Quasi-solvable lattice models for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ Demazure atoms and characters

TL;DR

This paper introduces quasi-solvable lattice models for symplectic and odd orthogonal Demazure characters, establishing their partition functions as these characters through functional equations derived from Yang--Baxter solutions.

Contribution

The authors construct new lattice models for Demazure characters of symplectic and orthogonal groups and develop methods to compute their partition functions despite the models being not fully solvable.

Findings

Partition functions represent Demazure characters and atoms.

Existence of solutions to Yang--Baxter equations enables functional equations.

Provides algorithms for reverse King and Sundaram tableaux.

Abstract

We construct colored lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang--Baxter equation that allows us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.