Stiefel-Whitney classes for symmetric groups

We prove several results about Stiefel-Whitney Classes (SWCs) $w_k(\pi )$ of representations $\pi$ of $S_n$. First, each SWC is polynomial in the character values of $\pi$ at involutions. Next, for a fixed $k$, the proportion of irreducible $\pi$ for which $w_k(\pi )=0$ approaches $100\mathrm{\%}$ as $n\to \mathrm{\infty }$. A similar result holds for the top SWCs. We also provide a simple criterion which determines the first nonvanishing SWC for a representation. The first four SWCs are computed explicitly. Finally, we give analogues for alternating groups.