Stiefel-Whitney classes of representations of dihedral and symmetric groups

Abstract

We compute Stiefel-Whitney classes of irreducible representations of dihedral groups and symmetric groups $S_4$ and $S_5$. We give character formulas for all Stiefel-Whitney classes of representations of the cyclic group of order $2$, the Klein four-group, and odd dihedral groups. For representations of even dihedral groups, we give a character formula for the first and second Stiefel-Whitney class. We also give a new proof of previously known character formula for the second Stiefel-Whitney class of a representation of $S_n$ for $n\geq 4$.