Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

1995 ◽  
Vol 47 (5) ◽  
pp. 929-945 ◽  
Author(s):  
Harald Ellers
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Quotient Groups

AbstractAssume k is an algebraically closed field of characteristic p and G is a finite group. If P is a p-subgroup of G such that G = PCG(P), and if H is a normal subgroup of G with P ≤ H, then the number of H-cliques of irreducible k[G]-modules is the same as the number of H/P-cliques of irreducible k[G/P]-modules.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

scholarly journals On the degree of an indecomposable representation of a finite group

1979 ◽  
Vol 28 (3) ◽  
pp. 321-324 ◽  
Author(s):  
Gerald H. Cliff
Keyword(s):  
Finite Group ◽  
Indecomposable Representation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group

AbstractLet k be an algebraically closed field of characteristic p, and G a finite group. Let M be an indecomposable kG-module with vertex V and source X, and let P be a Sylow p-subgroup of G containing V. Theorem: If dimkX is prime to p and if NG(V) is p-solvable, then the p-part of dimkM equals [P:V]; dimkX is prime to p if V is cyclic.

scholarly journals Block Idempotents and Normal p-Subgroups

1966 ◽  
Vol 28 ◽  
pp. 1-13 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Reduction Theorem ◽  
Closed Field ◽  
Modular Representations ◽  
Algebraically Closed Field ◽  
A Finite Group

In the theory of modular representations of a finite group G in an algebraically closed field Ω of characteristic p, Brauer has proved a useful reduction theorem for blocks [2, §§11, 12], [5, (88.8)], which can be reformulated as follows:THEOREM 1 (Brauer). Let P bean arbitraryp-subgroup of G; let N = NG(P) and W = PCG(P).

scholarly journals Factor Ideals of Some Representation Algebras

1969 ◽  
Vol 9 (1-2) ◽  
pp. 109-123 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Isomorphism Class ◽  
Closed Field ◽  
Complex Field ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Representation Algebra

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

On the Character Rings of Finite Groups

1963 ◽  
Vol 15 ◽  
pp. 605-612 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Roots Of Unity ◽  
Closed Field ◽  
One Dimensional ◽  
Elementary Group ◽  
A Finite Group ◽  
Character Ring

The characters of the representations of a finite group G over a field K of characteristic zero generate a ring oK(G) of functions on G, the K-character ring of G, which is readily seen to be Zϕ1 + . . . + Zϕn, where Z is the ring of rational integers and ϕ1, . . . , ϕn are the characters of the different irreducible representations of G over K. The theorem that every irreducible representation of G over an algebraically closed field Ω of characteristic zero is equivalent to a representation of G over the subfield of Ω which is generated by the g0th roots of unity (g0 the exponent of G) was proved by Brauer (4) via the theorems that(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations (3).

scholarly journals Steinberg-like characters for finite simple groups

2020 ◽  
Vol 23 (1) ◽  
pp. 25-78
Author(s):  
Gunter Malle ◽  
Alexandre Zalesski
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Regular Representation ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Algebraically Closed Field ◽  
Characteristic 2 ◽  
A Finite Group

AbstractLet G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called {\mathrm{Syl}_{p}}-regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension {\lvert S\rvert}, where F is an algebraically closed field of characteristic p.

scholarly journals The centralizer of a subgroup in a group algebra

2012 ◽  
Vol 56 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Susanne Danz ◽  
Harald Ellers ◽  
John Murray
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Closed Field ◽  
List Type ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Centralizer Algebra

AbstractLet F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FGH. Among these, we provide examples to show that•the centre Z(FGH) can be larger than the F-algebra generated by Z(FG) and Z(FH),•FGH can have primitive central idempotents that are not of the form ef, where e and f are primitive central idempotents of FG and FH respectively,•it is not always true that the simple FGH-modules are the same as the non-zero FGH-modules HomFH(S, T ↓ H), where S and T are simple FH and FG-modules, respectively.

scholarly journals A decomposition of equivariant K-theory in twisted equivariant K-theories

2017 ◽  
Vol 28 (02) ◽  
pp. 1750016 ◽  
Author(s):  
José Manuel Gómez ◽  
Bernardo Uribe
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representation ◽  
Isomorphism Class ◽  
Irreducible Representations ◽  
Direct Sum ◽  
K Theory ◽  
Group 2 ◽  
A Finite Group

For [Formula: see text] a finite group and [Formula: see text] a [Formula: see text]-space on which a normal subgroup [Formula: see text] acts trivially, we show that the [Formula: see text]-equivariant [Formula: see text]-theory of [Formula: see text] decomposes as a direct sum of twisted equivariant [Formula: see text]-theories of [Formula: see text] parametrized by the orbits of the conjugation action of [Formula: see text] on the irreducible representations of [Formula: see text]. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of [Formula: see text] to the subgroup of [Formula: see text] which fixes the isomorphism class of the irreducible representation.

scholarly journals An Application of Finite Groups to Hopf algebras

2021 ◽  
Vol 14 (3) ◽  
pp. 816-828
Author(s):  
Tahani Al-Mutairi ◽  
Mohammed Mosa Al-shomrani
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Finite Groups ◽  
Hopf Algebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

scholarly journals A nine-dimensional algebra which is not a block of a finite group

2020 ◽  
Author(s):  
Markus Linckelmann ◽  
William Murphy
Keyword(s):  
Finite Group ◽  
Basic Algebra ◽  
Group Algebras ◽  
Closed Field ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Rule Out

Abstract We rule out a certain nine-dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most 12 of blocks of finite group algebras.

Sign in / Sign up

Export Citation Format

Share Document