# Oberseminar Algebra und Geometrie

# Sommersemester 2016:

Modular Representation Theory

Organised by B. Klopsch, S. Schröer and M. Zibrowius.

**All talks take place Fridays at 12:30 in 25.22.03.73.**

## Short description

The aim of the seminar is to get to know and learn basic concepts and techniques in modular representation theory. We assume a basic knowledge of ordinary representation theory (i.e., we are happy to quote anything needed in this regard) and will mainly follow selected chapters of the classical self-contained text [CR81] by Curtis and Reiner. Because of the time restrictions, we will*not*prove all, but only selected theorems. (The two volumes contain much more material than we will be able to cover. Of course, everybody is free to read up on topics that we do not cover in depth. In particular, Chapter 1 can be used for reference purposes.)

NB: The page numbers refer to Vol. 1 for Chapter 2 and Vol. 2 for Chapter 7.

## Talks and tentative dates

15.04.2016 | 1. Introduction (Benjamin Klopsch)
main source: [Hu75] Describe basic features of ordinary and modular representations, using the example SL(2,p). Note that the famous article of Deligne and Lusztig on representations of reductive groups over finite fields appeared later than Humphrey's gentle expository article. |

22.04.2016 | 2. The Decomposition Map (Tobias Hemmert)
main source: Ch 2, Sec 16 of [CR81] (402-416) p-modular systems, module categories, Grothendieck groups, projective class groups, Grothendieck ring, reduction mod p, decomposition homomorphism, decomposition matrix, examples, extensions of the ground field |

29.04.2016 | 3. Brauer Characters (Johannes Fischer)
main source: Ch 2, Sec 17 of [CR81] (417-426) splitting fields, Brauer characters, recall: projective cover (from Sec 6C), linear independence of Brauer characters, virtual Brauer characters, example |

06.05. and 13.05.2016 | 4/5. The Cartan-Brauer Triangle (André Schell, Benedikt Schilson)
main source: Ch 2, Sec 18 of [CR81] (427-447) summary, Cartan homomorphism, Cartan-Brauer triangle (commutative diagram), properties of the Cartan-Brauer triangle and equivalently the decomposition and Cartan matrices, Brauer lifts, orthogonality relations for Brauer characters, example: characters of defect 0, example: the Steinberg character |

20.05. and 27.05.2016 | 6/7. Vertices and Sources (Benno Kuckuck, Matthias Riepe)
main source: Ch 2, Sec 19 of [CR81] (448-469) relative projective and injective modules, vertices and sources, going-down theorem, going-up theorem, examples: p-blocks and defect groups, Green's Indecomposability Theorem, Green's Theorem on Zeros of Characters |

03.06.2016 | 8. Green Correspondence. Applications to Character Theory (Marcus Zibrowius)
main source: Ch 2, Sec 20 of [CR81] (470-490) admissible triples, Green correspondence: group-theoretical reduction, Thompson's application to groups involving Frobenius groups |

10.06.2016 | 9. The Induction Theorem for Arbitrary Fields (Sasa Novakovic)
main source: Ch 2, Sec 21 of [CR81] (491-512) Witt-Berman Induction Theorem, Induction Theorem over fields of positive characteristic, Cartan-Brauer triangle (general case) |

17.06.2016 | 10. Modular Representations of p-Solvable Groups (Matteo Vannacci)
main source: Ch 2, Sec 22 of [CR81] (513-519) p-solvable groups, Fong-Swan-Rukolaine Theorem |

24.06.2016 | 11. Introduction to Block Theory (Christoph Bärligea)
main source: Ch 7, Sec 56 of [CR81] (407-428) admissible p-modular systems, principal indecomposable modules, p-blocks, examples, linked principal indecomposable modules, central characters, defect |

01.07.2016 | 12. The Defect Group of a p-Block (Anitha Thillaisundaram)
main source: Ch 7, Sec 57 of [CR81] (429-444) G-algebras, trace map, defect groups, defect groups as vertices, defect groups as Sylow intersections |

08.07.2016 | 13. The Brauer Correspondence (Kevin Langlois)
main source: Ch 7, Sec 58 of [CR81] (445-461) Brauer map, Brauer's First Main Theorem, Brauer correspondence, Brauer correspondent, module-theoretic interpretations of the Brauer correspondence |

15.07.2016 | 14. Applications of Blocks to Character Theory (Steffen Kionke)
main source: Ch 7, Sec 59 of [CR81] (462-470) Nagao Decomposition Theorem, Brauer's Second Main Theorem, generalized decomposition numbers |

## Literature

[CR06] | Curtis and Reiner, Representation Theory of Finite Groups and Associative Algebras, AMS Chelsea, Providence, Rhode Island, 2006. |

[CR81] | Curtis and Reiner, Methods of Representation Theory - with Applications to Finite Groups and Orders, vol. 1 and 2, John Wiley and Sons, New York, 1981. |

[Hu75] | Humphreys, Representations of SL(2,p), Amer. Math. Monthly 82 (1975), 21-39. |

[Se96] | Serre, Linear Representations of Finite Groups, Springer, New York, 1996. |