Some computations in the modular representation ring of a finite group

1971 ◽  
Vol 69 (1) ◽  
pp. 163-166 ◽  
Author(s):  
John Santa Pietro
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Modular Representation ◽  
Multiplicative Structure ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Representation Ring ◽  
Ring Isomorphism ◽  
A Finite Group

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

scholarly journals The identity of certain representation algebra decompositions

1970 ◽  
Vol 3 (1) ◽  
pp. 73-74
Author(s):  
S. B. Conlon ◽  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Linear Algebra ◽  
Residue Field ◽  
Complex Field ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
A Finite Group ◽  
Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

scholarly journals Certain representation algebras

1965 ◽  
Vol 5 (1) ◽  
pp. 83-99 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Alternating Group ◽  
Complex Numbers ◽  
Direct Sum ◽  
Characteristic 2 ◽  
A Finite Group ◽  
Inequivalent Representations ◽  
Representation Algebra

Let Λ be the set of inequivalent representations of a finite group over a field . Λ is made the basis of an algebra over the complex numbers , called the representation algebra, in which multiplication corresponds to the tensor product of representations and addition to direct sum. Green [5] has shown that if char (the non-modular case) or if is cyclic, then is semi-simple, i.e. is a direct sum of copies of . Here we consider two modular, non-cyclic cases, viz, where is or 4 (alternating group) and is of characteristic 2.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

Extending Endo-monomial Modules

2013 ◽  
Vol 20 (01) ◽  
pp. 169-172
Author(s):  
Ziqun Lu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Characteristic P ◽  
Complete Discrete Valuation Ring ◽  
A Finite Group

Let G be a finite group with a normal Sylow p-subgroup P. Let [Formula: see text] be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial [Formula: see text]-module. In this paper we prove that M extends to an [Formula: see text]-module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

scholarly journals Group structure and properties of block ideals of the group algebra

1975 ◽  
Vol 16 (1) ◽  
pp. 22-28 ◽  
Author(s):  
Wolfgang Hamernik
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Group Structure ◽  
Arbitrary Field ◽  
Structure And Properties ◽  
Characteristic P ◽  
Principal Block ◽  
A Finite Group

In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:(i) J(B1) ≤ Z(B1)(ii) J(B1)is commutative,(iii) G is p-nilpotent with abelian Sylowp-subgroups.

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

scholarly journals On Modular Representation Algebras and a Class of Matrix Algebras

1982 ◽  
Vol 33 (3) ◽  
pp. 351-355 ◽  
Author(s):  
J-C. Renaud
Keyword(s):  
Tensor Product ◽  
Cyclic Group ◽  
Prime Order ◽  
Matrix Algebra ◽  
Modular Representation ◽  
Complex Field ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Standard Operations

AbstractLet G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

scholarly journals Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

1999 ◽  
Vol 1999 (511) ◽  
pp. 145-191 ◽  
Author(s):  
Richard Dipper ◽  
Jochen Gruber
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Parabolic Type ◽  
Modular Representation ◽  
Weyl Groups ◽  
Modular Representation Theory ◽  
Schur Algebras ◽  
Schur Algebra ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

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