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 .

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

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 THE DUAL STRUCTURE OF CROSSED PRODUCT -ALGEBRAS WITH FINITE GROUPS

2013 ◽  
Vol 88 (2) ◽  
pp. 243-249 ◽  
Author(s):  
FIRUZ KAMALOV
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Orbit Space ◽  
Crossed Product ◽  
Irreducible Representations ◽  
Natural Action ◽  
Dual Structure ◽  
Projective Representations ◽  
A Finite Group ◽  
Sigma G

AbstractWe study the space of irreducible representations of a crossed product ${C}^{\ast } $-algebra ${\mathop{A\rtimes }\nolimits}_{\sigma } G$, where $G$ is a finite group. We construct a space $\widetilde {\Gamma } $ which consists of pairs of irreducible representations of $A$ and irreducible projective representations of subgroups of $G$. We show that there is a natural action of $G$ on $\widetilde {\Gamma } $ and that the orbit space $G\setminus \widetilde {\Gamma } $ corresponds bijectively to the dual of ${\mathop{A\rtimes }\nolimits}_{\sigma } G$.

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Restricting and Inducing on Inner Products of Representations of Finite Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1349-1354
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Reciprocity Theorem ◽  
Frobenius Reciprocity ◽  
Inner Products ◽  
Starting Point ◽  
Representations Of Finite Groups ◽  
A Finite Group

Of recent years the author has been interested in developing a representation theory of the algebra of representations [5; 6] of a finite group G, and dually of its classes [7]. In this paper Frobenius’ Reciprocity Theorem provides a starting point for the introduction of the inverses R-1 and I-1 of the restricting and inducing operators R and I. The condition under which such inverse operations are available is that the classes of G do not splitin the subgroup Ĝ. When this condition is satisfied the application of these operations to inner products is of interest.

ON THE IDENTITIES OF REPRESENTATIONS OF FINITE GROUPS OVER COMMUTATIVE RINGS

1992 ◽  
Vol 02 (01) ◽  
pp. 103-116
Author(s):  
SAMUEL M. VOVSI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Finite Basis ◽  
Finitely Based ◽  
Commutative Noetherian Ring ◽  
Representations Of Finite Groups ◽  
A Finite Group ◽  
Finite Exponent

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

scholarly journals On minimal faithful permutation representations of finite groups

1988 ◽  
Vol 38 (2) ◽  
pp. 207-220 ◽  
Author(s):  
David Easdown ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Exceptional Groups ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

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 On the kernels of representations of finite groups

1981 ◽  
Vol 22 (2) ◽  
pp. 151-154 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Modular Representation ◽  
Complex Representation ◽  
Representations Of Finite Groups ◽  
A Finite Group

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:(1) G is p-solvable of p-length 1.(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

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