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A. E. Zalesski

Publications and source records attributed to A. E. Zalesski.

9 recordsLinked to original sources
arXiv2026

Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type

Let $r, p$ be primes and $k$ be a positive integer. We show that if $G$ is a subgroup of $SL_r(p^k)$ lying only in the non-geometric Aschbacher class $\mathscr{C}_9$, then $|G|$ is bounded above by a function of $r$ and $k$, independently of $p$. We also show that, up to conjugacy in $GL_r(p^k)$, the number of such $G$ is bounded above by a function of $r$ that is independent of $p$ and $k$. Apart from being of interest in their own right, these results have an application in a computational version of the strong approximation theorem for finitely generated Zariski-dense subgroups of $SL_r(\mathbb{P})$, where $\mathbb{P}$ is a number field.

math.GR↗
arXiv2024

Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy prime

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field $\mathbb{F}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro.

math.RT↗
arXiv2021

Hall-Higman type theorems for exceptional groups of Lie type, I

The paper studies the minimum polynomial degrees of $p$-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the order of the element.

math.RT↗
arXiv2016

Almost cyclic elements in Weil representations of finite classical groups

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix. (A square matrix $M$ is called almost cyclic if it is similar to a block-diagonal matrix with two blocks, such that one block is scalar and another block is a matrix whose minimum and characteristic polynomials coincide. Reflections and transvections are examples of almost cyclic matrices. The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture.

math.GR↗
arXiv2013

On characters of Chevalley groups vanishing at the non-semisimple elements

Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

math.GR↗
arXiv2012

On generators and representations of the sporadic simple groups

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable a in F, M is similar to diag(a Id_h, M_1), where M_1 is cyclic and 0 <= h <= n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.

math.RT↗
arXiv2007

Large dimensional classical groups and linear spaces

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least 25, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than 25. The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.

math.GR↗