The Schützenberger involution and colored lattice models

Seminar on Solvable Lattice Models.

Solvable lattice models can be used to describe and study variousfunctions in p-adic representation theory. For instance, a recent paperby Brubaker, Buciumas, Bump and Gustafsson used lattice models todiscover an unexpected correspondence between so-called metaplectic andIwahori Whittaker functions. There are actually two dual variants of thelattice model describing the former function, called the Gamma and Deltavariants. Because of the correspondence one may therefore expect asimilar duality of variants for the Iwahori lattice model, but only onesuch model (a Delta variant) is known in the literature.

In this talk I will present a new Gamma variant for the Iwahori latticemodel and show that it is dual to the existing Delta variant. Thesedualities are highly non-trivial in the sense that they are in generalnot given by bijections of the individual states. However, I will showthat in the crystal limit of these lattice models the duality can berefined to a weight-preserving bijection of states given by theSchützenberger involution on Young tableaux.

This is joint work with Henrik Gustafsson.