## Topology Seminar: Jenny Wilson

- Date: 04/02/2012
- Time: 14:00

University of British Columbia

The cohomology groups of the pure string motion group are uniformly representation stable

Abstract

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the degree k rational cohomology of P\Sigma_n tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n -- that is, in some precise sense, the description of the decomposition of these cohomology groups into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.