Representations Of Finite Groups Research Articles

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic ℓ≠p has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for GLn.

Interdisciplinary applications of finite group theory - From epidemic modeling to financial market analysis

Finite group theory has a long history. Many mathematicians have done researches on it. As time goes on, it becomes a useful tool for human as it has a lot of applications. Finite group theory becomes more than just mathematical. As a result, its applications on epidemic modeling and financial market analysis are introduced respectively. These focus on SIR (Susceptible-Infective-Recovered) model and stock markets. Whats more, there are some other applications in different areas. What we came to the conclusion is that wired groups have developed in many fields, not only in financial markets and infectious disease models, but also in many other fundamental disciplines, chemistry, physics, computer science. They are going to be showed in the following essay.

Unisingular subgroups of symplectic groups over [formula omitted]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2n<250, specifically, the question remains open only 7 values of n.

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function f on a finite abelian group G can be written as a linear combination of characters of irreducible representations of G by f(x)=∑χ∈Gˆfˆ(χ)χ(x), where Gˆ is the dual group of G consisting of all characters of G and fˆ(χ) is the Fourier coefficient of f at χ∈Gˆ. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of f on a finite abelian group G with complexity that is quasi-linear in the order of G and polynomial in the FSOS sparsity of f. Moreover, for a nonnegative function f on a finite abelian group G and a subset S⊆Gˆ, we give a lower bound of the constant M such that f+M admits an FSOS supported on S. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 107. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.

A note on transfer in finite group theory

A note on transfer in finite group theory

On oriented m $m$‐semiregular representations of finite groups

AbstractA finite group admits an oriented regular representation if there exists a Cayley digraph of such that it has no digons and its automorphism group is isomorphic to . Let be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented ‐semiregular representations using ‐Cayley digraphs. Given a finite group , an ‐Cayley digraph of is a digraph that has a group of automorphisms isomorphic to acting semiregularly on the vertex set with orbits. We say that a finite group admits an oriented ‐semiregular representation (OSR for short) if there exists an ‐Cayley digraph of such that it has no digons and is isomorphic to its automorphism group. Moreover, if is regular, that is, each vertex has the same in‐ and out‐valency, we say is a regular oriented ‐semiregular representation (regular OSR for short) of . In this paper, we classify finite groups admitting a regular OSR or an OSR for each positive integer .

On m-partite oriented semiregular representations of finite groups generated by two elements

In this paper we extend the notion of oriented regular representations in the context of m-partite oriented digraphs with an integer m≥2. A group G is said to admit an m-partite oriented semiregular representation (m-POSR for short) if there exists an m-partite oriented digraph such that its automorphism group is isomorphic to G and acts semiregularly on vertices with every part of the digraph as an orbit. In this paper, we proved that G admits an m-POSR of valency two except for several cases when G is a finite group generated by at most two elements and m≥2 is an integer.

Matrices for finite group representations that respect Galois automorphisms

AbstractWe are given a finite group H, an automorphism $$\tau $$ τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group $$\langle \sigma \rangle $$ ⟨ σ ⟩ of order r, and an absolutely irreducible representation $$\rho :H\rightarrow \textsf {GL} (n,L)$$ ρ : H → GL ( n , L ) such that the action of $$\tau $$ τ on the character of $$\rho $$ ρ is the same as the action of $$\sigma $$ σ . Then the following are equivalent. $$\bullet $$ ∙ $$\rho $$ ρ is equivalent to a representation $$\rho ':H\rightarrow \textsf {GL} (n,L)$$ ρ ′ : H → GL ( n , L ) such that the action of $$\sigma $$ σ on the entries of the matrices corresponds to the action of $$\tau $$ τ on H, and $$\bullet $$ ∙ the induced representation $$\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )$$ ind H , H ⋊ ⟨ τ ⟩ ( ρ ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of $$A_5$$ A 5 over $$\mathbb {Q}[\sqrt{5}]$$ Q [ 5 ] and a four dimensional irreducible representation of the double cover of $$A_7$$ A 7 over $$\mathbb {Q}[\sqrt{-7}]$$ Q [ - 7 ] .

On some classes of finite σ-soluble &lt;i&gt;PσT&lt;/i&gt;-groups

Let X be a class of groups. Suppose that with each group G ∈ X we associate some system of its subgroups τ(G). Then τ is said to be a subgroup functor on X if the following conditions are hold: (1) G ∈ τ(G) for each group G ∈ X; (2) for any epimorphism φ: A → B, where A, B ∈ X, and for any groups H ∈ τ(A) and T ∈ τ(B) we have Hφ ∈ τ(B) and Tφ-1 ∈ τ( A). In this paper, were considered some applications of such subgroup functors in the theory of finite groups in which generalized normality for subgroups is transitive.

Some inequalities and equations of g-frame operator multipliers for finite group representations

Some inequalities and equations of g-frame operator multipliers for finite group representations

On gamma factors for representations of finite general linear groups

On gamma factors for representations of finite general linear groups

Representations of Finite Groups

The workshop Representations of Finite Groups was organised by Olivier Dudas (Marseille), Meinolf Geck (Stuttgart), Radha Kessar (Manchester), and Gabriel Navarro (Valencia). It covered a wide variety of aspects of the representation theory of finite groups and related topics, and showcased several recent breakthrough results.

Remarks on Conjectures in Block Theory of Finite Groups

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results.

Modular Conjectures for Direct Product of Finite Groups

The representation theory of a finite group, G, is an important area of research currently. This paper studied the modular representation of finite groups, which are direct products. There are three approaches to studying this representation: the ring approach, the character approach, and the module approach. Moreover, we learned some of the important conjectures in this representation, which link a representation of a finite group and its local subgroups, which are normalizer non-trivial p-subgroups. These conjectures are the McKay conjecture, Alperin’s weight conjecture, and the ordinary weight conjecture. The main aim of the proposed paper was to investigate these conjectures of direct products, the direct summands of which satisfy these conjectures for the associated tensor product of the p-block. We obtained the results by assuming the conjectures are true. Then, we used the properties of the direct products.

Projective joint spectra and characters of representations of [formula omitted

For a tuple of square complex-valued N×N matrices A1,…,An the determinant of their linear combination x1A1+⋯+xnAn, which is called a pencil, is a homogeneous polynomial of degree N in C[x1,...xn]. Zero-set of this polynomial is an algebraic set in the projective space CPn−1. This set is called the determinantal hypersurface or determinantal manifold of the tuple (A1,...,An). It was shown in [7] that determinantal hypersurfaces contain substantial information about representations of finite Coxeter groups. Namely, if G is a non-special Coxeter group of type A,B, or D, ρ1 and ρ2 are two linear representations of G, and the determinantal hypersurfaces of images of the Coxeter generators of G under ρ1 and ρ2 coincide as divisors in the projective space, the characters of ρ1 and ρ2 are equal, and, therefore, ρ1 and ρ2 are equivalent. In [30] this result was extended in the characters part to affine Coxeter groups of types B,C, and D. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of A type are not covered by this result, as their combinatorics is quite different. In this paper we explicitly construct a finite set in A˜n having the same property. We also show that every group which is a semidirect product of a finite group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensional representation of the group, the determinantal hypersurface of images of the set determines the representation character.

A Formalization of Finite Group Theory: Part III

A Formalization of Finite Group Theory: Part III

A Formalization of Finite Group Theory: Part II

A Formalization of Finite Group Theory: Part II

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Representations of Finite Groups by Posets of Small Height

Representations of Finite Groups by Posets of Small Height