Homogeneous spaces of semidirect products and finite Gelfand pairs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909201591173120 |
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| author | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo |
| author_facet | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo |
| contents | Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a $K$-invariant subgroup. We study the homogeneous space $X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the decomposition of the permutation representation of $H\ltimes A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl's results on the $q$-analog of the nonbinary Johnson scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08371 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogeneous spaces of semidirect products and finite Gelfand pairs Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo Representation Theory Combinatorics Functional Analysis Group Theory 20C15, 20G40, 33C55, 43A90, 20E22 Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a $K$-invariant subgroup. We study the homogeneous space $X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the decomposition of the permutation representation of $H\ltimes A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl's results on the $q$-analog of the nonbinary Johnson scheme. |
| title | Homogeneous spaces of semidirect products and finite Gelfand pairs |
| topic | Representation Theory Combinatorics Functional Analysis Group Theory 20C15, 20G40, 33C55, 43A90, 20E22 |
| url | https://arxiv.org/abs/2405.08371 |