arXiv subjects

Alexandre Zalesski

Publications and source records attributed to Alexandre Zalesski.

24 records · Page 2Linked to original sources

arXiv2018

Steinberg-like characters for finite simple groups

Let $G$ be a finite group and, for a prime $p$, let $S$ be a Sylow $p$-subgroup of $G$. A character $χ$ of $G$ is called $\Syl_p$-regular if the restriction of $χ$ to $S$ is the character of the regular representation of $S$. If, in addition, $χ$ vanishes at all elements of order divisible by $p$, $χ$ is said to be Steinberg-like. For every finite simple group $G$ we determine all primes $p$ for which $G$ admits a Steinberg-like character, except for alternating groups in characteristic~2. Moreover, we determine all primes for which $G$ has a projective $FG$-module of dimension $|S|$, where $F$ is an algebraically closed field of characteristic~$p$.

math.RT↗

arXiv2017

Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $μ$ a dominant weight and $W$ the Weyl group of ${\mathbf G}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $μ$ of ${\mathbf G}$ the sum $s_μ$ of distinct weights $w(μ)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit characters and every maximal torus $T$ of $G$, the multiplicity of the trivial character $1_T$ of $T$ in $s_μ$. The main case is where $μ=(q-1)ω$ and $ω$ is a fundamental weight of ${\mathbf G}$. Let $St$ denote the Steinberg character of $G$. Then we determine the unipotent characters occurring as constituents of $s_μ\cdot St$ defined to be 0 at the $p$-singular elements of $G$. Let $β_μ$ denote the Brauer character of a representation of $SL_{n}(q)$ arising from an irreducible representation of ${\mathbf G}$ with highest weight $μ$. Then we determine the unipotent constituents of the characters $β_μ\cdot St$ for $μ=(q-1)ω$, and also for some other $μ$ (called strongly $q$-restricted). In addition, for strongly restricted weights $μ$, we compute the \mult of $1_T$ in the restriction $β_μ|_T$ for every maximal torus $T$ of $G$.

math.GR↗

arXiv2017

On p-stability in groups and fusion systems

The aim of this paper is to generalise the notion of p-stability to fusion systems. We study the question how Qd(p) is involved in finite simple groups. We show that with a single exception a simple group involving Qd(p) has a subgroup isomorphic to either Qd(p) or a central extension of it by a cyclic group of order p. We define p-stability for fusion systems, characterise some of its properties and prove a fusion theoretic version of Thomson's maximal subgroup theorem. We introduce the notion of section p-stability both for groups and fusion systems and prove a version of Glauberman's theorem to fusion systems.

math.GR↗

arXiv2016

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $ϕ$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $ϕ(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim ϕ\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I.D. Suprunenko (Unipotent elements of non-prime order in representations of the classical algebraic groups: two big Jordan blocks, J. Math. Sci. 199(2014), 350 -- 374.

math.GR↗

arXiv2014

Subgroups of simple algebraic groups containing regular tori, and irreducible representations with multiplicity 1 non-zero weights

Our main goal is to determine, under certain restrictions, the maximal closed connected subgroups of simple algebraic groups containing a regular torus. We call a torus regular if its centralizer is abelian. We also obtain some results of independent interest. In particular, we determine the irreducible representations of simple algebraic groups whose non-zero weights occur with multiplicity 1.

math.GR↗

arXiv2013

A uniform upper bound for the character degree sums and Gelfand-Graev-like characters for finite simple groups

Let G be a finite non-abelian simple group and let p be a prime. We classify all pairs (G,p) such that the sum of the complex irreducible character degrees of G is greater than the index of a Sylow p-subgroup of G. Our classification includes all groups of Lie type in defining characteristic p (because every Gelfand-Graev character of G is multiplicity free and has degree equal to the above index), and a handful of well-described examples.

math.GR↗