Algorithms for Twisted Involutions in Weyl Groups

2012 ◽  
Vol 19 (02) ◽  
pp. 263-282 ◽  
Author(s):  
R. Haas ◽  
A. G. Helminck
Keyword(s):  
Root System ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Coxeter System ◽  
Special Cases

Let (W, Σ) be a finite Coxeter system, and θ an involution such that θ (Δ) = Δ, where Δ is a basis for the root system Φ associated with W, and [Formula: see text] the set of θ-twisted involutions in W. The elements of [Formula: see text] can be characterized by sequences in Σ which induce an ordering called the Richardson-Spinger Bruhat poset. The main algorithm of this paper computes this poset. Algorithms for finding conjugacy classes, the closure of an element and special cases are also given. A basic analysis of the complexity of the main algorithm and its variations is discussed, as well experience with implementation.

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.

Admissible Sequences for Twisted Involutions in Weyl Groups

2011 ◽  
Vol 54 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Ruth Haas ◽  
Aloysius G. Helminck
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Weyl Groups ◽  
One To One ◽  
Admissible Sequences

AbstractLetW be a Weyl group, Σ a set of simple reflections inW related to a basis Δ for the root system Φ associated with W and θ an involution such that θ(Δ) = Δ. We show that the set of θ- twisted involutions in W, = {w ∈ W | θ(w) = w–1} is in one to one correspondence with the set of regular involutions . The elements of are characterized by sequences in Σ which induce an ordering called the Richardson–Springer Poset. In particular, for Φ irreducible, the ascending Richardson–Springer Poset of , for nontrivial θ is identical to the descending Richardson–Springer Poset of .

scholarly journals Local finiteness of the twisted Bruhat orders on affine Weyl groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Weijia Wang
Keyword(s):  
Root System ◽  
Coxeter Group ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Oriented Matroid ◽  
Positive System ◽  
Locally Finite ◽  
Affine Weyl Groups ◽  
Affine Weyl Group ◽  
Weak Bruhat Order

AbstractIn this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra163 (1994), 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set 𝐵 in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed 𝐵-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

scholarly journals Nichols algebras over classical Weyl groups

2020 ◽  
pp. 2150187
Author(s):  
Zhengtang Tan ◽  
Weicai Wu ◽  
Shouchuan Zhang
Keyword(s):  
Conjugacy Classes ◽  
Weyl Groups ◽  
Infinite Dimensional ◽  
Type Formula ◽  
Nichols Algebras

We show that except in several cases conjugacy classes of classical Weyl groups [Formula: see text] and [Formula: see text] are of type [Formula: see text]. We prove that except in three cases Nichols algebras of irreducible Yetter–Drinfeld (YD) modules over the classical Weyl groups are infinite dimensional.

scholarly journals The distribution of descents and length in a Coxeter group

10.37236/1219 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Victor Reiner
Keyword(s):  
Rational Function ◽  
Coxeter Group ◽  
Generating Functions ◽  
Length Function ◽  
Weyl Groups ◽  
Coxeter System ◽  
Affine Weyl Groups ◽  
System L

We give a method for computing the $q$-Eulerian distribution $$ W(t,q)=\sum_{w \in W} t^{{\rm des}(w)} q^{l(w)} $$ as a rational function in $t$ and $q$, where $(W,S)$ is an arbitrary Coxeter system, $l(w)$ is the length function in $W$, and ${\rm des}(w)$ is the number of simple reflections $s \in S$ for which $l(ws) < l(w)$. Using this we compute generating functions encompassing the $q$-Eulerian distributions of the classical infinite families of finite and affine Weyl groups.

Second nilpotent BFC groups

1970 ◽  
Vol 11 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Iain. M. Bride
Keyword(s):  
Upper Bound ◽  
Nilpotent Groups ◽  
Conjugacy Classes ◽  
Unpublished Result ◽  
Special Cases ◽  
Image Position

The BFC number of a group G is defined to be the least upper bound n of the cardinals of the conjugacy classes of G, provided this is finite, and we then say that G is n-BFC. It was shown by B. H. Neumann [2] that the derived group G′ of such a group is finite, and J. Wiegold [5] proved that.This bound was sharpened by I. D. Macdonald [1] to, and P. M. Neumann has recently communicated the (unpublished) result that G′ ≦ nq(n) with q(n) a quadratic in log2w, an immense improvement on the above. J. A. H. Shepperd and J. Wiegold [4] improved the bound in two special cases, showing that if G is soluble, G′ ≦ np(n) with p(n) a quintic in Iog2n, and that if G is nilpotent of class 2, , It is conjectured that for any n-BFC group G, , Wiegold [5] having shown that this bound is attained by certain nilpotent groups of class 2.

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