Brauer trees in classical groups.

*(English)*Zbl 0704.20011In this important paper the authors complete the determination of the Brauer trees in classical groups (i.e., general linear, unitary, symplectic and orthogonal groups) of odd characteristic, initiated by them [in Bull. Am. Math. Soc., New Ser. 3, 1041-1044 (1980; Zbl 0452.20008) and Math. Z. 187, 81-88 (1984; Zbl 0545.20006)]. The results are based upon earlier work of the authors on the classification of blocks of classical groups [Invent. Math. 69, 109-153 (1982; Zbl 0507.20007) and J. Reine Angew. Math. 396, 122-191 (1989; Zbl 0656.20039)]. The work is a contribution to the program of determining the decomposition matrices of all finite simple groups.

The problem of finding decomposition numbers in a block is considerably easier if the block has a cyclic defect group, which is equivalent to saying that there are only finitely many indecomposable modules in the block. The information on the decomposition numbers can then be encoded in a certain graph, the so-called Brauer tree. The explicit knowledge of the Brauer trees for the simple groups will have many applications in group theory and other branches of mathematics.

The authors show that in a classical group the Jordan decomposition of characters induces graph automorphisms of Brauer trees. Hence only the Brauer trees for the unipotent blocks have to be determined. In such a block, the non-exceptional characters are exactly the unipotent characters. These are parametrized by certain combinatorial objects, the so-called symbols, which can be considered as generalizations of partitions. The authors determine the location of the unipotent characters on the Brauer tree in terms of the symbols. In order to facilitate the work with the symbols they introduce the abacus diagram corresponding to a symbol and give an excellent introduction to the somewhat complicated theory of symbols. The principal method of proof is Harish-Chandra induction from subparabolic subgroups.

The problem of finding decomposition numbers in a block is considerably easier if the block has a cyclic defect group, which is equivalent to saying that there are only finitely many indecomposable modules in the block. The information on the decomposition numbers can then be encoded in a certain graph, the so-called Brauer tree. The explicit knowledge of the Brauer trees for the simple groups will have many applications in group theory and other branches of mathematics.

The authors show that in a classical group the Jordan decomposition of characters induces graph automorphisms of Brauer trees. Hence only the Brauer trees for the unipotent blocks have to be determined. In such a block, the non-exceptional characters are exactly the unipotent characters. These are parametrized by certain combinatorial objects, the so-called symbols, which can be considered as generalizations of partitions. The authors determine the location of the unipotent characters on the Brauer tree in terms of the symbols. In order to facilitate the work with the symbols they introduce the abacus diagram corresponding to a symbol and give an excellent introduction to the somewhat complicated theory of symbols. The principal method of proof is Harish-Chandra induction from subparabolic subgroups.

Reviewer: G.Hiss

##### MSC:

20C20 | Modular representations and characters |

20G40 | Linear algebraic groups over finite fields |

20G05 | Representation theory for linear algebraic groups |

20C33 | Representations of finite groups of Lie type |

##### Keywords:

Brauer trees in classical groups; decomposition matrices; finite simple groups; decomposition numbers; defect groups; indecomposable modules; Jordan decompositions; characters; unipotent blocks; unipotent characters; symbols; Harish-Chandra induction
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\textit{P. Fong} and \textit{B. Srinivasan}, J. Algebra 131, No. 1, 179--225 (1990; Zbl 0704.20011)

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