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Gunter Malle

Publications and source records attributed to Gunter Malle.

At least 37 records · Page 2Linked to original sources
arXiv2024

Minimal heights and defect groups with two character degrees

Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.

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arXiv2023

Weight conjectures for $\ell$-compact groups and spetses

Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect the modular representation theory of finite groups of Lie type to the theory of $\ell$-compact groups. Under some mild conditions we prove that the fusion systems associated to homotopy fixed points of $\ell$-compact groups satisfy an equation which for finite groups of Lie type is equivalent to Alperin's Weight Conjecture. For finite reductive groups, Robinson's Ordinary Weight Conjecture is closely related to Lusztig's Jordan decomposition of characters and the corresponding results for Brauer $\ell$-blocks. Motivated by this, we define the principal block of a spets attached to a spetsial ${\mathbb Z}_\ell$-reflection group, using the fusion system related to it via $\ell$-compact groups, and formulate an analogue of Robinson's conjecture for this block. We prove this formulation for an infinite family of cases as well as for some groups of exceptional type. Our results not only provide further strong evidence for the validity of the weight conjectures, but also point toward some yet unknown structural explanation for them purely in the framework of fusion systems.

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arXiv2023

Weights for compact connected Lie groups

Let $\ell$ be a prime. If ${\mathbf G} $ is a compact connected Lie group, or a connected reductive algebraic group in characteristic different from $\ell$, and $\ell$ is a good prime for ${\mathbf G}$, we show that the number of weights of the $\ell$-fusion system of ${\mathbf G}$ is equal to the number of irreducible characters of its Weyl group. The proof relies on the classification of $\ell$-stubborn subgroups in compact Lie groups.

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arXiv2022

Height Zero Conjecture with Galois Automorphisms

We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô--Michler theorem for the prime~2, again with Galois automorphisms. We close, this time for odd primes $p$, with a new characterisation of $p$-closed groups via the decomposition numbers of certain characters.

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arXiv2022

Projective indecomposable permutation modules

We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.

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arXiv2022

Quasi-isolated blocks and Brauer's height zero conjecture

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups. In the supplement we treat a configuration for the $5$-blocks of the finite simple groups $E_8(q)$ that was inadvertently omitted in the first version and clarify and correct the meaning of some entries in Tables 3 and 4. The main theorems of the original version are not affected.

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arXiv2021

The principal block of a $\mathbb{Z}_\ell$-spets and Yokonuma type algebras

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact groups which may be of independent interest.

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arXiv2021

On almost $p$-rational characters of $p'$-degree

Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2πi/n})$ for some $n\in \mathbb{Z}^+$ prime to $p$ or precisely divisible by $p$. We prove that, in contrast to usual $p$-rational characters, there are always "many" almost $p$-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost $p$-rational irreducible characters of $G$ in terms of $p$. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for the number of such characters of $p'$-degree and prove that, in the minimal situation, the number of almost $p$-rational irreducible $p'$-characters of $G$ coincides with that of $N_G(P)$ for $P\in\mathrm{Syl}_p(G)$. Lastly, we propose a new way to detect the cyclicity of Sylow $p$-subgroups of a finite group $G$ from its character table, using almost $p$-rational irreducible $p'$-characters and the blockwise refinement of the McKay-Navarro conjecture.

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arXiv2020

On Willems' conjecture on Brauer character degrees

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.

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arXiv2020

The 2-parameter Green functions for 8-dimensional spin groups

The 2-parameter Green functions occur as a crucial ingredient in the character formula for Lusztig induction in finite reductive groups. Still, very little is known about these functions, in particular in the case of groups arsing from algebraic groups with disconnected centre. We collect some basic properties and then apply these, together with some explicit computations, to determine all 2-parameter Green functions for 8-dimensional spin groups in odd characteristic, whose centre is disconnected of order~4.

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arXiv2020

The inductive blockwise Alperin weight condition for type $\mathsf C$ and the prime $2$

We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$. As a main step, we derive a labelling set for the irreducible $2$-Brauer characters of the finite symplectic groups Sp$_{2n}(q)$ (with odd $q$), together with the action of automorphisms. As a further important ingredient we prove a Jordan decomposition for weights.

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arXiv2020

Overgroups of regular unipotent elements in simple algebraic groups

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

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arXiv2019

Low-dimensional representations of finite orthogonal groups

We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

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arXiv2018

On the number of characters in blocks of quasi-simple groups

We prove, for primes $p\ge5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero characters, and the number of conjugacy classes of a defect group and of its derived subgroup. For this, we determine these invariants explicitly, or at least give bounds for them for several classes of classical groups.

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arXiv2018

On a minimal counterexample to Brauer's $k(B)$-conjecture

We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect. For $p=3$ we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.

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