begin once more by considering Brauer algebras over a polynomial ring over Z. Take z to be an indeterminate over Z; we write R = Z[z] and define the Brauer algebra Bn(z) over R as the associative unital R-algebra generated by the transpositions s1, s2, . . . , sn - 1, together with elements E1, E2, . . . , En - 1, which satisfy the defining relations: s2 = 1 for 1 \leq i < n; i sisi+1si = si+1sisi+1 for 1 \leq i < n - 1; sisj = sj si for 2 \leq |i - j |; E2 = zE i i for 1 \leq i < n; siEj = Ej si for 2 \leq |i - j |; EiEj = Ej Ei for 2 \leq |i - j |; Eisi = siEi = Ei for 1 \leq i < n; Eisi\pm 1si = si\pm 1siEi\pm 1 = EiEi\pm 1 for 1 \leq i, i \pm 1 < n; Eisi\pm 1Ei = EiEi\pm 1Ei = Ei for 1 \leq i, i \pm 1 < n. Regard the group ring RSn as the subring of Bn(z) generated by the transpositions {si = (i, i + 1) : for 1 \leq i < n}. If f is an integer, 0 \leq f \leq [n/2], and λis a partition of n - 2f , define the elements xλ= w and mλ= E1E3 \cdot \cdot \cdot E2f - 1xλ, w\in Sλwhere Sλis the row stabiliser in s2f +1, s2f +2, . . . , sn - 1 of the superstandard tableau tλ\in Stdn(λ). Let Bλn be the two sided ideal of Bn(z) generated by mλand write \check Bλ= n Bμn . μλA cellular basis in terms of dangles has been given for the Brauer algebra in [Replacing cellular bases for Hn(q2) with cellular bases for RSn, the process used to construct cellular bases the B-M-W algebras in [4] will produce also cellular bases for Bn(z) as follows. If f is an integer, 0 \leq f \leq [n/2], and λa partition of n - 2f , then In(λ) retains the meaning assigned in (3.3). Theorem 9.1 The algebra Bn(z) is freely generated as an R-module by the collection (s, v), (t, u) \in I (d( n(λ) for λa partition s)v) - 1mλd(t)u of n - 2f and 0 \leq f \leq [n/2] . failed to look up btsu:2.label link to page 51 failed to look up formdef:1.label link to page 12 link to page 49 failed to look up bcell:2.label link to page 51 link to page 18 Moreover, the following statements hold.
1. The R-linear map determined by (d(s)v) - 1mλd(t)u \rightarrow (d(t)u) - 1mλd(s)v is an algebra anti-involution of Bn(z).
2. Suppose that b \in Bn(z) and let f be an integer, 0 \leq f \leq [n/2]. If λis a partition of n - 2f and (t, u) \in In(λ), then there exist a(u,w) \in R, for (u, w) \in In(λ), such that for all (s, v) \in In(λ), (d(s)v) - 1mλd(t)ub \equiv a(u,w)(d(s)v) - 1mλd(u)w mod \check Bλn . (9.1) (u,w) As a consequence of the above theorem, \check Bλn is the R-module freely generated by (d(s)v) - 1mμd(t)u : (s, v), (t, u) \in In(μ), for μλ. If f is an integer, 0 \leq f \leq [n/2], and λis a partition of n - 2f , the cell module Sλis defined to be the R-module freely generated by mλd(t)u + \check Bλ| n (t, u) \in In(λ) (9.2) with right Bn(z) action (mλd(t)u)b + \check Bλ= n a(u,w)mλd(u)w + \check Bλn for b \in Bn(z), (u,w) where the coefficients a(u,w) \in R, for (u, w) in In(λ), are determined by the expression (9.1). The construction of cellular algebras [5] equips the Bn(z)-module Sλwith a sym- metric associative bilinear form (compare (3.6) above). Following is the counterpart to Example 3.3, stated for reference in Sect.
