Elsevier

Advances in Mathematics

Volume 299, 20 August 2016, Pages 543-600
Advances in Mathematics

Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons

https://doi.org/10.1016/j.aim.2016.05.010Get rights and content
Under an Elsevier user license
open archive

Abstract

We study Hall–Littlewood polynomials using an integrable lattice model of t-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall–Littlewood polynomials (of the An root system) within the framework of this model. Introducing appropriate double-row transfer matrices, we extend this formalism to Hall–Littlewood polynomials based on the BCn root system, and obtain a new combinatorial formula for them. We then apply our methods to prove a series of refined Cauchy and Littlewood identities involving Hall–Littlewood polynomials. The last two of these identities are new, and relate infinite sums over hyperoctahedrally symmetric Hall–Littlewood polynomials with partition functions of the six-vertex model on finite domains.

Keywords

Cauchy and Littlewood identities
Symmetric functions
Alternating sign matrices
Six-vertex model

Cited by (0)

The authors are supported by ARC grant DP140102201 and ERC grant 278124 “LIC”. They would like to acknowledge hospitality and support from the Galileo Galilei Institute, where part of this work was carried out during the program “Statistical Mechanics, Integrability and Combinatorics”.