Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

TL;DR

This paper studies an integrable loop model with mixed boundary conditions, solving the associated qKZ equation, and connects the ground state sum to the enumeration of symmetric alternating sign matrices, providing new combinatorial insights.

Contribution

It introduces a specific integrable loop model with mixed boundaries, solves the qKZ equation, and links the ground state sum to symmetric alternating sign matrices.

Findings

Ground state sum equals the count of symmetric alternating sign matrices.

Derived a minimal degree solution to the qKZ equation.

Established a connection between loop models and combinatorial enumeration.

Abstract

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered.