Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

Abstract

We prove two identities of Hall---Littlewood polynomials, which appeared recently in Betea and Wheeler (2014). We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in Betea and Wheeler (2014), via the introduction of additional parameters. The left-hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right-hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.