Explanipedia

Hyperpolygonal arrangements Open

Lorenzo Giordani, Paul Mücksch, Gerhard Röhrle, Johannes Schmitt · 2025

In [5], a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call hyperpolygonal arrangements ${\mathscr H}_n$ . In this note, we stud…

G-complete reducibility and saturation Open

Michael Bate, Sören Böhm, Alastair Litterick, Benjamin Martin, Gerhard Röhrle · 2025

On good A1 subgroups, Springer maps, andovergroups of distinguished unipotent elements in reductive groups Open

Michael Bate, Sören Böhm, Benjamin Martin, Gerhard Röhrle · 2025

We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript, and to A. Thomas for providing the G2 example in Example 4.13. We thank the referee for a number of comments clarifying some poin…

Gary Seitz (1943–2023) : In memoriam Open

Martin W. Liebeck, Gerhard Röhrle, Donna Testerman · 2025

Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement Open

J. Matthew Douglass, Götz Pfeiffer, Gerhard Röhrle · 2025

Suppose V is a finite dimensional, complex vector space, $${\mathscr {A}}$$ is a finite set of codimension one subspaces of V , and G is a finite subgroup of the general linear group $$\operatorname {GL}(V)$$ that permutes the hype…

Arrangements, Matroids and Logarithmic Vector Fields Open

Takuro Abe, Graham Denham, Eva Maria Feichtner, Gerhard Röhrle · 2024

The focus of this workshop was on the ongoing interaction between geometric aspects of matroid theory with various directions in the study of hyperplane arrangements. A hyperplane arrangement is exactly a linear realization of a (loop-free…

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Open

Takuro Abe, Gerhard Röhrle, Christian Stump, Masahiko Yoshinaga · 2024

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…

The subgroup structure of pseudo-reductive groups Open

Michael Bate, Benjamin Martin, Gerhard Röhrle, Damian Sercombe · 2024

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 defin…

Inductive Freeness of Ziegler’s Canonical Multiderivations Open

Torsten Hoge, Gerhard Röhrle · 2024

Let $${{\mathscr {A}}}$$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $${{\mathscr {A}}}''$$ of $${{\mathscr {A}}}$$ to any hyperplane endowed with the natural multiplicity $$\kappa $$ is then…

Complete reducibility for Lie subalgebras and semisimplification Open

Michael Bate, Sören Böhm, Benjamin Martin, Gerhard Röhrle, Laura Voggesberger · 2023

Let G be a connected reductive linear algebraic group over a field k . Using ideas from geometric invariant theory, we study the notion of G -complete reducibility over k for a Lie subalgebra $${\mathfrak {h}}$$ of the Lie algebra $${\ma…

On Formality and Combinatorial Formality for Hyperplane Arrangements Open

Tilman Möller, Paul Mücksch, Gerhard Röhrle · 2023

A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the …

A construction of pseudo-reductive groups with non-reduced root system Open

Michael Bate, Gerhard Röhrle, Damian Sercombe, David I. Stewart · 2022

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type $BC_n$; these exist only in characteristic $2$. We also give a formula for the dimensions …

Overgroups of regular unipotent elements in reductive groups Open

Michael Bate, Benjamin Martin, Gerhard Röhrle · 2022

We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G . We show that under suitable hypotheses, such subgroups are G -irreducible in the sense…

Accurate Arrangements Open

Paul Mücksch, Gerhard Röhrle · 2020

Let $\\mathscr A$ be a Coxeter arrangement of rank $\\ell$. In 1987 Orlik,\nSolomon and Terao conjectured that for every $1\\leq d \\leq \\ell$, the first $d$\nexponents of $\\mathscr A$ -- when listed in increasing order -- are realized a…

RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS Open

Maike Gruchot, Alastair Litterick, Gerhard Röhrle · 2020

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ …

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Open

Takuro Abe, Gerhard Röhrle, Christian Stump, Masahiko Yoshinaga · 2018

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…

Nice Restrictions of Reflection Arrangements Open

Tilman Möller, Gerhard Röhrle · 2017

In a recent paper, Hoge and the second author classified all nice and all inductively factored reflection arrangements. In this note we extend this classification by determining all nice and all inductively factored restrictions of reflect…

Cocharacter-closure and the rational Hilbert–Mumford Theorem Open

Michael Bate, Sebastian Herpel, Benjamin Martin, Gerhard Röhrle · 2016

Cocharacter-closure and spherical buildings Open

Michael Bate, Sebastian Herpel, Benjamin Martin, Gerhard Röhrle · 2015

Peer reviewed

On a question of Külshammer for representations of finite groups in reductive groups Open

Michael Bate, Benjamin Martin, Gerhard Röhrle · 2015

Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Γ$ with Sylow $2$-subgroup $Γ_2$ and an infinite family of pairwise non-conjugate homomorphis…

On a question of K\\"ulshammer for representations of finite groups in\n reductive groups Open

Michael Bate, Benjamin Martin, Gerhard Röhrle · 2015

Let $G$ be a simple algebraic group of type $G_2$ over an algebraically\nclosed field of characteristic $2$. We give an example of a finite group\n$\\Gamma$ with Sylow $2$-subgroup $\\Gamma_2$ and an infinite family of pairwise\nnon-conjug…

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