Irreducible Group Research Articles

Zero fibers of quaternionic quotient singularities

We propose a generalization of Haiman’s conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group W acting on a quaternionic vector space V , by regarding V as a complex vector space, we consider the scheme-theoretic fiber over zero of the quotient map \pi\colon V \to V/W . For W an irreducible reflection group of (quaternionic) rank at least 6 , we show that the ring of functions on this fiber admits a (g+1)^{n} -dimensional quotient arising from an irreducible representation of a symplectic reflection algebra, where g=2N/n , with N the number of reflections in W and n=\mathrm{dim}_{\mathbf{H}}(V) , and we conjecture that this holds in general. We observe that in fact the degree of the zero fiber is precisely g+1 for the rank one groups (corresponding to the Kleinian singularities). In an appendix, we give a proof that three variants of the Coxeter number, including g , are integers.

Algebraic Enumeration of Polypolyhedra

Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order to enumerate the 3-dimensional polypolyhedra.

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

Automorphisms of Graph Products of Groups and Acylindrical Hyperbolicity

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if Γ \Gamma is a finite graph which contains at least two vertices and is not a join and if G \mathcal {G} is a collection of finitely generated irreducible groups, then either Γ G \Gamma \mathcal {G} is infinite dihedral or A u t ( Γ G ) \mathrm {Aut}(\Gamma \mathcal {G}) is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it. For instance, we show that the automorphism groups of most graph products verify vastness properties such as being SQ-universal; we show that many automorphism groups of graph products do not satisfy Kazhdan’s property (T); we solve the isomorphism problem between graph products in some cases; and we show that a graph product of coarse median groups, as defined by Bowditch, is coarse median itself.

Unisingular subgroups of symplectic groups over [formula omitted]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2n<250, specifically, the question remains open only 7 values of n.

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

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 of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions

Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.Comment: 21 pages, 1 figure. Final version typeset in the EPIGA style

A note on the inverse problem for finite differential Galois groups

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to H72, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of PSL3(ℂ) are the projective monodromy of a hypergeometric equation.

Factoring strongly irreducible group shift actions onto full shifts of lower entropy

We show that if G is a countable amenable group G with the comparison property and X is a strongly irreducible G -shift satisfying certain aperiodicity conditions, then X factors onto the full G -shift on N symbols, so long as the logarithm of N is less than the topological entropy of G .

Finite 3-orbit polyhedra in ordinary space, II

We enumerate the 188 3-orbit skeletal polyhedra in E3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}^3$$\\end{document} with irreducible symmetry group. The analysis is carried out by determining the polyhedra having each irreducible finite group of isometries as their symmetry group. Relevant information of every polyhedron is also organized in tables.

Root cycles in Coxeter groups

Abstract For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

On the Σ$\Sigma$‐invariants of Artin groups satisfying the K(π,1)$K(\pi,1)$‐conjecture

AbstractWe consider ‐invariants of Artin groups that satisfy the ‐conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC‐type, giving a sufficient condition for a character to belong to . We also prove some partial converses. As applications, we prove that the ‐conjecture holds true when there is a prime that divides for any edge with even label , we generalize to Artin groups the homological version of the Bestvina–Brady theorem and we compute the ‐invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the and properties of their derived subgroup.

On quotients of Coxeter groups

A group G is said to be just infinite if G itself is infinite but all proper quotients of G are finite. It was proven in [18] that irreducible affine Coxeter groups are just infinite. We show that the converse statement is also true, that is a Coxeter group is just infinite only if it is an irreducible affine one.Moreover, we show that just infinite Coxeter groups are profinitely rigid among all Coxeter groups.

IBIS soluble linear groups

Let G be a finite permutation group on An ordered sequence of elements of Ω is an irredundant base for G if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same cardinality, G is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.

Tower equivalence and Lusztig’s truncated Fourier transform

If f f denotes the truncated Lusztig Fourier transform, we show that the image by f f of the normalized characteristic function of a Coxeter element is the alternate sum of the exterior powers of the reflection representation, and that any class function is tower equivalent to its image by f f . In particular this gives a proof of the results of Chapuy and Douvropoulos on “Coxeter factorizations with generalized Jucys-Murphy weights and matrix tree theorems for reflection groups” for irreducible spetsial reflection groups, based on Deligne-Lusztig combinatorics.

On m-ovoids of finite classical polar spaces with an irreducible transitive automorphism group

On m-ovoids of finite classical polar spaces with an irreducible transitive automorphism group

On the centre of Iwahori-Hecke algebras

We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If W is any infinite irreducible Coxeter group and w∈W is a nontrivial element that is assumed not to be a translation in case W is affine, then there is an infinite sequence of conjugates of w by Coxeter generators whose length is non-decreasing and tends to infinity.

Counting isomorphism classes of groups of Fibonacci type with a prime power number of generators

Cavicchioli, O'Brien, and Spaggiari studied the number of isomorphism classes of irreducible groups of Fibonacci type as a function σ(n) of the number of generators n. In the case n=pl, where p is prime and l≥1, n≠2,4, they conjectured a function C(pl), that is polynomial in p, for the value of σ(pl). We prove that C(pl) is an upper bound for σ(pl). We introduce a function τ(n) for the number of abelianised groups and conjecture a function D(pl), that is polynomial in p, for the value of τ(pl), when pl≠2,4,5,7,8,13,23. We prove that D(pl) is an upper bound for τ(pl). We pose three questions that ask if particular pairs of groups with common abelianisations are non-isomorphic. We prove that if τ(pl)=D(pl) and each of these questions has a positive answer then σ(pl)=C(pl).

Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if Y subset mathbb {T}= mathbb {R}/mathbb {Z} is infinite then for each epsilon > 0 there exists an integer n such that nY is epsilon -dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on mathbb {T}^d also satisfy such a Glasner property where each infinite set (in fact, sufficiently large finite set) will have an epsilon -dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski connected Zariski closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the Cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.

A new approach to the generalized Springer correspondence

The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig’s generalized Springer correspondence.