Algebra and Topology Seminar

4pm Tuesday 7 October 2003

Karin Erdmann
Ocford University

Type A Hecke algebras, Clifford algebras, rank varieties

Let H = H_q(r) be the Hecke algebra of a symmetric group over some field K where q in K is a primitive l-th root of unity and where K has characteristic p > 0. One expects that the representation theory of H should be similar to that of a group algebra. Work of Dipper and Du shows that standard parabolic subalgebras behave similar to subgroup algebras of group algebras; and suggests that l-p parabolic subalgebras should be an analogue of elementary abelian p-groups.

In this lecture we will try to explain that these algebras control projectivity of H-modules, and that at least over characteristic zero, projectivity is controled by algebras of the form K [X₁,...,X_m]/<X_i²>. To a module of such algebra, over arbitrary characteristic, we associate a rank variety, whose vanishing characterizes projectivity of the module. This variety generalizes Carlson's rank variety in the group situation; and it constructed in terms of Clifford algebra representations.

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