A Unified Combinatorial Approach to Several Poincare Series Identities

2010/11/10 by Paul Levande, Levande, Paul
Mathematics · Physics and Astronomy · #05A05 #05A17 #05A19 #20C33 #20F55 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.CO #math.GR #math.RT #msc:05A05 #msc:05A17 #msc:05A19 #msc:20C33 #msc:20F55

paper · pdf · doi:10.48550/arxiv.1011.2409

This paper has been withdrawn by the author due to learning of an alternative, simpler way to prove the identities

openalex publication_date 2010/11/10 · arxiv created 2010/11/11 · arxiv updated 2010/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Mendes recently conjectured an identity simplifying the Poincaré series of the space of equivariant polynomial maps from ℝⁿ to a subrepresentation of Sym²(ℝⁿ). 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.

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