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Burkhard Külshammer

Publications and source records attributed to Burkhard Külshammer.

15 recordsLinked to original sources
arXiv2026

On the total character of a finite group

The total character $τ_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := τ_G(1)$. A proper subgroup $H$ of $G$ is rich if $τ_G$ 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↗
arXiv2024

On Frobenius graphs of diameter 3 for finite groups

For a subgroup $H$ of a finite group $G$, the Frobenius graph $Γ(G, H)$ 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

On a Theorem of Knörr

Knörr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his construction to symmetric algebras.

math.GR↗
arXiv2022

Group algebras in which the socle of the center is an ideal

Let $F$ be a field of characteristic $p > 0$. We study the structure of the finite groups $G$ for which the socle of the center of $FG$ is an ideal in $FG$ and classify the finite $p$-groups $G$ with this property. Moreover, we give an explicit description of the finite groups $G$ for which the Reynolds ideal of $FG$ is an ideal in $FG$.

math.GR↗
arXiv2022

Ideals in the center of symmetric algebras

We study symmetric algebras $A$ over an algebraically closed field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.

math.RT↗
arXiv2022

Congruences for sums of powers of an integer

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where $m(q,e)$ is "large". We also pay special attention to the situation where $e$ is a prime power.

math.NT↗
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↗
arXiv2017

Loewy lengths of centers of blocks II

Let ZB be the center of a p-block B of a finite group with defect group D. We show that the Loewy length LL(ZB) of ZB is bounded by $\frac{|D|}{p}+p-1$ provided D is not cyclic. If D is non-abelian, we prove the stronger bound $LL(ZB)<\min\{p^{d-1},4p^{d-2}\}$ where $|D|=p^d$. Conversely, we classify the blocks B with $LL(ZB)\ge\min\{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.

math.RT↗
arXiv2016

Loewy lengths of centers of blocks

Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. In a recent paper, Otokita gave an upper bound for the Loewy length LL(ZB) of the center ZB of B in terms of a defect group D of B. We refine his methods in order to prove the optimal bound LL(ZB)\le LL(FD) whenever D is abelian. We also improve Otokita's bound for non-abelian defect groups. As an application we classify the blocks B such that LL(ZB)\ge |D|/2.

math.RT↗
arXiv2015

Anchors of irreducible characters

Given a prime number $p$, every irreducible character $χ$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $χ$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.

math.GR↗
arXiv2014

Blocks with transitive fusion systems

Suppose that all nontrivial subsections of a $p$-block $B$ are conjugate (where $p$ is a prime). By using the classification of the finite simple groups, we prove that the defect groups of $B$ are either extraspecial of order $p^3$ with $p \in \{3,5\}$ or elementary abelian.

math.RT↗
arXiv2013

Central idempotents of the bifree and left-free double Burnside ring

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring $B^Δ(G,G)$ and the left-free double Burnside ring $B^{\trl}(G,G)$, as well as the primitive central idempotents of the algebras arising from scalar extension to $\QQ$.

math.RT↗
arXiv2013

2-blocks with abelian defect groups

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form $C_{2^m} \times C_{2^m}$ for $m \geq 2$ has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan's conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.

math.GR↗
arXiv2006

Depth two, normality and a trace ideal condition for Frobenius extensions

We review the depth two and Hopf algebroid-Galois theory in math.RA/0108067 and specialize to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse we observe that normal Hopf subalgebras over a field are depth two extensions. We introduce a generalized Miyashita-Ulbrich action on the centralizer of a ring extension, and apply it to a study of depth two and separable extensions, providing new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.

math.GR↗