
Karin Erdmann
University of Oxford
Type A Hecke algebras, Clifford algebras, rank varieties
Friday 24th October, 12:0512:55pm,
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 $\ellp$ parabolic subalgebras should be an analogue
of elementary abelian pgroups.
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.







