Topology Seminar: Patricia Hersh

In recent years, important families of symmetric group representations have come to be better understood through the perspective of representation stability, a viewpoint introduced and developed by Thomas Church, Jordan Ellenberg, and Benson Farb, among others. A fundamental example of representation stability is the $S_n$-module structure for the $i$-th cohomology of the configuration space of $n$ distinct, labeled points in the plane, or more generally in a connected, orientable manifold, as $i$ is held fixed and $n$ grows. For the plane, this translates to Whitney homology of the partition lattice via an $S_n$-equivariant version of the Goresky-MacPherson formula. This talk will survey the combinatorial literature regarding the partition lattice and discuss what new things this can tell us about representation stability for configuration spaces. In particular, we deduce new, sharp stability bounds and representation theoretic structure through a combination of symmetric function technology and poset topology. This is a joint work with Vic Reiner.