Finite groups generated in low real codimension

Abstract

We study the intersection lattice of the arrangement A G of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).If G is generated in codimension two, we show that the intersection lattice of A G is atomic. We prove that the alternating subgroup Alt(W ) of a reflection group W is strictly generated in codimension two; moreover, the subspace arrangement of Alt(W ) is the truncation at rank two of the reflection arrangement A W .Further, we compute the intersection lattice of all finite subgroups of GL 3 (R), and moreover, we emphasize the groups that are "minimally generated in real codimension two", i.e, groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension.

A rotation group whose subspace arrangement is not from a real reflection group

“…The author would like to express gratitude to Victor Reiner who directed him to Martino and Singh's paper [MS19], and to Ivan Martino for the fruitful discussion that led to this note. Additionally, he would like to acknowledge Yves de Cornulier who in [Cor14] gave the above group G (in the special case that m is prime) as an example of a finite subgroup of O n pRq that is not contained in a reflection group.…”

“…By and large we follow the notation of [MS19]. Fix a vector space V and a finite subgroup G of the general linear group GLpVq on V. For any subgroup H Ă G, V H denotes the subspace of fixed points of H: V H " tv P V : hv " v, @h P Hu.…”

“…We mention that what we here call bireflection groups, following [Kem99], [LP01], [GK03], [DEK09], are referred to by Martino and Singh in [MS19] as groups generated in codimension two. If a group is generated by bireflections and contains no reflections, Martino and Singh say it is strictly generated in condimension two.…”

“…The terminology of reflection-rotation groups, i.e. bireflection groups over R, is used by [LM16], [Lan16], and [MS19] as well.…”

“…Thus bireflection groups have emerged as a natural generalization of reflection groups and a natural object of study in their own right. It is in this context that Martino and Singh [MS19] studied the subspace arrangements of such groups. For a finite subgroup G Ă GLpVq, the subspace arrangement A G of G is the arrangement of subspaces (in V) of fixed points of nontrivial isotropy subgroups of G. For finite (real or complex) reflection groups W, the subspace arrangement A W is just the set of reflecting hyperplanes and their intersections.…”