A Unified Combinatorial Approach to Several Poincare Series Identities

TL;DR

This paper introduces a unified combinatorial method using integer partition bijections to prove and generalize identities related to Poincaré series in representation theory.

Contribution

It provides a simple bijective proof for Mendes' conjectured identity and extends it to a broader class of Poincaré series identities.

Findings

Proved Mendes' Poincaré series identity using bijections

Generalized the bijection to prove additional identities

Demonstrated the versatility of combinatorial methods in representation theory

Abstract

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 $Sym^{2}(\mathbb{R}^{n})$. 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 Poincar\'e series identities, including a version of the identity conjectured by Mendes as well as refinements of it.