scholarly journals Representations of Weyl groups of type B induced from centralisers of involutions

1991 ◽  
Vol 44 (2) ◽  
pp. 337-344 ◽  
Author(s):  
Philip D. Ryan
Keyword(s):  
Weyl Group ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Type B ◽  
Complex Character ◽  
Linear Character ◽  
Complex Linear

Let G be a Weyl group of type B, and T a set of representatives of the conjugacy classes of self-inverse elements of G. For each t in T, we construct a (complex) linear character πt of the centraliser of t in G, such that the sum of the characters of G induced from the πt contains each irreducible complex character of G with multiplicity precisely 1. For Weyl groups of type A (that is, for the symmetric groups), a similar result was published recently by Inglis, Richardson and Saxl.

Fischer Matrices for Projective Representations of Generalized Symmetric Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 449-462
Author(s):  
Mohammed S. Almestady ◽  
Alun O. Morris
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Symmetric Groups ◽  
Type B ◽  
Projective Representations ◽  
Ordinary Case ◽  
Covering Groups

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

On the restriction of cuspidal representations to unipotent elements

2002 ◽  
Vol 132 (1) ◽  
pp. 35-56 ◽  
Author(s):  
DIPENDRA PRASAD ◽  
NILABH SANAT
Keyword(s):  
Conjugacy Class ◽  
Maximal Torus ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Cuspidal Representation ◽  
Linear Character ◽  
Maximal Tori ◽  
Complex Linear ◽  
Cuspidal Representations

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

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 Gamma Positivity of the Descent Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

10.37236/9037 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Hiranya Kishore Dey ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Alternating Group ◽  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Type D ◽  
Positive Elements

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$. We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively.  We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$.  We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

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.

INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

scholarly journals On flag varieties, hyperplane complements and Springer representations of Weyl groups

1990 ◽  
Vol 49 (3) ◽  
pp. 449-485 ◽  
Author(s):  
G. I. Lehrer ◽  
T. Shoji
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Nilpotent Element ◽  
Parabolic Type ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Flag Varieties ◽  
Group Representations ◽  
Definition Of

AbstractLet G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

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