The McKay Conjecture on character degrees
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915268646666240 |
|---|---|
| author | Cabanes, Marc Späth, Britta |
| author_facet | Cabanes, Marc Späth, Britta |
| contents | We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay.
The conjecture was reduced by Isaacs--Malle--Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups.
We study mainly characters of normalizers N$_{\mathbf G}({\mathbf S})^F$ of Sylow $d$-tori ${\mathbf S}$ ($d\geq 3$) in a simply-connected algebraic group ${\mathbf G}$ of type D$_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type some D$_{k}\times\ $D$_{l-k}$. The finite groups ${\mathbf M}^F$ are an efficient substitute for N$_{\mathbf G}({\mathbf S})^F$ or the $\ell$-local subgroups of ${\mathbf G}^F$ relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of Aut$({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that Irr$({\mathbf M}^F)$ and Irr$({\mathbf G}^F)$ share some key features in that regard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20392 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The McKay Conjecture on character degrees Cabanes, Marc Späth, Britta Representation Theory Group Theory 20C15 (20C33 20G40) We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay. The conjecture was reduced by Isaacs--Malle--Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N$_{\mathbf G}({\mathbf S})^F$ of Sylow $d$-tori ${\mathbf S}$ ($d\geq 3$) in a simply-connected algebraic group ${\mathbf G}$ of type D$_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type some D$_{k}\times\ $D$_{l-k}$. The finite groups ${\mathbf M}^F$ are an efficient substitute for N$_{\mathbf G}({\mathbf S})^F$ or the $\ell$-local subgroups of ${\mathbf G}^F$ relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of Aut$({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that Irr$({\mathbf M}^F)$ and Irr$({\mathbf G}^F)$ share some key features in that regard. |
| title | The McKay Conjecture on character degrees |
| topic | Representation Theory Group Theory 20C15 (20C33 20G40) |
| url | https://arxiv.org/abs/2410.20392 |