Bethe roots and refined enumeration of alternating-sign matrices

TL;DR

This paper links the ground state properties of a specific XXZ spin chain to the refined enumeration of alternating-sign matrices, revealing a deep combinatorial connection through Bethe roots.

Contribution

It proves that certain symmetric polynomials in Bethe roots correspond to refined counts of alternating-sign matrices, establishing a novel mathematical relationship.

Findings

Bethe roots relate to elementary symmetric polynomials

Polynomials match refined enumeration numbers

Connection between quantum spin chains and combinatorics

Abstract

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.