  We compute cup product pairings in the integral cohomology ring of the moduli
space of rank two stable bundles with odd determinant over a Riemann surface
using methods of Zagier. The resulting formula is related to a generating
function for certain skew Schur polynomials. As an application, we compute the
nilpotency degree of a distinguished degree two generator in the mod two
cohomology ring. We then give descriptions of the mod two cohomology rings in
low genus, and describe the subrings invariant under the mapping class group
action.
