On a Combinatorial Identity

TL;DR

This paper provides a straightforward and elementary proof of a combinatorial identity related to vertex representations of quantum Kac-Moody algebras, using a method involving identities of distributions.

Contribution

The paper introduces a simple, direct proof of a recently discovered combinatorial identity in the context of quantum algebra representations.

Findings

Proof of the combinatorial identity is elementary and direct.

The method involves demonstrating a related identity of distributions.

The result simplifies understanding of the combinatorial structure in quantum algebra representations.

Abstract

Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of distributions.