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Bernhard Mühlherr

Publications and source records attributed to Bernhard Mühlherr.

15 recordsLinked to original sources
arXiv2025

Presentation and uniqueness of Kac-Moody groups over local rings

To any generalised Cartan matrix (GCM) $A$ and any ring $R$, Tits associated a Kac-Moody group $\mathfrak{G}_A(R)$ defined by a presentation à la Steinberg. For a domain $R$ with field of fractions $\mathbb{K}$, we explore the question of whether the canonical map $φ_R\colon\thinspace \mathfrak{G}_A(R)\to \mathfrak{G}_A(\mathbb{K})$ is injective. This question for Cartan matrices has a long history, and for GCMs was already present in Tits' foundational papers on Kac-Moody groups. We prove that for any $2$-spherical GCM $A$, the map $φ_R$ is injective for all valuation rings $R$ (under an additional minor condition (co)). To the best of our knowledge, this is the first such injectivity result beyond the classical setting.

math.GR↗
arXiv2024

A classification of generalized root systems

Dimitrov and Fioresi introduced an object that they call a generalized root system. This is a finite set of vectors in a euclidean space satisfying certain compatibilities between angles and sums and differences of elements. They conjecture that every generalized root system is equivalent to one associated to a restriction of a Weyl arrangement. In this note we prove the conjecture and provide a complete classification of generalized root systems up to equivalence.

math.CO↗
arXiv2023

Isometries of wall-connected twin buildings

We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affne twin buildings of rank at least 3.

math.GR↗
arXiv2023

On isometries of twin buildings

A twin building consists of two buildings that are twinned by a codistance function. We prove that the local structure of a twin building uniquely determines the two buildings up to isomorphism. This has been known for twin buildings satisfying a technical condition (co).

math.GR↗
arXiv2021

Veldkamp quadrangles and polar spaces

Veldkamp polygons are certain graphs $Γ=(V,E)$ such that for each $v\in V$, $Γ_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$. These relations are all trivial if and only if $Γ$ is a thick generalized polygon. A Veldkamp polygon is called flat if no two vertices have the same set of vertices that are opposite in a natural sense. We explore the connection between Veldkamp quadrangles and polar spaces. Using this connection, we give the complete classification of flat Veldkamp quadrangles in which some but not all of the relations $\equiv_v$ are trivial.

math.CO↗
arXiv2018

The cone topology on masures

Masures are generalizations of Bruhat--Tits buildings and the main examples are associated with almost split Kac--Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure $Δ$ as well as on the building at infinity of $Δ$, which is the twin building associated with G. The aim of this article is twofold: firstly, to introduce and study the cone topology on the twin building at infinity of a masure. It turns out that this topology has various favorable properties that are required in the literature as axioms for a topological twin building. Secondly, by making use of the cone topology, we study strongly transitive actions of a group G on a masure $Δ$. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on $Δ$ if and only if it acts strongly transitively on the twin building at infinity $\partial$$Δ$. Along the way a criterion for strong transitivity is given and the existence and good dynamical properties of strongly regular hyperbolic automorphisms of the masure are proven.

math.GR↗
arXiv2018

On the Tits cone of a Weyl groupoid

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.

math.CO↗
arXiv2016

Intrinsic reflections in Coxeter systems

Let $(W,S)$ be a Coxeter system and let $s \in S$. We call $s$ a right-angled generator of $(W,S)$ if $st = ts$ or $st$ has infinite order for each $t \in S$. We call $s$ an intrinsic reflection of $W$ if $s \in R^W$ for all Coxeter generating sets $R$ of $W$. We give necessary and sufficient conditions for a right-angled generator $s \in S$ of $(W,S)$ to be an intrinsic reflection of $W$.

math.GR↗
arXiv2015

Simplicial arrangements on convex cones

We introduce the notion of a Tits arrangement on a convex open cone as a special case of (infinite) simplicial arrangements. Such an object carries a simplicial structure similar to the geometric representation of Coxeter groups. The standard constructions of subarrangements and restrictions, which are known in the case of finite hyperplane arrangements, work as well in this more general setting.

math.CO↗
arXiv2012

Descent of affine buildings - I. Large minimal angles

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits. This first part lays the foundations for our approach and deals with the `large minimal angle' case.

math.MG↗
arXiv2011

Codistances of 3-spherical buildings

We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.

math.GR↗
arXiv2008

Angle-deformations in Coxeter groups

The isomorphism problem for Coxeter groups has been reduced to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system (W,R) all Coxeter generating sets S of W which are contained in R^W, the set of reflections of (W,R). In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets S in R^W which are sharp-angled with respect to R.

math.GR↗
arXiv2005

The isomorphism problem for Coxeter groups

By a recent result obtained by R. Howlett and the author considerable progress has been made towards a complete solution of the isomorphism problem for Coxeter groups. In this paper we give a survey on the isomorphism problem and explain in particular how the result mentioned above reduces it to its `reflection preserving' version. Furthermore we desrcibe recent developments concerning the solution of the latter.

math.GR↗