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Algebra Seminar
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Karin Erdmann
University of Oxford

Type A Hecke algebras, Clifford algebras, rank varieties

Friday 24th October, 12:05-12:55pm, Carslaw 373.

Let ${\cal H} = H_q(r)$ be the Hecke algebra of a symmetric group over some field $K$ where $q \in K$ is a primitive $\ell$-th root of unity and where $K$ has characteristic $p \geq 0$. One expects that the representation theory of ${\cal H}$ should be similar to that of a group algebra. Work of Dipper and Du shows that standard parabolic subalgebras behave similarly to subgroup algebras of group algebras; and suggests that $\ell-p$ parabolic subalgebras should be an analogue of elementary abelian p-groups. In this lecture we will try to explain how these algebras control projectivity of ${\cal H}$-modules, and that at least over characteristic zero, projectivity is controlled by algebras of the form $K[X_1, \ldots X_m]/\langle X_i^2 \rangle$. To a module of such an 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 is constructed in terms of Clifford algebra representations.