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Gerhard Roehrle

Publications and source records attributed to Gerhard Roehrle.

At least 37 records · Page 2Linked to original sources
arXiv2021

Overgroups of regular unipotent elements in reductive groups

We study reductive subgroups $H$ of a reductive linear algebraic group $G$ -- possibly non-connected -- such that $H$ contains a regular unipotent element of $G$. We show that under suitable hypotheses, such subgroups are $G$-irreducible in the sense of Serre. This generalizes results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

math.GR↗
arXiv2021

Complete reducibility: Variations on a theme of Serre

In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--Röhrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $σ$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

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arXiv2021

Accurate Arrangements

Let $\mathscr A$ be a Coxeter arrangement of rank $\ell$. In 1987 Orlik, Solomon and Terao conjectured that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ -- when listed in increasing order -- are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free. We call a general free arrangements with this natural property involving their free restrictions accurate. In this paper we initialize their systematic study. Our principal result shows that MAT-free arrangements, a notion recently introduced by Cuntz and Mücksch, are accurate. This theorem in turn directly implies this special property for all ideal subarrangements of Weyl arrangements. In particular, this gives a new, simpler and uniform proof of the aforementioned conjecture of Orlik, Solomon and Terao for Weyl arrangements which is free of any case-by-case considerations. Another application of a slightly more general formulation of our main theorem shows that extended Shi arrangements, extended Catalan arrangements and ideal-Shi arrangements share this property as well. We also study arrangements that satisfy a slightly weaker condition, called almost accurate arrangements, where we simply disregard the ordering of the exponents involved. This property in turn is implied by many well established concepts of freeness such as supersolvability and divisional freeness.

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arXiv2020

Semisimplification for Subgroups of Reductive Algebraic Groups

Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H'$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serre's "$G$-analogue" of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.

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arXiv2020

Relative complete reducibility and normalised subgroups

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 $G$ which normalise the identity component $K^{\circ}$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ}$ is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as 'rational' versions over non-algebraically closed fields.

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arXiv2019

On the $K(π, 1)$-problem for restrictions of complex reflection arrangements

Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$ is a $K(π,1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr A(W)$, let $X(\mathscr A(W)^Y)$ be the complement in $Y$ of the hyperplanes in $\mathscr A(W)$ not containing $Y$. We hope that $X(\mathscr A(W)^Y)$ is always a $K(π,1)$. We prove it in case of the monomial groups $W = G(r,p,\ell)$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(π,1)$ property remains to be proved.

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arXiv2019

On relative complete reducibility

Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange.

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arXiv2019

Freeness of multi-reflection arrangements via primitive vector fields

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

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arXiv2019

Some remarks on free arrangements

We exhibit a particular free subarrangement of a certain restriction of the Weyl arrangement of type $E_7$ and use it to give an affirmative answer to a recent conjecture by T.~Abe on the nature of additionally free and stair-free arrangements.

math.CO↗
arXiv2019

Arrangements of ideal type are inductively free

Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type $\mathcal{A}_\mathcal{I}$ stemming from an ideal $\mathcal{I}$ in the set of positive roots of a reduced root system is free. Recently, Röhrle showed that a large class of the $\mathcal{A}_\mathcal{I}$ satisfy the stronger property of inductive freeness and conjectured that this property holds for all $\mathcal{A}_\mathcal{I}$. In this article, we confirm this conjecture.

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arXiv2018

The topology of arrangements of ideal type

In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne's seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a $K(π,1)$-arrangement. In this paper we study the $K(π,1)$-property for a certain class of subarrangements of Weyl arrangements, the so called arrangements of ideal type ${\mathscr A}_I$. These stem from ideals $I$ in the set of positive roots of a reduced root system. We show that the $K(π,1)$-property holds for all arrangements ${\mathscr A}_I$ if the underlying Weyl group is classical and that it extends to most of the ${\mathscr A}_I$ if the underlying Weyl group is of exceptional type. Conjecturally this holds for all ${\mathscr A}_I$. In general, the ${\mathscr A}_I$ are neither simplicial, nor is their complexification fiber type.

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arXiv2018

Restrictions of aspherical arrangements

In this note we present examples of $K(π,1)$-arrangements which admit a restriction which fails to be $K(π,1)$. This shows that asphericity is not hereditary among hyperplane arrangements.

math.AT↗
arXiv2018

On unipotent radicals of pseudo-reductive groups

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k'/k}(G') of a reductive k'-group G'. When G= \R_{k'/k}(G') we also provide some results on the orders of elements of the unipotent radical \RR_u(G_{\bar k}) of the extension of scalars of G to the algebraic closure \bar k of k.

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arXiv2018

Inductive Freeness of Ziegler's Canonical Multiderivations for Reflection Arrangements

Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A''$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let $A = A(W)$ be the reflection arrangement of a complex reflection group $W$. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction $A''$ of $A$ to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if $A''$ itself is inductively free.

math.CO↗
arXiv2018

The modality of a Borel subgroup in a simple algebraic group of type $E_8$

Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}\, k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B : \mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $\mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $\mathrm{mod}(B:\mathfrak u) = 20$, when $G$ is of type $\mathrm E_8$. This completes the determination of the values for $\mathrm{mod}(B:\mathfrak u)$ for $G$ of exceptional type.

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arXiv2017

Divisionally free Restrictions of Reflection Arrangements

We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao's conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products, surprisingly, it is not closed under taking localizations. In addition, we determine all divisionally free restrictions of all reflection arrangements.

math.CO↗
arXiv2017

Counting chambers in restricted Coxeter arrangements

Solomon showed that the Poincaré polynomial of a Coxeter group $W$ satisfies a product decomposition depending on the exponents of $W$. This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type $E_8$.

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