I will present a 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. I have 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.

Vertex models are a tool used in statistical mechanics to model particle systems. A notable example is the six-vertex or square-ice model. I build on a vertex model for permuted-basement Demazure atoms, inspired by Borodin and Wheeler's model for non-symmetric Macdonald polynomials. The proof is completely combinatorial; gluing two vertex models together gives a product and the resulting picture is transformed into a summation. A notable feature of this construction is that the underlying Yang–Baxter equation does not hold for all boundary conditions. However, it holds in enough cases to show the result.