On the source algebra equivalence class of blocks with cyclic defect groups, II
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912750441070592 |
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| author | Hiss, Gerhard Lassueur, Caroline |
| author_facet | Hiss, Gerhard Lassueur, Caroline |
| contents | This series of papers is a contribution to the program of classifying $p$-blocks of finite groups up to source algebra equivalence, starting with the case of cyclic blocks.
To any $p$-block $\mathbf{B}$ of a finite group with cyclic defect group $D$, Linckelmann associated an invariant $W( \mathbf{B} )$, which is an indecomposable endo-permutation module over $D$, and which, together with the Brauer tree of $\mathbf{B}$, essentially determines its source algebra equivalence class. In Parts II-IV of our series of papers, we classify, for odd $p$, those endo-permutation modules of cyclic $p$-groups arising from $p$-blocks of quasisimple groups.
In the present Part II, we reduce the desired classification for the quasisimple classical groups of Lie type $B$, $C$, and $D$ to the corresponding objective for the general linear and unitary groups; the classification is completed for the latter groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the source algebra equivalence class of blocks with cyclic defect groups, II Hiss, Gerhard Lassueur, Caroline Representation Theory Group Theory 20C20, 20C15, 20C33 This series of papers is a contribution to the program of classifying $p$-blocks of finite groups up to source algebra equivalence, starting with the case of cyclic blocks. To any $p$-block $\mathbf{B}$ of a finite group with cyclic defect group $D$, Linckelmann associated an invariant $W( \mathbf{B} )$, which is an indecomposable endo-permutation module over $D$, and which, together with the Brauer tree of $\mathbf{B}$, essentially determines its source algebra equivalence class. In Parts II-IV of our series of papers, we classify, for odd $p$, those endo-permutation modules of cyclic $p$-groups arising from $p$-blocks of quasisimple groups. In the present Part II, we reduce the desired classification for the quasisimple classical groups of Lie type $B$, $C$, and $D$ to the corresponding objective for the general linear and unitary groups; the classification is completed for the latter groups. |
| title | On the source algebra equivalence class of blocks with cyclic defect groups, II |
| topic | Representation Theory Group Theory 20C20, 20C15, 20C33 |
| url | https://arxiv.org/abs/2502.09176 |