11. Example 9.1 Let n = 3 and λ= (1) so that \check Bλ= n (0) and mλ= E1. We order the basis (9.2) for the module Sλas v1 = E1, v2 = E1s2 and v3 = E1s2s1 and, with respect to this ordered basis, the Gram matrix vi, vj of the bilinear form on the Bn(z)-module Sλis \? \? z 1 1 \?1 z 1\? . 1 1 z The determinant of the Gram matrix given above is (z - 1)2(z + 2). By Theorem 2.3 of [12], the Bratteli diagram associated with Bn(z) is identical to the Bratteli diagram for Bn(q, r). Thus μ\rightarrow λretains the meaning assigned in Sect. 5. failed to look up wkdef.label failed to look up ydef:2.label link to page 32 link to page 33 failed to look up bcell:2.label failed to look up b-murphy.label link to page 4 link to page 51 link to page 51 link to page 51 J Algebr Comb (2007) 26: 291-341 Let f be an integer, 0 \leq f \leq [n/2], and λbe a partition of n - 2f with t removable nodes and (p - t) addable nodes. Suppose that μ(1) μ(2) \cdot \cdot \cdot μ(p) is the ordering of {μ: μ\rightarrow λ} by dominance order on partitions. If 1 \leq k \leq t, define yλ= m μ(k) λd(s) + \check Bλn where s|n - 1 = tμ(k) \in Stdn - 1(μ(k)) and, if t < k \leq p define wk by (5.20) and, by analogy with (5.21), write yλ= E m μ(k) 2f - 1w - 1 k μ(k) + \check Bλn . Given the elements yλμin Sλfor each partition μ\rightarrow λ, define N μto be the Bn - 1(z)- submodule of Sλgenerated by {yλ: νν\rightarrow λand νμ} and let \check N μbe the Bn - 1(z)-submodule of Sλgenerated by {yλ: νν\rightarrow λand νμ}. Theorem 5.9 and the construction given for the B-M-W algebras in Sect. 6 have analogues in the context of Bn(z). Thus the cell module (9.2) has a basis over R, {mt = mλbt + \check Bλ: n t \in Tn(λ)} indexed by the paths Tn(λ) of shape λin the Bratteli diagram associated with Bn(z), and defined in the same manner as the basis (6.5). 10 Jucys-Murphy operators for the Brauer algebras Define the operators Li , for i = 1, . . . , n, in Bn(z) by L1 = 0 and Li = si - 1 - Ei - 1 + si - 1Li - 1si - 1 for 1 < i \leq n. Remark 10.1 The elements Li as defined above bear an obvious analogy with the elements \~ Di defined in Sect. 2.2; thus we refer to the elements Li as the “Jucys- Murphy operators” in Bn(z). In [10], M. Nazarov made use of operators xi with are related to the Li defined above by xi = z - 1 + L[10] is a scalar multiple of the identity, we derive the next statement from results in Sect. 2 of [10]. Proposition 10.1 Let i and k be integers, 1 \leq i < n and 1 \leq k \leq n. 1. si and Lk commute if i = k - 1, k.
2. Li and Lk commute. 3. si commutes with Li + Li+1.
4. L2 + L3 + \cdot \cdot \cdot + Ln belongs to the centre of Bn(z). link to page 47 (j ) (j ) For integers j, k with 1 \leq j, k \leq n, we define the elements L by L = 0 and k 1 (j ) (j ) L = s s k+1 j +k - 1 - Ej +k - 1 + sj +k - 1Lk j +k - 1, for k \geq
1. In particular, (1) L = L k k , for k = 1, . . . , n, are the Jucys-Murphy elements for Bn(z). The objective now is to show that mtλis a common eigenvector for the action of the Jucys-Murphy elements Lk on the cell module Sλ. Proposition 10.2 Let i, k be integers with 1 \leq i \leq n and 1 < k \leq n. Then \? \? \?(1 - z)Ei if k = 2; (i) EiL = 0 if k = 3; k \? \? (i+2) EiL if k \geq 4. k - 2 Proof If (i) k = 2 then EiL = E k i (si - Ei ) = (1 - z)Ei . For k = 3 we have (i) EiL = E 3 i (si+1 - Ei+1 + si+1si si+1 - si+1Ei si+1) = Ei(si+1 - Ei+1) + Ei(Ei+1si+1 - si+1) =
