University of Sydney Algebra Seminar

Jennifer Wilson (University of Chicago)

Monday 22 August, 12:05-12:55pm, Carslaw 375

Representation stability for the cohomology of the groups of pure string motions

The string motion group \(\Sigma_n\), the group of motions of \(n\) disjoint, unlinked, unknotted circles in 3-space, is a generalization of the braid group. It can be identified with the symmetric automorphism group of the free group. The pure string motion group \(P\Sigma_n\), the analogue of the pure braid group, admits an action by the hyperoctahedral group \(W_n\). The rational cohomology of \(P\Sigma_n\) is not stable in the classical sense -- the dimension of the \(k^{\mbox{th}}\) cohomology group tends to infinity as \(n\) grows -- however, Church and Farb have recently developed a notion of stability for a sequence of vector spaces with a group action, which they call representation stability. Inspired by their recent work on the cohomology of the pure braid group, they conjectured that for each \(k>0\), the \(k^{\mbox{th}}\) rational cohomology of \(P\Sigma_n\) is uniformly representation stable with respect to the induced action of \(W_n\), that is, the description of the decomposition of the cohomology group into irreducible \(W_n\)-representations stabilizes for \(n >> k\). In this talk, I will give an overview of the theory of representation stability, and outline a proof verifying this conjecture. This result has implications for the cohomology of the string motion group, and the permutation-braid group.