The Many Faces of a Character

TL;DR

This paper establishes identities between different polynomial families related to affine algebra characters, connecting combinatorial sums with Bethe-Ansatz solutions, and proposes a new fermionic sum representation.

Contribution

It proves identities among polynomial families linked to affine $su(2)$ characters and introduces a novel fermionic sum derived from Bethe-Ansatz solutions.

Findings

Identities between bosonic, fermionic, and configuration sum polynomials.

Different expressions for affine $su(2)$ characters at level one.

A conjectured fermionic sum representation from Bethe-Ansatz.

Abstract

We prove an identity between three infinite families of polynomials which are defined in terms of `bosonic', `fermionic', and `one-dimensional configuration' sums. In the limit where the polynomials become infinite series, they give different-looking expressions for the characters of the two integrable representations of the affine $su(2)$ algebra at level one. We conjecture yet another fermionic sum representation for the polynomials which is constructed directly from the Bethe-Ansatz solution of the Heisenberg spin chain.