Bigraded Poincaré polynomials and the equivariant cohomology of Rep(C2)-complexes

We are interested in computing the Bredon cohomology with coefficients in the constant Mackey functor F2̲ for equivariant Rep(C2) spaces, in particular for Grassmannian manifolds of the form Grk(V) where V is some real representation of C2. It is possible to create multiple distinct Rep(C2) constructions of (and hence multiple filtration spectral sequences for) a given Grassmannian. For sufficiently small examples one may exhaustively compute all possible outcomes of each spectral sequence and determine if there exists a unique common answer. However, the complexity of such a computation quickly balloons in time and memory requirements. We introduce a statistic on M2-modules valued in the polynomial ring Z[x,y] which makes cohomology computation of Rep(C2)-complexes more tractable, and we present some new results for Grassmannians.