scholarly journals A Two-Sided Analogue of the Coxeter Complex

10.37236/8015 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Function ◽  
Joint Distribution ◽  
Coxeter Groups ◽  
Double Coset ◽  
Minimal Length ◽  
Coxeter System ◽  
Coxeter Complex ◽  
Eulerian Polynomial ◽  
Left And Right ◽  
Simplicial Poset

For any Coxeter system $(W,S)$ of rank $n$, we study an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. For finite $W$, this complex is first described in work of Hultman. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.

scholarly journals A two-sided analogue of the Coxeter complex

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Function ◽  
Joint Distribution ◽  
Coxeter Groups ◽  
Minimal Length ◽  
Coxeter System ◽  
Coxeter Complex ◽  
Eulerian Polynomial ◽  
International Audience ◽  
Left And Right ◽  
Simplicial Poset

International audience For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .

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 Automorphisms of Coxeter Groups and Lusztig’s Conjectures for Hecke Algebras with Unequal Parameters

2009 ◽  
Vol 195 ◽  
pp. 153-164
Author(s):  
Cédric Bonnafé
Keyword(s):  
Weight Function ◽  
Solvable Group ◽  
Coxeter Groups ◽  
Hecke Algebras ◽  
Finite Solvable Group ◽  
Coxeter System ◽  
Group Of Automorphisms

AbstractLet (W,S) be a Coxeter system, let G be a finite solvable group of automorphisms of (W, S) and let ϕ be a weight function which is invariant under G. Let ϕG denote the weight function on WG obtained by restriction from ϕ. The aim of this paper is to compare the a-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated with (W, ϕ) and to (WG,ϕG), provided that Lusztig’s Conjectures hold.

scholarly journals Minimal length elements of finite Coxeter groups

2012 ◽  
Vol 161 (15) ◽  
pp. 2945-2967 ◽  
Author(s):  
Xuhua He ◽  
Sian Nie
Keyword(s):  
Coxeter Groups ◽  
Minimal Length

scholarly journals Boolean complexes and boolean numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Coxeter Group ◽  
Bruhat Order ◽  
Graph Invariant ◽  
Coxeter System ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Order Ideals ◽  
Simplicial Poset ◽  
The Ideal

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

scholarly journals Minimal Overlapping Patterns in Colored Permutations

10.37236/2021 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Adrian Duane ◽  
Jeffrey Remmel
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Minimal Length ◽  
Special Cases ◽  
Maximum Packings ◽  
Colored Permutations

A pattern $P$ of length $j$ has the minimal overlapping property if two consecutive occurrences of the pattern can overlap in at most one place, namely, at the end of the first consecutive occurrence of the pattern and at the start of the second consecutive occurrence of the pattern. For patterns $P$ which have the minimal overlapping property, we derive a general formula for the generating function for the number of consecutive occurrences of $P$ in words, permutations and $k$-colored permutations in terms of the number of maximum packings of $P$ which are patterns of minimal length which has $n$ consecutive occurrences of the pattern $P$. Our results have as special cases several results which have appeared in the literature. Another consequence of our results is to prove a conjecture of Elizalde that two permutations $\alpha$ and $\beta$ of size $j$ which have the minimal overlapping property are strongly $c$-Wilf equivalent if $\alpha$ and $\beta$ have the same first and last elements.

scholarly journals Automorphisms of Right-Angled Coxeter Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-10
Author(s):  
Mauricio Gutierrez ◽  
Anton Kaul
Keyword(s):  
Automorphism Group ◽  
Semidirect Product ◽  
Coxeter Groups ◽  
Sufficient Conditions ◽  
Coxeter System ◽  
Group Extensions ◽  
Semidirect Products ◽  
Mild Conditions

If is a right-angled Coxeter system, then is a semidirect product of the group of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, is a semidirect product of by the quotient . We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence and consequently, all group extensions are trivial.

scholarly journals Counting Stirling permutations by number of pushes

2020 ◽  
Vol 29 (1) ◽  
pp. 17-27
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Joint Distribution ◽  
Inverse Matrix ◽  
Infinite Matrix ◽  
Do So

AbstractLet 𝒯(k)n denote the set of k-Stirling permutations having n distinct letters. Here, we consider the number of steps required (i.e., pushes) to rearrange the letters of a member of 𝒯(k)n so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯(k)n for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When k = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the LU-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.

scholarly journals Zero excess and minimal length in finite Coxeter groups

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Sarah B. Hart ◽  
Peter J. Rowley
Keyword(s):  
Coxeter Groups ◽  
Minimal Length

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