MINIMAL EXCEPTIONAL -GROUPS

We investigate the question which \(\mathbb {Q}\)-valued characters and characters of \(\mathbb {Q}\)-representations of finite groups are \(\mathbb {Z}\)-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.

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.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

Weights for symmetric and general linear groups

A programme for the calculation of characters and representations of finite groups

In the present paper, we discuss a major project whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as such are no longer conclusive. We describe the basics of calculations with elements of exceptional groups in their minimal representations, which allow one to reduce calculations in the group itself to calculations in subgroups of smaller rank. For all practical purposes, such calculations are much more efficient than localization methods. Bibliography: 147 titles.

Editorial

Epsilon factors of representations of finite general linear groups

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

Regular orbits of cyclic subgroups in permutation representations of certain simple groups

Motivated by applications to equivariant neural networks and cryo-electron microscopy we consider the problem of recovering the generic orbit in a representation of a finite group from invariants of low degree. The main result proved here is that invariants of degree at most three separate generic orbits in the regular representation of a finite group defined over any infinite field. This answers a question posed in [4]. We also discuss this problem for subregular reprsentations of the dihedral and symmetric groups.

Finite Groups With Many Conjugate Elements

Minimal irreducible linear groups and representation of finite groups

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

The complete classes of orthogonal Latin squares can be constructed from Galois fields. The complete Y-matrix in (1) presents a new representation of finite group. A complete Y- matrix decides a finite group, and a finite group induces a complete-matrix. Based on this idea, we can construct Galois fields by constructing a pair of complete Y-matrixes, one is associated to additive group, and another is associated to multiplication group of the Galois field. The complete Y-matrix corresponding to multiplication group is constructed by some proper cyclic permutations since a cyclic group can be constructed a cyclic permutation. In this paper, we present a method to generate orthogonal Latin squares based on the construction of fields by complete-matrixes.