Georgia Institute of Technology College of Sciences

Real sub-varieties and more generally semi-algebraic subsets of $\mathbb{R}^n$ that are stable under the action of the symmetric group on $n$ elements acting on $\mathbb{R}^n$ by permuting coordinates, are expected to be topologically better behaved than arbitrary semi-algebraic sets. In this talk I will quantify this statement by showing polynomial upper bounds on the multiplicities of the irreducible $\mathfrak{S}_n$-representations that appear in the rational cohomology groups of such sets. I will also discuss some algorithmic results on the complexity of computing the equivariant Betti numbers of such sets and sketch some possible connectios with the recently developed theory of FI-modules. (Joint work with Cordian Riener).