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Gunter Malle

Publications and source records attributed to Gunter Malle.

At least 55 records · Page 3Linked 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

The Projective Height Zero Conjecture

We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.

math.RT↗
arXiv2017

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\neq 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p'}(G)$ having odd order.

math.RT↗
arXiv2017

Cuspidal characters and automorphisms

We investigate the action of outer automorphisms of finite groups of Lie type on their irreducible characters. We obtain a definite result for cuspidal characters. As an application we verify the inductive McKay condition for some further infinite families of simple groups at certain primes.

math.RT↗
arXiv2017

Modular irreducibility of cuspidal unipotent characters

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\ell$-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied in \cite{GM96}.

math.RT↗
arXiv2017

Bounding Harish-Chandra series

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.

math.RT↗
arXiv2016

On the number of simple modules in a block of a finite group

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian defect groups and $\ell(B) = p^r - 1$. We conjecture that the inequality $\ell(B) \leq p^r$ holds for an arbitrary $p$-block with defect group of sectional rank $r$. We show this to hold for a large class of $p$-blocks of various families of quasi-simple and nearly simple groups.

math.RT↗
arXiv2016

On defects of characters and decomposition numbers

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

math.RT↗
arXiv2016

A Murnaghan--Nakayama rule for values of unipotent characters in classical groups

We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell\ge3$ the first Cartan invariant in the principal $\ell$-block is larger than~2 unless Sylow $\ell$-subgroups are cyclic.

math.RT↗
arXiv2016

A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there exists a cyclic $p'$-subgroup of $F^*(G)$ which does not normalise any non-trivial $p$-subgroup of $G$ that is generated by conjugates of~$A$.

math.GR↗
arXiv2016

Reductive groups and Steinberg maps

This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.

math.RT↗
arXiv2016

Defect zero characters predicted by local structure

Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.

math.RT↗
arXiv2015

Brauer's height zero conjecture for quasi-simple goups

We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.

math.RT↗
arXiv2015

On Blocks with One Modular Character

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $φ$. We prove that $φ$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms the basic set conjecture for these blocks.

math.RT↗
arXiv2015

Characters of odd degree

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime~$2$, hence for characters of odd degree.

math.RT↗