0. If k = 4 then, (i) (i) EiL = E s 4 i (si+2 - Ei+2) + si+2Ei L3 i+2 = (i+2) Ei(si+2 - Ei+2) = EiL , 2 and when k > 4, (i) (i) EiL = E s k i (si+k - 2 - Ei+k - 2) + si+k - 2Ei Lk - 1 i+k - 2 = (i+2) Ei(si+k - 2 - Ei+k - 2) + si+k - 2EiL s k - 3 i+k - 2 = (i+2) (i+2) Ei(si+k - 2 - Ei+k - 2 + si+k - 2L s k - 3 i+k - 2) = Ei Lk - 2 by induction. Corollary 10.3 Let f, k be integers, 0 < f \leq [n/2] and 1 \leq k \leq n. Then \? \? \?0, if k is odd, 1 \leq k \leq 2f + 1; E1E3 \cdot \cdot \cdot E2f - 1Lk = \?(1 - z)E1E3 \cdot \cdot \cdot E2f - 1, if k is even, 1 < k \leq 2f ; \? (2f +1) E1E3 \cdot \cdot \cdot E2f - 1L , if 2f + 1 < k \leq n. k - 2f Proof If k is odd, 1 < k \leq 2f + 1, then by Proposition 10.2, (1) (3) E1E3 \cdot \cdot \cdot EkLk = E1E3 \cdot \cdot \cdot EkL = E = \cdot \cdot \cdot k 1E3 \cdot \cdot \cdot Ek Lk - 2 \cdot \cdot \cdot = (k) E1E3 \cdot \cdot \cdot EkL = 0. (10.1) 1 link to page 47 link to page 40 link to page 48 link to page 40 link to page 40 link to page 41 link to page 48 J Algebr Comb (2007) 26: 291-341 Since Ek+2Ek+3 \cdot \cdot \cdot E2f - 1 commutes with Lk, the first case follows. If k is even and 1 < k \leq 2f , then the relations Eisi = Ei and E2 = zE i i , together with (10.1), show that E1E3 \cdot \cdot \cdot E2f - 1Lk = E1E3 \cdot \cdot \cdot E2f - 1(sk - 1 - Ek - 1 + sk - 1Lk - 1sk) = (1 - z)E1E3 \cdot \cdot \cdot E2f - 1 + E1E3 \cdot \cdot \cdot E2f - 1Lk - 1sk - 1 = (1 - z)E1E3 \cdot \cdot \cdot E2f -
1. The final case follows in a similar manner. Let f be an integer, 0 \leq f \leq [n/2], and λbe a partition of n - 2f . For each path t \in Tn(λ), define the polynomial Pt(k) = j - i if [λ(k)] = [λ(k - 1)] \cup {(i, j )} i - j + 1 - z if [λ(k)] = [λ(k - 1)] \ {(i, j )}. The proof of the next statement is identical to the proof of Proposition 7.5 given above; for the proof of Proposition 10.5, we refer to the proof of Proposition 7.6. Proposition 10.4 If λis a partition of n and k is an integer with 1 \leq k \leq n, then mtλLk = Ptλ(k)mtλ. Proposition 10.5 Let f be an integer, 0 < f \leq [n/2], and λbe a partition of n - 2f . Then mtλLk = Ptλ(k)mtλ. Proposition 10.6 Let f be an integer, 0 \leq f \leq [n/2], and λbe a partition of n - 2f . Then there exists an invariant α\in R such that L2 + L3 + \cdot \cdot \cdot + Ln acts on Sλas a scalar multiple by αof the identity. Proof As in the proof of Proposition 7.7, we obtain α= n k=2 Ptλ(k). Theorem 10.7 Let f be an integer 0 \leq f \leq [n/2] and λbe a partition of n - 2f . If t \in Tn(λ), then there exist av \in R, for v \in Tn(λ) with v t, such that mtLk = Pt(k)mt + avmv. v\in Tn(λ) v t Proof By repeating word for word the argument given in the proof of Theorem 7.8, we show that the statement holds true when 1 \leq k < n. That Ln acts triangularly on Sλ, can then be observed using Proposition 10.6: n mtLn = Pt(k)mt - mt(L2 + L3 + \cdot \cdot \cdot + Ln - 1). k=1 Thus the generalised eigenvalue for Ln acting on mt is Pt(n). link to page 42 link to page 43 link to page 49 link to page 51 link to page 43 link to page 48 11 Semisimplicity criteria for Brauer algebras Below are analogues for the Brauer algebras of the results of Sect.
