A Unified Combinatorial Approach to Several Poincare Series Identities

Mendes recently conjectured an identity simplifying the Poincar'e series of the space of equivariant polynomial maps from ${\mathbb{R}}^n$ to a subrepresentation of $Sy{m}^2({\mathbb{R}}^n)$. We show how to prove this identity using a fairly simple integer partition bijection. First, we give a bijective proof of a similar, well-known identity from representation theory. We then show that this bijection can be generalized to prove other Poincar'e series identities, including a version of the identity conjectured by Mendes as well as refinements of it.