Special Functions for Hyperoctahedral Groups Using Bosonic Lattice Models

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 Cartan type 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 two and three, but not for higher rank. In ranks two and three, this may be regarded as a new generating function formula for zonal spherical functions (also known as Hall–Littlewood polynomials) in type C.