Asymptotics of Plethysm
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918137420578816 |
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| author | Kuppel, Tim |
| author_facet | Kuppel, Tim |
| contents | We study multiplicities $a^{dλ}_{μ,(dk)}$ of highest weight representations $\mathbb S_{dλ}(\mathbb C^n)$, $λ\vdash pk$, of length at most $p$, in $\mathbb{S}_μ(S^{dk}(\mathbb C^n))$, $μ\vdash p$, so called plethysm coefficients, as $d$ tends to $\infty$. These are given by quasi-polynomials, which in the case of $S^p(S^{dk}(\mathbb C^n))$ can explicitly be computed by Pieri's rule. We show that for all but a finite, explicit list of $λ$'s the leading term is in fact constant and that $$
a^{dλ}_{μ,(dk)}\sim \frac{\dim V_μ}{p!}c^{dλ}_{p,dk} $$ as $d\to\infty$. In particular, we answer a conjecture of Kahle and Michałek, going back to Howe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Plethysm Kuppel, Tim Representation Theory Combinatorics 05A16, 05E10, 20G05 We study multiplicities $a^{dλ}_{μ,(dk)}$ of highest weight representations $\mathbb S_{dλ}(\mathbb C^n)$, $λ\vdash pk$, of length at most $p$, in $\mathbb{S}_μ(S^{dk}(\mathbb C^n))$, $μ\vdash p$, so called plethysm coefficients, as $d$ tends to $\infty$. These are given by quasi-polynomials, which in the case of $S^p(S^{dk}(\mathbb C^n))$ can explicitly be computed by Pieri's rule. We show that for all but a finite, explicit list of $λ$'s the leading term is in fact constant and that $$ a^{dλ}_{μ,(dk)}\sim \frac{\dim V_μ}{p!}c^{dλ}_{p,dk} $$ as $d\to\infty$. In particular, we answer a conjecture of Kahle and Michałek, going back to Howe. |
| title | Asymptotics of Plethysm |
| topic | Representation Theory Combinatorics 05A16, 05E10, 20G05 |
| url | https://arxiv.org/abs/2509.06424 |