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Thomas Breuer

Publications and source records attributed to Thomas Breuer.

At least 19 recordsLinked to original sources
arXiv2026

On the total character of a finite group

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$. A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ''contained'' in the permutation character $(1_H)^G$. In the first part of this paper, we investigate rich subgroups whose index is a product of two primes. We also consider rich subgroups of symmetric and alternating groups. In the second part we establish a formula for $T(G)$ in the case where the order of $G$ is a prime power. This result is analogous to a formula for the class number of $G$ proved by P. Hall, and it confirms a conjecture by Heffernan and MacHale from 2008. In the last part of the paper, we investigate finite groups $G$ where $T(G)$ is small, in a certain sense.

math.GR↗
arXiv2025

Counting Irreducible Representations of a Finite Abelian Group

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension $n$ over the field of order $q$, up to equivalence, using Brauer characters. We also provide a formula for such $n$ using the prime decomposition of the exponent of $G$ and an algorithm to compute the irreducible degrees and their multiplicities.

math.GR↗
arXiv2024

Verification of the conjugacy classes and ordinary character table of the Monster

As part of the programme to re-compute the character tables of all the groups in the Atlas we re-compute the character table of $\mathbb M$, the Monster simple group. We operate under the uniqueness hypotheses of $\mathbb M$ and the existence of an ordinary faithful representation of degree $196883 = 47.59.71$ and determine the conjugacy classes and centralizer orders of the elements of $\mathbb M$. Along the way we re-compute the character tables of centralizers of $p$-elements for $p < 11$ as well as fusions of conjugacy classes of these centralizers in $\mathbb M$.

math.GR↗
arXiv2024

Application-Driven Exascale: The JUPITER Benchmark Suite

Benchmarks are essential in the design of modern HPC installations, as they define key aspects of system components. Beyond synthetic workloads, it is crucial to include real applications that represent user requirements into benchmark suites, to guarantee high usability and widespread adoption of a new system. Given the significant investments in leadership-class supercomputers of the exascale era, this is even more important and necessitates alignment with a vision of Open Science and reproducibility. In this work, we present the JUPITER Benchmark Suite, which incorporates 16 applications from various domains. It was designed for and used in the procurement of JUPITER, the first European exascale supercomputer. We identify requirements and challenges and outline the project and software infrastructure setup. We provide descriptions and scalability studies of selected applications and a set of key takeaways. The JUPITER Benchmark Suite is released as open source software with this work at https://github.com/FZJ-JSC/jubench.

cs.DC↗
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

On Frobenius graphs of diameter 3 for finite groups

For a subgroup $H$ of a finite group $G$, the Frobenius graph $\Gamma(G, H)$ records the constituents of the restrictions to $H$ of the irreducible characters of $G$. We investigate when this graph has diameter 3.

math.GR↗
arXiv2023

Finite groups can be generated by a pi-subgroup and a pi'-subgroup

Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi'-subgroup. This gives a free product description of a countably generated free profinite group.

math.GR↗
arXiv2020

Subgroups of arbitrary even ordinary depth

We show that for each positive integer $n$, there are a group $G$ and a subgroup $H$ such that the ordinary depth is $d(H, G) = 2n$. This solves the open problem posed by Lars Kadison whether even ordinary depth larger than $6$ can occur.

math.GR↗
arXiv2019

The Loewy Structure of Certain Fixpoint Algebras, Part II

In Part I of this paper, we introduced a class of certain algebras of finite dimension over a field. All these algebras are split, symmetric and local. Here we continue to investigate their Loewy structure. We show that in many cases their Loewy length is equal to an upper bound established in Part I, but we also construct examples where we have a strict inequality.

math.RT↗
arXiv2019

Finite simple groups with short Galois orbits on conjugacy classes

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.

math.GR↗
arXiv2015

Almost Worst Case Distributions in Multiple Priors Models

A worst case distribution is a minimiser of the expectation of some random payoff within a family of plausible risk factor distributions. The plausibility of a risk factor distribution is quantified by a convex integral functional. This includes the special cases of relative entropy, Bregman distance, and $f$-divergence. An ($ε$-$γ$)-almost worst case distribution is a risk factor distribution which violates the plausibility constraint at most by the amount $γ$ and for which the expected payoff is not better than the worst case by more than $ε$. From a practical point of view the localisation of almost worst case distributions may be useful for efficient hedging against them. We prove that the densities of almost worst case distributions cluster in the Bregman neighbourhood of a specified function, interpreted as worst case localiser. In regular cases, it coincides with the worst case density, but when the latter does not exist, the worst case localiser is perhaps not even a density. We also discuss the calculation of the worst case localiser, and its dependence on the threshold in the plausibility constraint.

math.OC↗