8. Let κbe a field and take \hat z \in κ. Then z \rightarrow \hat z determines a homomorphism R \rightarrow κ, giving κan R- module structure. A Brauer algebra over κis a specialisation Bn(\hat z) = Bn(z) \otimes R κ. For t \in Tn(λ) and k = 1, . . . , n, let \hat Pt(k) denote the evaluation of the monomial Pt(k) at \hat z, \hat Pt(k) = j - i if [λ(k)] = [λ(k - 1)] \cup {(i, j )} i - j + 1 - \hat z if [λ(k)] = [λ(k - 1)] \ {(i, j )}, and as previously, define the ordered n-tuple \hat P (t) = ( \hat Pt(1), . . . , \hat Pt(n)). The operators Li provide conditions necessary for the existence of a homomorphic image of one cell module for Bn(\hat z) in another cell module for Bn(\hat z). Theorem 11.1 Let κbe a field. Suppose that for each pair of partitions λof n - 2f and μof n - 2f , for integers f, f with 0 \leq f, f \leq [n/2], and for each pair of partitions s \in Tn(λ) and t \in Tn(μ), the conditions λμand \hat P (s) = \hat P (t) together imply that λ= μ. Then Bn(\hat z) is a semisimple algebra over κ. By an analogous statement to Lemma 8.3, the Jucys-Murphy elements do in fact distinguish between the cell modules of Bn(z) in Theorem 11.1. The results of this section can be used to derive the next statement which is Theorem 3.3 of [3]. As in Theorem 8.5, the statement may be generalised to the setting where |λ| > |μ|. Theorem 11.2 Let λbe a partition of n and μbe a partition of n - 2f , where f >
0. If HomBn(\hat z)(Sλ, Sμ) = 0, then (j - i) - (j - i) = f (1 - \hat z). (i,j )\in [λ] (i,j )\in [μ] Proof Suppose that u \in Sλ, v \in Sμare non-zero and that u \rightarrow v under some element in HomBn(\hat z)(Sλ, Sμ). Then, using Proposition 10.6, n uLi = (j - i)u i=1 (i,j )\in [λ] while n vLi = f (1 - \hat z)v + (j - i)v. i=1 (i,j )\in [μ] Since v is the homomorphic image of u, it follows that (j - i) = f (1 - \hat z) + (j - i). (i,j )\in [λ] (i,j )\in [μ] link to page 49 link to page 51 link to page 49 link to page 45 link to page 18 link to page 18 failed to look up formdef:1.label failed to look up formdef:1.label J Algebr Comb (2007) 26: 291-341 Hence the result. Theorem 11.1 gives a sufficient but not the necessary condition for Bn(\hat z) to be a semisimple algebra over κ. Necessary and sufficient conditions on the semisimplicity of Bn(\hat z) have been given by H. Rui in [11]. Example 11.1 Let κ= Q and \hat z =
4. Take n = 3, λ= (1) and μ= (1, 1, 1). In characteristic zero the cell modules corresponding to the partitions (3), (2, 1) and (1, 1, 1) are absolutely irreducible. But, taking t = (\emptyset , , , ) \in Tn(λ) and u = \emptyset , , , \in Tn(μ), then \hat P (t) = (0, - 1, 2 - \hat z) = (0, - 1, - 2) and \hat P (u) = (0, - 1, - 2). Since \hat P (t) = \hat P (u) whilst λμ, the pair t, u violates the hypotheses of Theorem 11.1. However, by reference to the determinant of Gram matrix associated to Sλin Example 9.1, it follows that Sλis absolutely irreducible and hence that B3(\hat z) remains semisimple by appeal to appropriate analogues of Theorems 4.3 and 4.4. 12 Conjectures Define a sequence of polynomials (pi(z) | i = 1, 2, . . . , ) by p1(z) = (z + 2)(z - 1) and pi(z) = (z + 2i)(z - i)(z + i - 2)pi - 1(z) if i is odd; (z + 2i)(z - i)pi - 1(z) if i is even. Conjecture 12.1 For κa field, \hat z \in κand an algebra over κ, with n \geq 2, the following statements hold: (i) If n = 2k + 1, then the bilinear form on the Bn(\hat z)-module S(1) determined by (3.6) is non-degenerate if and only if pk(\hat z) = 0. (ii) If n = 2k, then the bilinear form on the Bn(\hat z)-module S\emptyset determined by (3.6) is non-degenerate if and only if \hat z = 0 and pk(\hat z) =
0. Conjecture 12.2 For κa field, \hat z \in κand an algebra over κ, with n \geq 2, the following statements hold: (i) If n = 2k + 1, then Bn(\hat z) is semisimple and only if κSn is semisimple and p2k - 1(\hat z) = 0. (ii) If n = 2k, then Bn(\hat z) is semisimple and only if κSn is semisimple, \hat z = 0 and p2k - 2(\hat z) =
0. References