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Gunter Malle

Publications and source records attributed to Gunter Malle.

At least 19 recordsLinked to original sources
arXiv2026

Harish-Chandra theories, Ennola $d$-ality and Rouquier blocks for spetses

It has been shown that the theory of unipotent characters of finite reductive groups admits a generalisation to objects whose Weyl group is a spetsial complex reflection group, called spetses. In this paper we prove several natural properties satisfied by the unipotent characters of spetses, in particular the validity of all Harish-Chandra theories as well as the existence of Ennola $d$-alities for all integers $d$, Alvis--Curtis duality, and compatibility with Rouquier blocks of relative Hecke algebras.

math.RT↗
arXiv2026

The subnormaliser conjecture and unipotent characters

We prove instances of the subnormaliser conjecture on character bijections for finite groups by deriving generic versions of such bijections for unipotent characters of nearly simple groups of Lie type. For this, we formulate an extension of $d$-Harish-Chandra theory to what we call \emph{generic subnormalisers}, which are certain, usually disconnected, reductive subgroups of a simple algebraic group. For very good primes the generic bijections give rise to bijections to certain subgroups that should contain subnormalisers and thus be suitable for an eventual inductive approach. If the group in question has abelian Sylow $\ell$-subgroups for some prime~$\ell$ then our bijections satisfy the properties predicted by the subnormaliser conjecture, and moreover preserve character values up to sign, Brauer $\ell$-blocks, and are Galois equivariant.

math.RT↗
arXiv2026

Zeros of characters and orders of elements in finite groups

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes $p>5$ we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.

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arXiv2025

Subnormalisers of semisimple elements in finite groups of Lie type

We determine subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. For this, we determine the maximal overgroups of normalisers of Sylow tori. This is motivated by the recent character correspondence conjecture by Moret\'o and Rizo as well as by the question of existence of quasi-semiregular elements in finite permutation groups.

math.GR↗
arXiv2025

The Picky Conjecture for groups of Lie type

Recently, Moret\'o and Rizo proposed a conjecture, known as the Picky Conjecture, proposing new character correspondences extending the McKay Conjecture. We prove the Picky Conjecture for all quasi-simple groups of Lie type for non-defining primes. In favourable situations, we also obtain the stronger version postulating preservation of character values up to sign, and we show this stronger version holds in general when assuming certain natural properties of Lusztig's Jordan decomposition. Along the way, we complete the determination of semisimple picky elements in these groups by classifying picky 2- and 3-elements.

math.RT↗
arXiv2025

Partial character tables for $\mathbb{Z}_\ell$-spetses

Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$. Secondly, when $\ell > 2$ is a very good prime for ${\mathbb{G}}$, the Weyl group $W$ of ${\mathbb{G}}$ has order coprime with $\ell$, and $q\equiv1\pmod\ell$ we introduce a formula for the values of characters in the principal block of ${\mathbb{G}}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality. In both cases we formulate and provide evidence for several conjectures concerning the proposed values.

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arXiv2025

Intersections of blocks of cyclotomic Hecke algebras

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from $E_8$. We also propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.

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arXiv2025

Generic weights for finite reductive groups

This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of $e$-cuspidality in the $e$-Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.

math.RT↗
arXiv2025

Picky elements, subnormalisers, and character correspondences

We gather evidence on a new local-global conjecture of Moret\'o and Rizo on values of irreducible characters of finite groups. For this we study subnormalisers and picky elements in finite groups of Lie type and determine them in many cases, for unipotent elements as well as for semisimple elements of prime power order. We also discuss subnormalisers of unipotent and semisimple elements in connected as well as in disconnected reductive linear algebraic groups.

math.GR↗
arXiv2024

The continuity of $p$-rationality of characters and the principal block

We study rationality properties of irreducible characters of finite groups. We show that the continuity of $2$-rationality is a phenomenon that can be detected in the principal $2$-block, thus refining a recent result of N. N. Hung. We also propose a conjectural group theoretical criterion for the continuity gap at level $1$ for all primes

math.RT↗
arXiv2024

On minimal positive heights for blocks of almost quasi-simple groups

The Eaton--Moretó conjecture extends the recently-proven Brauer height zero conjecture to blocks with non-abelian defect group, positing equality between the minimal positive heights of a block of a finite group and its defect group. Here we provide further evidence for the inequality in this conjecture that is not implied by Dade's conjecture. Specifically, we consider minimal counter-examples and show that these cannot be found among almost quasi-simple groups for $p\ge5$. Along the way, we observe that most such blocks have minimal positive height equal to~1.

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arXiv2024

Zeros of $S$-characters

The concept of $S$-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. We present examples that this does not generalise to $S$-characters, thereby answering a question posed by J-P. Serre.

math.GR↗
arXiv2024

Brauer's Height Zero Conjecture

We complete the proof of Brauer's Height Zero Conjecture from 1955 by establishing the open implication for all odd primes.

math.RT↗
arXiv2024

Computational data on ${\mathfrak S}_n$-extensions of $\mathbb Q$

We discuss computational results on field extensions $K/{\mathbb Q}$ of degree $n\le11$ with Galois group of the Galois closure isomorphic to the full symmetric group ${\mathfrak S}_n$. More precisely, we present statistics on the number of such extensions as a function of the field discriminant and compare them to the known predictions by Bhargava and the author. We also investigate the numbers of fields with equal discriminant and tabulate class numbers and class groups to compare them against Cohen--Lenstra--Martinet type of heuristics and their proposed improvements.

math.NT↗
arXiv2024

A Brauer--Galois height zero conjecture

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the absolute Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.

math.RT↗
arXiv2024

Rationality of extended unipotent characters

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof we use realisations of characters in $\ell$-adic cohomology groups of Deligne--Lusztig varieties as well as block theoretic considerations.

math.RT↗
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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