Prof. Dr. Gerhard Röhrle

Publications on zbMATH.

Publications on MathSciNet.

Recent preprints on the ArXiv.

Latest preprints:

On connected subgraph arrangements
with L. Giordani, T. Möller, and P. Mücksch

Abstract:
Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph G, so called connected subgraph arrangements A_G. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if A_G is an aspherical connected subgraph arrangement, then A_G is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.
math.CO/2502.18144

Hyperpolygonal arrangements
with L. Giordani, P. Mücksch, and J. Schmitt

Abstract:
In 2024, Bellamy, Craw, Rayan, Schedler, and Weiss introduced a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties which we thus call hyperpolygonal arrangements H_n. In this note we study these arrangements and investigate their properties systematically. Remarkably the arrangements H_n discriminate between essentially all local properties of arrangements. In addition we show that hyperpolygonal arrangements are projectively unique and combinatorially formal.
We note that the arrangement H_5 is the famous counterexample of Edelman and Reiner from 1993 of Orlik's conjecture that the restriction of a free arrangement is again free.
math.CO/2502.02274

Free multiderivations of connected subgraph arrangements
with Paul Mücksch and Sven Wiesner

Abstract:
Cuntz and Kühne introduced the class of connected subgraph arrangements A_G, depending on a graph G, and classified all graphs G such that the corresponding arrangement A_G is free. We extend their result to the multiarrangement case and classify all graphs G for which the corresponding arrangement A_G supports some multiplicity m such that the multiarrangement (A_G,m) is free.
math.CO/2406.19866

The subgroup structure of pseudo-reductive groups
with Michael Bate, Ben Martin, and Damian Sercombe

Abstract:
Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.
math.GR/2406.11286

G-complete reducibility and saturation
Michael Bate, Sören Böhm, Alastair Litterick, and Benjamin Martin

Abstract:
Let H < G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p> 0. In our first principal theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V). Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V). We also study saturation of finite subgroups of Lie type in G. Here we generalise a result due to Nori from 1987 in case G = GL(V).
math.RT/2401.16927