Finite Group Algebras and their Modules

Landrock, Peter

doi:10.1017/cbo9781107325524

Cited by 168 publications

(86 citation statements)

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“…We deduce that Extßk{G)(L, M) S Ex\llk(NG(C))(f(L),f(M)) as in [14,II 5.9]. Since the closure of any non-projective object under the equivalence relation generated by the property of having a non-zero Ext group gives all non-projectives in the block, we deduce for non-projective Mackey functors that Af is in b if and only if f(M) is in ê.…”

Section: Brauer Treesmentioning

confidence: 83%

Untitled

1995

Trans. Amer. Math. Soc.

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Abstract.Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological K-theory of classifying spaces, the algebraic K-theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.

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Section: Brauer Treesmentioning

confidence: 83%

Untitled

1995

Trans. Amer. Math. Soc.

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“…Mackey's tensor product decomposition [14,Corollary II.6.4] implies that B χ 6 is induced by an irreducible character of degree q …”

Section: M-groups and Characters Of Sylow-2-subgroupsmentioning

confidence: 99%

Character Table of a Borel Subgroup of the Ree Groups ²F₄(q²)

Himstedt¹,

Huang²

2009

LMS J. Comput. Math.

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We compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2 F 4 (2 2n+1 ) for all n 1 and prove that these Borel subgroups are M -groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2 F 4 (2 2n+1 ) and show that the IsaacsMalle-Navarro version of the McKay conjecture holds for 2 F 4 (2 2n+1 ) in characteristic 2. For most of the calculations we use CHEVIE.

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“…Since L is a trivial source module (being a direct summand of lB), Y has trivial source, and so all its endomorphisms are liftable (see [11,Theorem II 12.4]). Every irreducible /cG-module in 53 is self-dual, since it has a realvalued Brauer character.…”

Section: The Blocks B2b Replace a By B Inmentioning

confidence: 99%

2-blocks and 2-modular characters of the Chevalley groups 𝐺₂(𝑞)

Hiß¹,

Shamash²

1992

Math. Comp.

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Abstract. We first determine the distribution of the ordinary irreducible characters of the exceptional Chevalley group Gj(q), q odd, into 2-blocks. This is done by using the method of central characters. Then all but two of the irreducible 2-modular characters are determined. The results are given in the form of decomposition matrices. The methods here involve concepts from modular representation theory and symbolic computations with the computer algebra system MAPLE. As a corollary, the smallest degree of a faithful representation of G2(q), q odd, over a field of characteristic 2 is obtained.

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Finite Group Algebras and their Modules

Cited by 168 publications

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Character Table of a Borel Subgroup of the Ree Groups ²F₄(q²)

2-blocks and 2-modular characters of the Chevalley groups 𝐺₂(𝑞)

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Finite Group Algebras and their Modules

Cited by 168 publications

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Character Table of a Borel Subgroup of the Ree Groups 2F4(q2)

2-blocks and 2-modular characters of the Chevalley groups 𝐺₂(𝑞)

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Character Table of a Borel Subgroup of the Ree Groups ²F₄(q²)