A Polyhedral Proof of a Wreath Product Identity

TL;DR

This paper provides a geometric proof for a wreath product identity related to permutation statistics, extending the discrete geometric approach to identities previously resistant to such methods.

Contribution

It offers a polyhedral geometric proof for a specific wreath product identity, expanding the applicability of geometric techniques in algebraic combinatorics.

Findings

Geometric proof of a wreath product identity

Extension of discrete geometric methods to new identities

Validation of Biagioli and Zeng's wreath product identity

Abstract

In 2013, Beck and Braun proved and generalized multiple identities involving permutation statistics via discrete geometry. Namely, they recognized the identities as specializations of integer point transform identities for certain polyhedral cones. They extended many of their proof techniques to obtain identities involving wreath products, but some identities were resistant to their proof attempts. In this article, we provide a geometric justification of one of these wreath product identities, which was first established by Biagioli and Zeng.