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Gunter Malle

Publications and source records attributed to Gunter Malle.

At least 73 records · Page 4Linked to original sources
arXiv2015

Simple endotrivial modules for linear, unitary and exceptional groups

Motivated by a recent result of Robinson showing that simple endotrivial modules essentially come from quasi-simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of Lie type. Our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods. As one application we prove that the $\ell$-rank of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As a second application we complete the proof that principal blocks of finite simple groups cannot have Loewy length 4, thus answering a question of Koshitani, Külshammer and Sambale. Our results also imply a vanishing result for irreducible characters of special linear and unitary groups.

math.RT↗
arXiv2014

On the number of $p'$-degree characters in a finite group

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for infinitely many groups $G$.

math.GR↗
arXiv2014

Decomposition matrices for exceptional groups at d=4

We determine the decomposition matrices of unipotent $\ell$-blocks of defect $Φ_4^2$ for exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first author, together with properties of generalized Gelfand-Graev characters which were recently shown to hold whenever the underlying characteristic is good.

math.RT↗
arXiv2014

Invariant blocks under coprime actions

If a finite group $A$ acts coprimely as automorphisms on a finite group $G$, then the $A$-invariant Brauer $p$-blocks of $G$ are exactly those that contain $A$-invariant irreducible characters.

math.RT↗
arXiv2014

Characters of positive height in blocks of finite quasi-simple groups

Eaton and Moretó proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~$p$. We provide some further evidence for blocks of groups of Lie type considered in cross characteristic.

math.RT↗
arXiv2014

A class group heuristic based on the distribution of 1-eigenspaces in matrix groups

We propose a modification to the Cohen--Lenstra prediction for the distribution of class groups of number fields, which should also apply when the base field contains non-trivial roots of unity. The underlying heuristic derives from the distribution of 1-eigenspaces in certain generalized symplectic groups over finite rings. The motivation for that heuristic comes from the function field case. We also give explicit formulas for the new predictions in several important cases. These are in close accordance with known data.

math.NT↗
arXiv2014

Decomposition matrices for low rank unitary groups

We study the decomposition matrices for the unipotent $\ell$-blocks of finite special unitary groups SU$_n(q)$ for unitary primes $\ell$ larger than $n$. Up to very few unknown entries, we give a complete solution for $n=2,\ldots,10$. We also prove a general result for two-column partitions when $\ell$ divides $q+1$. This is achieved using projective modules coming from the $\ell$-adic cohomology of Deligne--Lusztig varieties.

math.RT↗
arXiv2013

Variations on the Baer--Suzuki Theorem

The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $\langle x, x^g \rangle$ is a $p$-group for all $g \in G$, then the normal closure of $x$ in $G$ is a $p$-group. We consider the case where $x^g$ is replaced by $y^g$ for some other $p$-element $y$. While the analog of Baer--Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of $p$-elements.

math.GR↗
arXiv2013

Simple endotrivial modules for quasi-simple groups

We investigate simple endotrivial modules of finite quasi-simple groups and classify them in several important cases. This is motivated by a recent result of Robinson showing that simple endotrivial modules of most groups come from quasi-simple groups.

math.GR↗
arXiv2013

Products and commutators of classes in algebraic groups

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki theorem for finite groups. We also answer a question of Pavel Shumyatsky on commutators of pairs of conjugacy classes in simple algebraic groups. It turns out that the resulting examples are exactly those for which the product also consists of only finitely many classes.

math.GR↗
arXiv2012

Rational rigidity for E_8(p)

We prove the existence of certain rationally rigid triples E8 in good characteristic and thereby show that these groups over the prime field occur as Galois groups over the field of rational numbers. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. As a byproduct we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a classification of possible overgroups in exceptional groups containing regular unipotent elements.

math.GR↗
arXiv2012

Split Spetses for primitive reflection groups

Let $(V,W)$ be an exceptional spetsial irreducible reflection group $W$ on a complex vector space $V$, that is a group $G_n$ for $n \in \{4, 6, 8, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37\}$ in the Shephard-Todd notation. We describe how to determine some data associated to the corresponding (split) "spets", given complete knowledge of the same data for all proper subspetses (the method is thus inductive). The data determined here is the set Uch$(\mathbb G)$ of "unipotent characters" of $\mathbb G$ and the associated set of Frobenius eigenvalues, and its repartition into families. The determination of the Fourier matrices linking unipotent characters and "unipotent character sheaves" will be given in another paper. The approach works for all split reflection cosets for primitive irreducible reflection groups. The result is that all the above data exist and are unique (note that the cuspidal unipotent degrees are only determined up to sign).

math.RT↗
arXiv2012

Frobenius--Schur indicators of unipotent characters and the twisted involution module

Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius--Schur indicators for "unipotent characters" of twisted dihedral groups.

math.RT↗
arXiv2012

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5.

math.GR↗
arXiv2011

Products of conjugacy classes and fixed point spaces

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).

math.GR↗