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Register a new userRepresentations of the general linear groups which are irreducible over subgroups
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We classify all triples $(G,V,H)$ such that $SL_n(q)leq Gleq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $Vdar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
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In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called {em PSH-algebras}. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bi-algebra that contains a large PSH-algebra that extends Zelevinskys algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.
Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of this largest degree for finite groups of Lie type. Moreover, we show that for groups of Lie type, the Steinberg character has largest degree among all unipotent characters.
Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.
We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $mathrm{F}_4$, where we observe that the parametrization is uniform over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
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