arXiv subjects

Götz Pfeiffer

Publications and source records attributed to Götz Pfeiffer.

11 recordsLinked to original sources

arXiv2026

Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products

We revisit the structure of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, and inspired by a recent construction of Serre for involution centralizers, we refine this understanding by interpreting $N_W(P)$ as a subdirect product via Goursat's Lemma. Central to our approach is a Galois connection on the lattice of parabolic subgroups, which leads to a new decomposition \begin{align*} N_W(P) \cong (P \times Q) \rtimes ((A \times B) \rtimes C)\text, \end{align*} where each subgroup reflects a structural feature of the ambient Coxeter system. This perspective yields a more symmetric description of $N_W(P)$, organized around naturally associated reflection subgroups on mutually orthogonal subspaces of the reflection representation of $W$. Our analysis provides new conceptual clarity and includes a case-by-case classification for all irreducible finite Coxeter groups.

math.GR↗

arXiv2025

Computing Young's Natural Representations for Generalized Symmetric Groups

We provide an algorithmic framework for the computation of explicit representing matrices for all irreducible representations of a generalized symmetric group $\Grin_n$, i.e., a wreath product of cyclic group of order $r$ with the symmetric group $\Symm_n$. The basic building block for this framework is the Specht matrix, a matrix with entries $0$ and $\pm1$, defined in terms of pairs of certain words. Combinatorial objects like Young diagrams and Young tableaus arise naturally from this setup. In the case $r = 1$, we recover Young's natural representations of the symmetric group. For general $r$, a suitable notion of pairs of $r$-words is used to extend the construction to generalized symmetric groups. Separately, for $r = 2$, where $\Grin_n$ is the Weyl group of type $B_n$, a different construction is based on a notion of pairs of biwords.

math.RT↗

arXiv2025

The BMM Symmetrising Trace Conjecture for Families of Complex Reflection Groups of Rank Two

The exceptional complex reflection groups of rank 2 are partitioned into three families. We construct explicit matrix models for the Hecke algebras associated to the maximal groups in the tetrahedral and octahedral family, and use them to verify the BMM symmetrising trace conjecture for all groups in these two families, providing evidence that a similar strategy might apply for the icosahedral family.

math.RT↗

arXiv2023

Centers of Hecke Algebras of Complex Reflection Groups

We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra $H$. These models enable concrete calculations inside $H$. For each of the eight groups, we compute an explicit integral basis of the center of $H$.

math.RT↗

arXiv2021

On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon's descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.

math.GR↗

arXiv2018

The Varchenko Determinant of a Coxeter Arrangement

The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.

math.CO↗

arXiv2018

On the Table of Marks of a Direct Product of Finite Groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group~$S_3$.

math.GR↗

arXiv2016

On the Complexity of Multiplication in the Iwahori--Hecke Algebra of the Symmetric Group

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.

math.GR↗

arXiv2013

On the Quiver Presentation of the Descent Algebra of the Symmetric Group

We describe a presentation for the descent algebra of the symmetric group $\sym{n}$ as a quiver with relations. This presentation arises from a new construction of the descent algebra as a homomorphic image of an algebra of forests of binary trees which can be identified with a subspace of the free Lie algebra. In this setting, we provide a new short proof of the known fact that the quiver of the descent algebra of $\sym{n}$ is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed $n\in\mathbb{N}$, the finite set of relations from these families that are relevant for the descent algebra of $\sym{n}$ generates the ideal of relations, and hence yields an explicit presentation by generators and relations of the algebra.

math.GR↗

arXiv2010

A Note on Element Centralizers in Finite Coxeter Groups

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.

math.GR↗

arXiv2008

Around Solomon's descent algebras

We study different problems related to the Solomon's descent algebra $Σ(W)$ of a finite Coxeter group $(W,S)$: positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if $W$ is irreducible and if the longest element is central, then the Loewy length of $Σ(W)$ is equal to $\lceil |S| / 2 \rceil$.

math.RT↗