Representations and characters of finite groups

In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over finite filed; III.A Law of Large Numbers for the characters of GL_n(k) over finite field k; IV.An outline of construction of factor representations of the group GLB(F_q).

This article has been replaced by arXiv:0807.3093

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is ...

This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes the precise structure of the critical group both for the regular representation of any finite group, and for the reflection representation of the symmetric group.

In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A, split or not.

We determine the decomposition of cyclic characters of alternating groups into irreducible characters. As an application, we characterize pairs $(w, V)$, where $w\in A_n$ and $V$ is an irreducible representation of $A_n$ such that $w$ admits a non-zero invariant vector in $V$. We also establish new global conjugacy classes for alternating groups, thereby giving a new proof of a result of Heide and Zalessky on the existence of such classes.

We investigate the question which Q-valued characters and characters of Q-representations of finite groups are Z-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large.

In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a non-trivial intersection of the kernels of all but one irreducible characters or, equivalently, finite groups with an irreducible character vanishing on all but two conjugacy classes. We investigate such groups and in particular we characterize...

Let G be a finite group and ? be an irreducible character of G, the number cod(?) = jG : Let $ G $ be a finite group and $ \chi $ be an irreducible character of $ G $, the number $ \cod(\chi) = |G: \kernel(\chi)|/\chi(1) $ is called the codegree of $ \chi $. Also, $ \cod(G) = \{ \cod(\chi) \ | \ \chi \in \Irr(G) \} $. For $d\in\cod(G)$, the multiplicity of $d$ in $G$, denoted by $m'_G(d)$, is the number of irreducible characters of $G$ having codegree $d$. A finite group $G...