The Schützenberger involution and colored lattice models

Abstract: Colored lattice models can be used to describe many different types of special functions of interest in both algebraic combinatorics and representation theory, for example Schur polynomials, nonsymmetric Macdonald polynomials, and characters and Whittaker functions for representations of p-adic groups. A notable example is the metaplectic ice model of which there are actually two different variants: a Gamma and a Delta variant. These variants differ in key aspects but surprisingly produce equal partition functions, which are weighted sums over admissible configurations, and this equality is called the Gamma-Delta duality. The duality was used to prove the analytic continuation of certain multiple Dirichlet series and is highly non-trivial, especially since the number of configurations on each side of the equality can differ. In this paper we construct a new family of solvable, colored lattice models and prove that they are dual to existing lattice models in the literature, including the above metaplectic case and the lattice model for (non-metaplectic) Iwahori Whittaker functions together with its crystal limit for Demazure atoms for Cartan type A. The equality of partition functions is shown using Yang-Baxter equations involving R-matrices mixing lattice model rows of types Gamma and Delta. For the crystal Demazure lattice model we show that the duality refines to a weight-respecting bijection of states given by the Sch\"utzenberger involution on the associated Gelfand-Tsetlin patterns or semistandard Young tableaux. We also show how the individual steps exchanging two rows in the proof of the duality for the partition functions refines to Berenstein-Kirillov, or Bender-Knuth involutions.