Functorial equivalence classes of $2$-blocks of tame representation type
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908545432158208 |
|---|---|
| author | Boltje, Robert Bouc, Serge Yılmaz, Deniz |
| author_facet | Boltje, Robert Bouc, Serge Yılmaz, Deniz |
| contents | For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over $\mathbb{F}$ if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over $\mathbb{F}$ must have isomorphic fusion systems. The converse is wrong in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functorial equivalence classes of $2$-blocks of tame representation type Boltje, Robert Bouc, Serge Yılmaz, Deniz Representation Theory 20C20 (Primary), 20J15, 19A22 (Secondary) For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over $\mathbb{F}$ if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over $\mathbb{F}$ must have isomorphic fusion systems. The converse is wrong in general. |
| title | Functorial equivalence classes of $2$-blocks of tame representation type |
| topic | Representation Theory 20C20 (Primary), 20J15, 19A22 (Secondary) |
| url | https://arxiv.org/abs/2509.14682 |