Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial

Abstract

This thesis is about a manifestly positive combinatorial rule for the expansion of the product of two polynomials: Schur polynomials and permuted-basement Demazure atoms. Special cases of the latter polynomials include Demazure atoms and characters; there are known tableau formulas for their expansions when multiplied by a Schur polynomial, due to Haglund, Luoto, Mason and van Willigenburg (2011). We find a vertex model formula, giving a new rule even in these special cases, extending a technique introduced by Zinn-Justin (2009) for calculating Littlewood–Richardson coefficients.

We derive a coloured vertex model for permuted-basement Demazure atoms. This model is inspired by Brubaker, Buciumas, Bump and Gustafsson's model for Demazure atoms (2021) and Borodin and Wheeler's model for permuted-basement nonsymmetric Macdonald polynomials (2022). We make this model compatible with an uncoloured vertex model for Schur polynomials, putting them in a single framework. Unlike previous work on structure coefficients via vertex models, a remarkable feature of our construction is that it relies on a Yang–Baxter equation that only holds for certain boundary conditions. However, this restricted Yang–Baxter equation is sufficient to show our result.