Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909599389450240 |
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| author | Miller, Timothy C. |
| author_facet | Miller, Timothy C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial Miller, Timothy C. Combinatorics 05E05 G.2.1 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. |
| title | Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial |
| topic | Combinatorics 05E05 G.2.1 |
| url | https://arxiv.org/abs/2503.09240 |