Orbits and characters associated with rook placements for Sylow $p$-subgroups of finite orthogonal groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913660389031936 |
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| author | Ignatev, Mikhail Venchakov, Mikhail |
| author_facet | Ignatev, Mikhail Venchakov, Mikhail |
| contents | Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field of characteristic $p$. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. Almost all important classes of orbits and characters studied to the moment can be uniformly described as the orbits and characters associated with so-called orthogonal rook placements. In the paper, we study such orbits for the orthogonal group. We construct a polarization for the canonical form on such an orbit and present a semi-direct decomposition for the corresponding irreducible characters in the spirit of the Mackey little group method. As a corollary, we compute the dimension of an orbit associated with an orthogonal rook placement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbits and characters associated with rook placements for Sylow $p$-subgroups of finite orthogonal groups Ignatev, Mikhail Venchakov, Mikhail Representation Theory Group Theory 20C15, 17B08, 20D15 Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field of characteristic $p$. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. Almost all important classes of orbits and characters studied to the moment can be uniformly described as the orbits and characters associated with so-called orthogonal rook placements. In the paper, we study such orbits for the orthogonal group. We construct a polarization for the canonical form on such an orbit and present a semi-direct decomposition for the corresponding irreducible characters in the spirit of the Mackey little group method. As a corollary, we compute the dimension of an orbit associated with an orthogonal rook placement. |
| title | Orbits and characters associated with rook placements for Sylow $p$-subgroups of finite orthogonal groups |
| topic | Representation Theory Group Theory 20C15, 17B08, 20D15 |
| url | https://arxiv.org/abs/2406.04436 |