Depth in Coxeter groups of type $B$

International audience The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length. La statistique profondeur a été introduite par Petersen et Tenner pour tout groupe de Coxeter $W$. Elle est définie pour tout $w \in W$ à partir de ses factorisations en produit de réflexions (non nécessairement simples). Pour le type $B$, nous introduisons un algorithme calculant la profondeur, et donnant une formule explicite pour cette statistique. On utilise par ailleurs cet algorithme pour caractériser tous les éléments ayant une profondeur égale à leur longueur. Ces derniers s’avèrent être les éléments pleinement commutatifs “hauts-et-bas” introduits par Stembridge. Nous donnons enfin une caractérisation des éléments dont la longueur absolue, la profondeur et la longueur coïncident.

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Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2484 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Soojin Cho ◽

Kyoungsuk Park

Keyword(s):

Bruhat Order ◽

Type B ◽

Coxeter System ◽

Covering Relation ◽

Signed Permutations ◽

Weak Bruhat Order ◽

International Audience

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.

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Les polynômes eul\IeC èriens stables de type B

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3040 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Mirkó Visontai ◽

Nathan Williams

Keyword(s):

Linear Operator ◽

Type B ◽

Multivariate Polynomial ◽

Eulerian Polynomial ◽

Eulerian Polynomials ◽

Signed Permutations ◽

International Audience ◽

Multiple Variables ◽

Real Stability

International audience We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Nous prèsentons un raffinement multivariè d'un polynôme eulèrien de type B dèfini par Brenti. En prouvant que ce polynôme est stable nous gènèralisons un rèsultat de Brenti selon laquel chaque racine du polynôme eulèrien de type B est rèelle. Notre preuve combine un raffinement de la statistique des descentes pour les permutations signèes avec la stabilitè—une gènèralisation de la propriètè d'avoir uniquement des racines rèelles aux polynômes en plusieurs variables. La connexion est que nos polynômes eulèriens raffinès satisfont une rècurrence donnèe par un opèrateur linèaire qui prèserve la stabilitè.

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Asymptotics of a Locally Dependent Statistic on Finite Reflection Groups

The Electronic Journal of Combinatorics ◽

10.37236/9454 ◽

2020 ◽

Vol 27 (2) ◽

Author(s):

Frank Röttger

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Coxeter Groups ◽

Random Permutation ◽

Wasserstein Distance ◽

Reflection Groups ◽

Signed Permutation ◽

Signed Permutations ◽

Dependency Structures

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was listed as an interesting direction by Chatterjee and Diaconis (2017). For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme to signed permutations, i.e. elements of Coxeter groups of type $ \mathtt{B}_n $, which are also known as the hyperoctahedral groups. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $\mathtt{D}_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cramér-Wold device.

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The facial weak order in finite Coxeter groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6399 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Aram Dermenjian ◽

Christophe Hohlweg ◽

Vincent Pilaud

Keyword(s):

Coxeter Group ◽

Coxeter Groups ◽

Geometric Interpretation ◽

Weak Order ◽

Maximal Length ◽

Lattice Congruence ◽

Finite Coxeter Group ◽

International Audience

International audience We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.

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Shortest path poset of finite Coxeter groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2721 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Saúl A. Blanco

Keyword(s):

Shortest Path ◽

Coxeter Group ◽

Coxeter Groups ◽

Shortest Paths ◽

Combinatorial Interpretation ◽

Nous Permet ◽

Finite Coxeter Group ◽

International Audience ◽

Absolute Length ◽

Bruhat Graph

International audience We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$. Nous définissons un poset en utilisant le plus court chemin entre l'identité et le plus long mot de $W, w_0$, dans le graph de Bruhat du groupe finie Coxeter, $W$. Nous prouvons que ce poset est l'union de posets Boolean du même rang que la longueur absolute de $w_0$; ça signifie que tous les plus courts chemins, étiquetés par réflexions $t_1,\ldots, t_m$ sont totalement commutatives. Ça nous permet de donner une interprétation combinatoire aux termes avec le moindre grade dans le $\textbf{cd}$-index complet de $W$.

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The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2510 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Arthur L.B. Yang ◽

Philip B. Zhang

Keyword(s):

Linear Operators ◽

Type B ◽

The Real ◽

Eulerian Polynomials ◽

Hurwitz Stability ◽

Type D ◽

International Audience

International audience Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach. Basé sur le théorème de Hermite–Biehler, nous prouvons simultanément les polynômes eulériens de type $D$ et les polynômes eulériens affine de type $B$ ont seulement racines réelle, qui sont d’abord obtenue par Savage et Visontai en utilisant le théorie des polynômes $s$-eulériens. Nous confirmons aussi les conjectures de Hyatt sur la propriété entrelacement de polynômes mi-eulériens. Le travail de Borcea et Brändén sur la caractérisation des opérateurs linéaires préservant la stabilité Hurwitz est essentielle à cette approche.

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Type B plactic relations for $r$-domino tableaux

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2700 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Müge Taşkın

Keyword(s):

Recent Work ◽

Equivalence Relation ◽

Type B ◽

Domino Tableaux ◽

Signed Permutations ◽

International Audience

International audience The recent work of Bonnafé et al. (2007) shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm (1990). Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tool for their proofs.

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Kazhdan-Lusztig polynomials of boolean elements

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2326 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Pietro Mongelli

Keyword(s):

Coxeter Groups ◽

Catalan Numbers ◽

Weyl Groups ◽

Schubert Varieties ◽

Intersection Homology ◽

Combinatorial Interpretation ◽

Coxeter Graph ◽

Signed Permutations ◽

International Audience ◽

Product Formulas

International audience We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type $q$ in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements. Nous donnons des formules combinatoires pour les polynômes de Kazhdan-Lusztig et leurs analogues paraboliques de type $q$ pour les éléments booléens, introduite dans [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], dans les groupes de Coxeter dont le graphe de Coxeter est un arbre. Ces formules utilisent les nombres de Catalan et une interprétation combinatoire des graphes du groupe de Coxeter. Dans le cas des groupes de Weyl classiques, cette interprétation combinatoire peut être reformulée en termes de statistiques de permutations avec signe. Avec ces formules, on peut calculer le polynôme de l’intersection homologie de Poincaré pour la variété de Schubert de éléments booléens.

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