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**Local fusions in block source algebras.**
*(English)*
Zbl 0606.20016

Almost three decades ago J. A. Green introduced vertices and sources to study the structure of finite groups G and their group rings. Guided by Green’s profound investigations the author considers the general concept of an interior G-algebra A to define so called source algebras of A.

He writes in his introduction: Obviously the most interesting new case where this generalization applies occurs when A is a block algebra of G. In this case, the source algebra B is likely to contain all the ”local” information about the block; for instance, the module categories of A and B are equivalent through restriction, and it is easy to see that vertices and sources of indecomposble A-modules can be computed from the corresponding B-modules; similarly, it is not difficult to show that the matrix of generalized decomposition numbers can be computed from B. Our main result here implies the analogous statement concerning the local category of the block where objects are the local pointed groups on A and morphisms are the G-exomorphisms, namely: the equivalence class of the local category can be computed from B. In order to prove that, we introduce the so called local fusion category of any interior G-algebra and we show that (i) local fusion categories of A and B are equivalent, (ii) local and local fusion categories of A coincide.

He writes in his introduction: Obviously the most interesting new case where this generalization applies occurs when A is a block algebra of G. In this case, the source algebra B is likely to contain all the ”local” information about the block; for instance, the module categories of A and B are equivalent through restriction, and it is easy to see that vertices and sources of indecomposble A-modules can be computed from the corresponding B-modules; similarly, it is not difficult to show that the matrix of generalized decomposition numbers can be computed from B. Our main result here implies the analogous statement concerning the local category of the block where objects are the local pointed groups on A and morphisms are the G-exomorphisms, namely: the equivalence class of the local category can be computed from B. In order to prove that, we introduce the so called local fusion category of any interior G-algebra and we show that (i) local fusion categories of A and B are equivalent, (ii) local and local fusion categories of A coincide.

Reviewer: W.Hamernik

### MSC:

20C20 | Modular representations and characters |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

16S34 | Group rings |

### Keywords:

vertices; sources; group rings; interior G-algebra; source algebras; block algebra; module categories; generalized decomposition numbers; local pointed groups; G-exomorphisms; local fusion category### References:

[1] | Broué, M.; Puig, L., A Frobenius theorem for blocks, Invent. Math., 56, 117-128 (1980) · Zbl 0425.20008 |

[2] | Green, J. A., Some remarks on defect groups, Math. Z., 107, 133-150 (1968) · Zbl 0164.34002 |

[3] | Puig, L., Pointed groups and construction of characters, Math. Z., 176, 265-292 (1981) · Zbl 0464.20007 |

[4] | Puig, L., The source algebra of a nilpotent block (1981), preprint |

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