Finite Complex Reflection Groups Research Articles

Combinatorial simpliciality of arrangements of hyperplanes

We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite complex reflection groups except for the Shephard–Todd group \(G_{31}\).

UNIFORMIZATION SYSTEMS OF EQUATIONS WITH SINGULARITIES ALONG THE DISCRIMINANT SETS OF COMPLEX REFLECTION GROUPS OF RANK THREE

K. Saito [RIMS Kokyuroku Bessatsu 287 (1977), 117-137] introduced uniformization systems of equations with singularities along free divisors. The purpose of this paper is to construct such systems for the case of finite irreducible complex reflection groups of rank three. Each of them is essentially a system of linear differential equations of Fuchsian type in two variables with three linearly independent solutions. As a special case, uniformization systems of equations whose monodoromy groups are the finite irreducible complex reflection groups themselves are obtained in a closed form. This case has already been treated by Haraoka and the first author in [Funkcial. Ekvac. 53 (2010), 435-488] and our study is a generalization of their result.

Homology computations for complex braid groups

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.

Multivariate diagonal coinvariant spaces for complex reflection groups

For finite complex reflection groups, we consider the graded W-modules of diagonally harmonic polynomials in r sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of r.

Krammer representations for complex braid groups

Let B be the generalized braid group associated to some finite complex reflection group W. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of B. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjecture its faithfulness when W has a single class of reflections. If true, this conjecture would imply various group-theoretic properties for these groups, that we prove separately to be true for the other groups.

Reflection subgroups of finite complex reflection groups

All simple extensions of the reflection subgroups of a finite complex reflection group G are determined up to conjugacy. As a consequence, it is proved that if the rank of G is n and if G can be generated by n reflections, then for every set R of n reflections which generate G, every subset of R generates a parabolic subgroup of G.

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.

Automorphisms and generalized involution models of finite complex reflection groups

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G ( r , p , n ) with gcd ( p , n ) = 1 ; G ( r , p , 2 ) with r / p odd; or G 23 , the Coxeter group of type H 3 . We additionally provide explicit formulas for all automorphisms of G ( r , p , n ) , and construct new Gelfand models for the groups G ( r , p , n ) with gcd ( p , n ) = 1 .

Deformed diagonal harmonic polynomials for complex reflection groups

arXiv : http://arxiv.org/abs/1011.3654 We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version. Nous introduisons une déformation de l'espace des polynômes harmoniques (multi-diagonaux) pour tout groupe de réflexions complexes de la forme W=G(m,p,n), et soutenons l'hypothèse que cet espace est toujours isomorphe, en tant que W-module gradué, à l'espace d'origine.

Automorphisms of complex reflection groups

Let G ⊂ G L ( C r ) G\subset \mathrm {GL}(\mathbb {C}^r) be a finite complex reflection group. We show that when G G is irreducible, apart from the exception G = S 6 G=\mathfrak {S}_6 , as well as for a large class of non-irreducible groups, any automorphism of G G is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of N G L ( C r ) ( G ) N_{\mathrm {GL}(\mathbb {C}^r)}(G) and of a “Galois” automorphism: we show that G a l ( K / Q ) \mathrm {Gal}(K/\mathbb {Q}) , where K K is the field of definition of G G , injects into the group of outer automorphisms of G G , and that this injection can be chosen such that it induces the usual Galois action on characters of G G , apart from a few exceptional characters; further, replacing K K if needed by an extension of degree 2 2 , the injection can be lifted to A u t ( G ) \mathrm {Aut}(G) , and every irreducible representation admits a model which is equivariant with respect to this lifting. Along the way we show that the fundamental invariants of G G can be chosen rational.

Quasi-invariants of complex reflection groups

AbstractWe introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

Constructing representations of Hecke algebras for complex reflection groups

AbstractWe investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of aW-graph to the situation of complex reflection groups. We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras.

REFLECTION SUBGROUPS OF THE MONODROMY GROUPS OF APPELL’S F4

Assume the system of differential equations E4(a, b, c, c′; X, Y) satisfied by Appell’s hypergeometric function F4(a, b, c, c′; X, Y) has a finite irreducible monodromy group M4(a, b, c, c′). The monodromy matrix Γ3∗ derived from a loop Γ3 surrounding once the irreducible component C ={(X, Y)|(X - Y)2 -2(X + Y) +1 =0} of the singular locus of E4 is a complex reflection. The minimal normal subgroup NC of M4 containing Γ3∗ is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation 2E1(a, b, c). It is known that NC is reducible if e :=c +c′-a -b -1 ∉Z or if e ∈Z and P(G) is a dihedral group. We prove that, if e ∈Z,then NC is the (irreducible) Coxeter group W(D4), W(F4)and W(H4) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.

Branching properties for the groups [formula omitted

We study general properties of the restriction of the representations of the finite complex reflection groups G ( d e , e , r + 1 ) to their maximal parabolic subgroups of type G ( d e , e , r ) , and focus notably on the multiplicity of components. In combinatorial terms, this amounts to the following question: which symmetries arise or disappear when one changes (exactly) one pearl in a combinatorial necklace?

The minimal degree for a class of finite complex reflection groups

We calculate the minimal faithful permutation degree for a class of finite complex reflection groups G ( p , p , q ) , for p and q primes, and demonstrate that they form examples where the minimal degree of a direct product is strictly less than the sum of the minimal degrees of the direct factors.

Q,t-Fuß–Catalan numbers for finite reflection groups

In type A, the q,t-Fuß–Catalan numbers can be defined as the bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with nonnegative integer coefficients. We prove the conjectures for the dihedral groups and for the cyclic groups. Finally, we present several ideas on how the q,t-Fuß–Catalan numbers could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras and thereby generalize known connections.

Reflection groups acting on their hyperplanes

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W ⊂ GL ( V ) and its reflection arrangement A , we prove that the action of W on A is canonically related with other natural representations of W, through a ‘periodic’ family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply | A | ⩾ n ( n + 1 ) / 2 for n = dim V , and relate the W-equivariance of the corresponding map with the period of our family.

$q,t$-Fuß-Catalan numbers for complex reflection groups

In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress. Dans le cas du type $A$, les $q,t$-nombres de Fuß-Catalan $\mathrm{Cat}_n^{(m)}(q,t)$ peuvent être définis comme la série de Hilbert bigraduée d'un certain module associé au groupe symétrique $\mathcal{S}_n$. Nous généralisons cette construction aux groupes de réflexion complexes (finis) et nous formulons de jolies propriétés (conjecturales) algébriques et combinatoires de ces polynômes en $q$ et $t$. Enfin, nous décrivons une idée sur la manière dont ces polynômes pourraient être liés à certaines séries de Hilbert de modules apparaissant dans le contexte des algèbres de Cherednik rationnelles. Ceci est un travail en cours.

Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.

Splitting fields for extended complex reflection groups and Hecke algebras

Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.