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.

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 Excedances in classical and affine Weyl groups

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Generating Functions ◽  
Weyl Groups ◽  
Nous Obtenons ◽  
Affine Weyl Groups ◽  
Basic Properties ◽  
Affine Groups ◽  
International Audience ◽  
Affine Permutations

International audience Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We obtain some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type $\widetilde{B}_n, \widetilde {C}_n$ and $\widetilde {D}_n$ and determine the generating functions of their distributions. This paper is inspired by one of Clark and Ehrenborg (2011) in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups. Basée sur la notion des excédances colorés et absolu introduits par Bagno and Garber, nous donnons un analogue des polynômes des dérangements. Nous obtenons quelques propriétés de base de ces polynômes. En outre, nous définissons une excédance statistique pour le groupes de Weyl affines de type $\widetilde{B}_n, \widetilde {C}_n$ et $\widetilde {D}_n$ et nous déterminons les fonctions génératrices de leurs distributions. Cet article est inspirè d'un article de Clark et Ehrenborg (2011) dans lequel les auteurs introduisent les excedances pour le groupe des permutations affine et demander si cette statistique peut être éèfinie pour les autres groupes affines.

scholarly journals Affine Weyl Groups as Infinite Permutations

10.37236/1356 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Henrik Eriksson ◽  
Kimmo Eriksson
Keyword(s):  
Poincaré Series ◽  
Weak Order ◽  
Length Function ◽  
Weyl Groups ◽  
Unified Theory ◽  
Affine Weyl Groups ◽  
Poincare Series ◽  
Permutation Models ◽  
Refined Version

We present a unified theory for permutation models of all the infinite families of finite and affine Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincaré series of these affine Weyl groups.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

Tits Indices

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Coxeter Group ◽  
Coxeter System ◽  
Relative Rank ◽  
Coxeter Diagram ◽  
The Absolute ◽  
Relative Type

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

scholarly journals Bijective projections on parabolic quotients of affine Weyl groups

2014 ◽  
Vol 41 (4) ◽  
pp. 911-948 ◽  
Author(s):  
Elizabeth Beazley ◽  
Margaret Nichols ◽  
Min Hae Park ◽  
XiaoLin Shi ◽  
Alexander Youcis
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

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