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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Centers of generic Hecke algebras


Author: Lenny K. Jones
Journal: Trans. Amer. Math. Soc. 317 (1990), 361-392
MSC: Primary 20C30; Secondary 20G05, 20G40, 22E50
MathSciNet review: 948191
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Abstract: Let $ W$ be a Weyl group and let $ W'$ be a parabolic subgroup of $ W$. Define $ A$ as follows:

$\displaystyle A = R{ \otimes _{{\mathbf{Q}}[u]}}\mathcal{A}(W)$

where $ \mathcal{A}(W)$ is the generic algebra of type $ {A_n}$ over $ {\mathbf{Q}}[u]$ an indeterminate, associated with the group $ W$, and $ R$ is a $ {\mathbf{Q}}[u]$-algebra, possibly of infinite rank, in which $ u$ is invertible. Similarly, we define $ A'$ associated with $ W'$. Let $ M$ be an $ A - A$ bimodule, and let $ b \in M$. Define the relative norm [14]

$\displaystyle {N_{W,W'}}(b) = \sum\limits_{t \in T} {{u^{ - l(t)}}{a_{{t^{ - 1}}}}b{a_t}} $

where $ T$ is the set of distinguished right coset representives for $ W'$ in $ W$. We show that if $ b \in {Z_M}(A') = \{ m \in M\vert ma' = a'm\quad \forall a' \in A'\} $, then $ {N_{W,W'}}(b) \in {Z_M}(A)$. In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Hecke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott.

Let $ \alpha = ({k_1},{k_2}, \ldots ,{k_z})$ be a partition of $ n$ and let $ {S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}}$ be a "left-justified" parabolic subgroup of $ {S_n}$ of shape $ \alpha $.

Define

$\displaystyle {b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal{N}_\alpha })$

, where

$\displaystyle {\mathcal{N}_\alpha } = \prod\limits_{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})} $

with $ {w_i}$ a $ {k_i}$-cycle of length $ {k_i} - 1$ in $ {S_{{k_i}}}$. Then the main result of this paper is

Theorem. The set $ \{ {b_\alpha }\vert\alpha \vdash n\} $ is a basis for $ {Z_{A({S_n})}}(A({S_n}))$ over $ {\mathbf{Q}}[u,{u^{ - 1}}]$.

Remark. The norms $ {b_\alpha }$ in $ {Z_{A({S_n})}}(A({S_n}))$ are analogs of conjugacy class sums in the center of $ {\mathbf{Q}}{S_n}$ and, in fact, specialization of these norms at $ u = 1$ gives the standard conjugacy class sum basis of the center of $ {\mathbf{Q}}{S_n}$ up to coefficients from $ {\mathbf{Q}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0948191-6
PII: S 0002-9947(1990)0948191-6
Keywords: Generic Hecke algebras, Weyl groups, parabolic subgroups
Article copyright: © Copyright 1990 American Mathematical Society