A geometric and generating function approach to plethysm
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2025
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| author | Gutiérrez, Álvaro Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| author_facet | Gutiérrez, Álvaro Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| contents | Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A geometric and generating function approach to plethysm Gutiérrez, Álvaro Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike Combinatorics 05E10 (Primary) 05A15, 05E05 (Secondary) Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients. |
| title | A geometric and generating function approach to plethysm |
| topic | Combinatorics 05E10 (Primary) 05A15, 05E05 (Secondary) |
| url | https://arxiv.org/abs/2511.02649 |