Edward Formanek (Pennsylvania State University)

Friday 24th November, 12.05-12.55pm, Carslaw 373

Braid group representations of low degree

Let G be a finitely generated group. For a fixed integer n, the equivalence classes of irreducible representations G -> GL(n,C) form an algebraic variety. There are only a few groups for which this variety is well understood. These include finite groups, abelian groups, abelian-by-finite groups and certain arithmetic groups, such as SL(n,Z) (n at least 3). Although classifying the irreducible representations for a general group is probably hopeless, the braid group B_n seems more tractable because its presentation is short and simple.

The combined work of E. Formanek, W. Lee, I. Sysoeva and M. Vazirani has classified the irreducible complex representations of B_n of degree <= n. Other than some exceptional representations when n <= 8, all such representations are either one-dimensional or the tensor product of a one-dimensional representation with a specialization of either the Burau representation or the standard representation. (The Burau and standard representations are representations B_n -> GL(r,C[t,t^-1]), where r = n-2, n-1, or n, and t is an indeterminate. A specialization is a representation B_n -> GL(r,C) obtained by setting t equal to a nonzero complex number.)