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 On the Degrees of Irreducible Representations of a Finite Group

1951 ◽  
Vol 3 ◽  
pp. 5-6 ◽  
Author(s):  
Noboru Itô
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Irreducible Representations ◽  
A Finite Group

In 1896 G. Frobenius proved: the degree of any (absolutely) irreducible representation of a finite group divides its order. This theorem was improved by I. Schur in 1904 as follows: the degree of any irreducible representation of a finite group divides the index of its centre.

scholarly journals A Gelfand Model for Weyl Groups of Type D2n

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
José O. Araujo ◽  
Luis C. Maiarú ◽  
Mauro Natale
Keyword(s):  
Finite Group ◽  
Weyl Algebra ◽  
Polynomial Model ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Irreducible Factor ◽  
Direct Sum ◽  
Finite Dimensional ◽  
A Finite Group

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

On The Number of Classes of a Finite Group Invariant for Certain Substitutions

1974 ◽  
Vol 26 (5) ◽  
pp. 1090-1097 ◽  
Author(s):  
A. J. van Zanten ◽  
E. de Vries
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Group Invariant ◽  
A Finite Group ◽  
Number Of Classes ◽  
Special Case

In this paper we consider representations of groups over the field of the complex numbers.The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

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 Splitting fields of real irreducible representations of finite groups

2021 ◽  
Vol 25 (31) ◽  
pp. 897-902
Author(s):  
Dmitrii Pasechnik
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Finite Groups ◽  
Irreducible Representations ◽  
Content Type ◽  
Cyclotomic Numbers ◽  
Representations Of Finite Groups ◽  
A Finite Group ◽  
Algorithmic Procedure

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

L'application des representations au calcul explicite sur certaines chaînes de Markov finies

1984 ◽  
Vol 16 (3) ◽  
pp. 656-666 ◽  
Author(s):  
Bernard Ycart
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Irreducible Representations ◽  
Permutation Groups ◽  
A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

scholarly journals On the determination of the ramification index in Clifford's theorem

1988 ◽  
Vol 31 (3) ◽  
pp. 469-474
Author(s):  
Robert W. van der Waall
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Ramification Index ◽  
A Finite Group

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.

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