Bigraded Poincaré polynomials and the equivariant cohomology of Rep\((C_2)\)-complexes

We are interested in computing the Bredon cohomology with coefficients in the constant Mackey functor \(\underline{{\mathbb {F}}_2}\) for equivariant \(\text {Rep}(C_2)\) spaces, in particular for Grassmannian manifolds of the form \(\operatorname {Gr}_k(V)\) where V is some real representation of \(C_2.\) It is possible to create multiple distinct \(\text {Rep}(C_2)\) 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 \(\mathbb {M}_2\)-modules valued in the polynomial ring \(\mathbb Z[x,y]\) which makes cohomology computation of Rep\((C_2)\)-complexes more tractable, and we present some new results for Grassmannians.