Representations of complex imprimitive reflection groups

1983 ◽  
Vol 94 (3) ◽  
pp. 425-436 ◽  
Author(s):  
M. C. Hughes
Keyword(s):  
Weyl Groups ◽  
Unified Theory ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Common Structure ◽  
Principal Character ◽  
Irreducible Components

In his attempt to give the irreducible characters of the Weyl groups of type A, B and D, Mayer [3] found a unified theory using their common structure as reflection groups. In this paper, we give the characters of the complex imprimitive reflection groups. We are able to give an algorithm which allows us to calculate the irreducible components of the principal character of a ‘Weyl’ subgroup induced up to the whole group. This is a generalization of the algorithm given by Mayer.

On the characters of the Weyl group of type D

1975 ◽  
Vol 77 (2) ◽  
pp. 259-264 ◽  
Author(s):  
S. J. Mayer
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Unified Theory ◽  
Reflection Groups ◽  
Type D ◽  
Common Structure ◽  
Principal Character ◽  
Similar Development

This paper is a continuation of (2), (3) in the development of a unified theory of the characters of the Weyl groups of the simple Lie algebras using their common structure as reflection groups; compare Carter (1) for a similar development for the conjugacy classes. We look at the Weyl group of type D, which is a subgroup of index two in the Weyl group of type C. It was first studied by Young (4), but rather less is known about the characters of this group than those of types A and C. Indeed, the situation is rather more complicated, but we are able to give, as before, an algorithm to determine irreducible constituents of the principal character of a Weyl subgroup induced up to the whole group. We shall also study the case where the rank of the Weyl group is even, when extra irreducible characters may arise, and after constructing these, we shall state some results on their occurrence in the induced principal character.

scholarly journals Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

scholarly journals Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

2010 ◽  
Vol 197 ◽  
pp. 175-212
Author(s):  
Maria Chlouveraki
Keyword(s):  
Infinite Series ◽  
Hecke Algebras ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

scholarly journals Specht modules for finite reflection groups

1995 ◽  
Vol 37 (3) ◽  
pp. 279-287 ◽  
Author(s):  
S. HalicioǦlu
Keyword(s):  
Present Author ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Young Tableaux ◽  
Reflection Groups ◽  
Specht Modules ◽  
Irreducible Modules ◽  
Original Approach

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

scholarly journals Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

2010 ◽  
Vol 197 ◽  
pp. 175-212 ◽  
Author(s):  
Maria Chlouveraki
Keyword(s):  
Infinite Series ◽  
Hecke Algebras ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series G(de, e, r), thus completing their calculation for all complex reflection groups.

scholarly journals SCHUR ELEMENTS AND BASIC SETS FOR CYCLOTOMIC HECKE ALGEBRAS

2011 ◽  
Vol 10 (05) ◽  
pp. 979-993 ◽  
Author(s):  
MARIA CHLOUVERAKI ◽  
NICOLAS JACON
Keyword(s):  
Hecke Algebras ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Schur Elements ◽  
Basic Sets ◽  
Cyclotomic Hecke Algebras

We study the Schur elements and the a-function for cyclotomic Hecke algebras. As a consequence, we show the existence of canonical basic sets, as defined by Geck–Rouquier, for certain complex reflection groups. This includes the case of finite Weyl groups for all choices of parameters (in characteristic 0).

Jacquet Modules of Parabolically Induced Representations and Weyl Groups

2001 ◽  
Vol 53 (4) ◽  
pp. 675-695 ◽  
Author(s):  
Dubravka Ban
Keyword(s):  
Weyl Groups ◽  
Supercuspidal Representation ◽  
Induced Representations ◽  
Irreducible Components

AbstractThe representation parabolically induced from an irreducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations.

scholarly journals SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

2018 ◽  
Vol 15 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Abeer Al-Siyabi
Keyword(s):  
Dynkin Diagram ◽  
Voronoi Cell ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Unified Theory ◽  
Grand Unified Theory ◽  
Root Lattice ◽  
Gauge Bosons ◽  
Aperiodic Tiling ◽  
Coxeter Plane

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to [Formula: see text] and even to [Formula: see text] by noting the Coxeter–Dynkin diagram embedding [Formula: see text]. Another embedding can be made through the relation [Formula: see text] for more popular [Formula: see text]. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups [Formula: see text] which can be obtained directly from [Formula: see text] by projection. This leads to relations of the [Formula: see text] polytopes with the quasicrystallography in 4D and [Formula: see text] polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group [Formula: see text].

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