Demystifying a divisibility property of the Kostant partition function

TL;DR

This paper provides a combinatorial proof for a divisibility property of the Kostant partition function, explaining its occurrence and suggesting possible generalizations, thus clarifying a previously mysterious phenomenon.

Contribution

It offers the first combinatorial proof of the divisibility identities and explains their natural origin, extending understanding of the Kostant partition function.

Findings

Proved divisibility identities combinatorially

Provided a natural explanation for the divisibility

Suggested multiple generalizations of the identities

Abstract

We study a family of identities regarding a divisibility property of the Kostant partition function which first appeared in a paper of Baldoni and Vergne. To prove the identities, Baldoni and Vergne used techniques of residues and called the resulting divisibility property "mysterious." We prove these identities entirely combinatorially and provide a natural explanation of why the divisibility occurs. We also point out several ways to generalize the identities.