A Unified Combinatorial Approach to Several Poincare Series Identities

Abstract

Mendes recently conjectured an identity simplifying the Poincaré series of the space of equivariant polynomial maps from $\mathbb{R}^{n}$ to a subrepresentation of $Sym^{2}(\mathbb{R}^{n})$. 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 Poincaré series identities, including a version of the identity conjectured by Mendes as well as refinements of it.

This paper has been withdrawn by the author due to learning of an alternative, simpler way to prove the identities