Special Functions for Hyperoctahedral Groups Using Bosonic, Trigonometric Six-Vertex Models

TL;DR

This paper explores new bosonic lattice models for type B/C root systems, demonstrating their partition functions match type C zonal spherical functions in low ranks but not in higher ranks.

Contribution

It introduces novel bosonic models for type B/C root systems and establishes their connection to zonal spherical functions in low ranks.

Findings

Partition functions match type C zonal spherical functions in rank 2 and 3.

Models do not generalize to higher ranks.

Provides a new approach to studying symmetric polynomials via lattice models.

Abstract

Recent works have sought to realize certain families of orthogonal, symmetric polynomials as partition functions of well-chosen classes of solvable lattice models. Many of these use Boltzmann weights arising from the trigonometric six-vertex model $R$-matrix (or generalizations or specializations of these weights). In this paper, we seek new variants of bosonic models on lattices designed for type B/C root systems, whose partition functions match the zonal spherical function in type C. Under general assumptions, we find that this is possible for all highest weights in rank $2$ and $3$, but not for higher rank.