Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918413762297856 |
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| author | Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark |
| author_facet | Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark |
| contents | Let $Δ^λ$ be the Weyl functor for the partition $λ$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing that any $\mathrm{SL}_2(\mathbb{F})$-plethysm $Δ^{(M,1^N)}\mathrm{Sym}^d E$ factors as a tensor product of two simpler $\mathrm{SL}_2(\mathbb{F})$-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$, obtaining a new family of $\mathrm{SL}_2(\mathbb{F})$-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of $\mathrm{SL}_2(\mathbb{F})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark Representation Theory Combinatorics 20C20 (Primary) 05E10, 05A30, 20G05, 05E05 (Secondary) Let $Δ^λ$ be the Weyl functor for the partition $λ$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing that any $\mathrm{SL}_2(\mathbb{F})$-plethysm $Δ^{(M,1^N)}\mathrm{Sym}^d E$ factors as a tensor product of two simpler $\mathrm{SL}_2(\mathbb{F})$-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$, obtaining a new family of $\mathrm{SL}_2(\mathbb{F})$-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of $\mathrm{SL}_2(\mathbb{F})$. |
| title | Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions |
| topic | Representation Theory Combinatorics 20C20 (Primary) 05E10, 05A30, 20G05, 05E05 (Secondary) |
| url | https://arxiv.org/abs/2509.01490 |