The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$
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| Format: | Preprint |
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2022
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| _version_ | 1866909274579402752 |
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| author | Turek, Pavel |
| author_facet | Turek, Pavel |
| contents | Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. We provide a practical algebraic description of the representation ring of $k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a family of modular plethysms of the natural $k\mathrm{SL}_2(\mathbb{F}_p)$-module $E$ of the form $\nabla^ν\mathrm{Sym}^l E$ for a partition $ν$ of size less than $p$ and $0\leq l\leq p-2$. Within this family we classify both the modular plethysms of $E$ which are projective and the modular plethysms of $E$ which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of $E$ are replaced by $k\mathrm{SL}_2(\mathbb{F}_p)$-modules of the form $\nabla^ν V$, where $V$ is a non-projective indecomposable $k\mathrm{SL}_2(\mathbb{F}_p)$-module and $|ν|<p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_08943 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$ Turek, Pavel Representation Theory Combinatorics 20C20 (Primary), 05E05, 05E10, 19A22, 20C33 (Secondary) Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. We provide a practical algebraic description of the representation ring of $k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a family of modular plethysms of the natural $k\mathrm{SL}_2(\mathbb{F}_p)$-module $E$ of the form $\nabla^ν\mathrm{Sym}^l E$ for a partition $ν$ of size less than $p$ and $0\leq l\leq p-2$. Within this family we classify both the modular plethysms of $E$ which are projective and the modular plethysms of $E$ which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of $E$ are replaced by $k\mathrm{SL}_2(\mathbb{F}_p)$-modules of the form $\nabla^ν V$, where $V$ is a non-projective indecomposable $k\mathrm{SL}_2(\mathbb{F}_p)$-module and $|ν|<p$. |
| title | The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$ |
| topic | Representation Theory Combinatorics 20C20 (Primary), 05E05, 05E10, 19A22, 20C33 (Secondary) |
| url | https://arxiv.org/abs/2210.08943 |