A Littlewood-type identity for Robbins polynomials

TL;DR

This paper generalizes the Littlewood identity by replacing Schur polynomials with Robbins polynomials, linking them to alternating sign matrices and six-vertex model configurations, thus extending combinatorial and algebraic identities.

Contribution

It introduces a new Littlewood-type identity involving Robbins polynomials and Pfaffian formulas, connecting combinatorics, algebra, and statistical mechanics.

Findings

Generalized Littlewood identity with Robbins polynomials

Pfaffian formula related to six-vertex model

Enumeration of alternating sign matrices

Abstract

We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the classical identity, Schur polynomials are replaced by so-called modified Robbins polynomials. These polynomials are a generalization of Schur polynomials and enumerate down-arrowed monotone triangles, and thus also alternating sign matrices. As an additional factor on the other side of the identity, we have a Pfaffian formula which we interpret in terms of the partition function of six-vertex model configurations corresponding to diagonally symmetric alternating sign matrices.