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arXiv · 2503.09240

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

Timothy C. Miller
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Abstract

We present the first positive combinatorial rule for expanding the product of a permuted-basement Demazure atom and a Schur polynomial. Special cases of permuted-basement Demazure atoms include Demazure atoms and characters. These cases have known tableau formulas for their expansions when multiplied by a Schur polynomial, due to Haglund, Luoto, Mason and van Willigenburg. We find a vertex model formula, giving a new rule even in these special cases, extending a technique introduced by Zinn-Justin for calculating Littlewood-Richardson coefficients. We derive a coloured vertex model for permuted-basement Demazure atoms, inspired by Borodin and Wheeler's model for non-symmetric Macdonald polynomials. 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.

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Timothy C. Miller. 2025-03-12. Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial. https://arxiv.org/abs/2503.09240

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