Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures
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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.
KeywordsCauchy and Littlewood identities Symmetric functions Alternating sign matrices Six-vertex model
We express our sincere thanks to Ole Warnaar, for many valuable insights and suggestions which motivated this work, and in particular for suggesting the idea of \(u\)-deformations of the original identities (1)–(3), and to Eric Rains, for helping us in arriving at Conjecture 1. M.W. would like to thank Frédéric Jouhet for his interest in this work and for pointing out the reference , and Jean-Christophe Aval, Philippe Nadeau and Eric Ragoucy for invitations to present related work at LaBRI, ICJ and LAPTh, respectively. We finally wish to acknowledge the open-source package Sage whose built-in functions for Hall–Littlewood and Macdonald polynomials were indispensable in verifying some of the conjectures. This work was done under the support of the ERC Grant 278124, “Loop models, integrability and combinatorics”.
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