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

TL;DR

The paper solves the long-standing problem of obtaining a positive combinatorial rule for the product of a permuted-basement Demazure atom $\mathcal{A}_\alpha^\sigma(x)$ and a Schur polynomial $s_\lambda(x)$ by developing a unified vertex-model framework. It constructs colour-encoded atom and Schur vertex models on rhombic lattices, couples them through a restricted Yang–Baxter identity (the Column Lemma), and proves a positive expansion $\mathcal{A}_\alpha^\sigma(x)s_\lambda(x)=\sum_\beta a_{\alpha\lambda}^\beta(\sigma)\mathcal{A}_\beta^\sigma(x)$ with structure coefficients $a_{\alpha\lambda}^\beta(\sigma)$. A key contribution is the atom–Schur model, whose partition function yields these coefficients, together with a bijection to skyline fillings and SSAFs, linking integrable-system techniques to combinatorial formulas. The work also outlines potential extensions to related polynomials and higher structures, such as Grothendieck analogues, while highlighting the role of boundary-restricted Yang–Baxter relations. This advances positive combinatorial rules beyond Demazure atoms/characters and unifies several perspectives within algebraic combinatorics and integrable models.

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.

Figures (3)

• Figure 5.2: Banned adjacencies where $1\leq i<j\leq n$.
• Figure 6.2: The full set of tiles where $f,g,h,i,j,k$ are in $[n]$, $i\leq j$, $\sigma(f)<\sigma(g)$ and $\sigma(f)<\sigma(h)$.
• Figure 6.7: Two tilings of weight $x_1^3x_2^2x_3^4$ from the configurations in \ref{['lem.prisms']} where $\lambda=(2,2,1)$, $\sigma=231$ and $\alpha=(2,2,1)$, so that $n=3$, $k=4$ and $\sigma^{-1}=312$. Regions are labelled to facilitate exposition.

Theorems & Definitions (40)

• Example 2.1
• Example 2.2
• Remark 2.3
• Example 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Definition 3.5