Irreducible Complex Reflection Groups Research Articles

Catalan numbers for complex reflection groups

We construct $(q,t)$-Catalan polynomials and $q$-Fuss-Catalan polynomials for any irreducible complex reflection group $W$. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik algebras of $W$, and Opdam's analysis of permutations of the irreducible representations of $W$ arising from the Knizhnik-Zamolodchikov connection.

Gelfand models and Robinson–Schensted correspondence

In F. Caselli (Involutory reflection groups and their models, J. Algebra 24:370---393, 2010), a uniform Gelfand model is constructed for all nonexceptional irreducible complex reflection groups which are involutory. Such models can be naturally decomposed into the direct sum of submodules indexed by S n -conjugacy classes, and we present here a general result that relates the irreducible decomposition of these submodules with the projective Robinson---Schensted correspondence. This description also reflects, in a very explicit way, the existence of split representations for these groups.

Projective reflection groups

We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of all non-exceptional irreducible complex reflection groups find a natural description in this wider setting.

The center of pure complex braid groups

Broué, Malle and Rouquier (1998) conjectured in [2] that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center.

Refined Gelfand models for wreath products

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370–393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with the generalized Robinson–Schensted correspondence for these groups.

Involutory reflection groups and their models

A finite subgroup G of GL ( n , C ) is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements g ∈ G such that g g ¯ = 1 , where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.

Models and refined models for involutory reflection groups and classical Weyl groups

A finite subgroup $G$ of $GL(n,\mathbb{C})$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements $g \in G$ such that $g \bar{g}=1$, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups. If $G$ is a classical Weyl group this result is much refined in a way which is compatible with the Robinson-Schensted correspondence on involutions. Un sous-groupe fini $G$ de GL(n,ℂ) est dit involutoire si la somme des dimensions de ses représentations irréductibles complexes est donné par le nombre de involutions absolues dans le groupe, c'est-a-dire le nombre de éléments $g \in G$ tels que $g \bar{g}=1$, où le bar dénote la conjugaison complexe. Un modèle combinatoire uniforme est construit pour tous les groupes de réflexions complexes irréductibles qui sont involutoires, en comprenant, toutes les familles de groupes de Coxeter finis irréductibles. Si $G$ est un groupe de Weyl ce résultat peut se raffiner d'une manière compatible avec la correspondance de Robinson-Schensted sur les involutions.

The Lappo-Danilevskii method and trivial intersections of radicals in lower central series terms for certain fundamental groups

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.

Towards spetses I

We present a formalization, using data uniquely defined at the level of the Weyl group, of the construction and combinatorial properties of unipotent character sheaves and unipotent characters for reductive algebraic groups over an algebraic closure of a finite field. This formalization extends to the case where the Weyl group is replaced by a complex reflection group, and in many cases we get families of unipotent characters for a mysterious object, a kind of reductive algebraic group with a nonreal Weyl group, the “spets”. In this first part, we present the general results about complex reflection groups, their associated braid groups and Hecke algebras, which will be needed later on for properties of “spetses”. Not all irreducible complex reflection groups will give rise to a spets (the ones which do so are called “spetsial”), but all of them afford properties which already allow us to generalize many of the notions attached to the Weyl groups through the approach of “generic groups” (see [BMM1]).

Homotopy commutativity of finite loop spaces

Let p be a fixed prime. We say that X is a finite loop space if X has the homotopy type of a 1-connected p-localized finite CW-complex and X has a classifying space BX. As examples of finite loop spaces we can consider: (1) The p-localizations of the compact 1-connected Lie groups. (2) For each irreducible complex reflection group G of order prime to p, whose unitary representation restricts to the p-adics, we can construct a space X(G) where mod p cohomology is the ring of invariants of G, and it is therefore a polynomial algebra. Since by [-2] this space can be "pulled back" to a space of finite type, then (2(X(G)(p)) is a finite loop space. These irreducible types are listed in Clark-Ewing list. (3) In [9] Quillen showed that there is a space X such that H*(X,F~)~-Fp[x I . . . . . xn] where deg(xl)=2mi and m is any divisor of p 1 . These spaces are called "p-adic Grasmannians" and, also by [2], f2(X(p)) are finite loop spaces. (4) Zabrodsky constructed in [13] two more spaces with polynomial mod p cohomology which also produce finite loop spaces in the same way as above. (5) In [-1] Aguad6 proved that the types 29 and 34 of the Clark-Ewing list are realizables for p = 5 and p = 7 respectively, and consequently their rood p cohomology is the ring of invariants. Furthermore, for the type 29 and p--5 , Aguad6 has also proved that its m o d p cohomology is a polynomial algebra, hence its loop space is a finite loop space. Since for the type 34 and p = 7 the proof of the same is in progress, we also include its loop space as a finite loop space. (6) Any product of the spaces above is also a finite loop space.