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 Bn 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 Bn 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 Bn -> GL(r,C[t,t-1]), where r = n-2, n-1, or n, and t is an indeterminate. A specialization is a representation Bn -> GL(r,C) obtained by setting t equal to a nonzero complex number.)
After the seminar we will take the speaker to lunch.
See the Algebra Seminar web page for information about other seminars in the series.
Anthony Henderson email@example.com.