scholarly journals Gallery Posets of Supersolvable Arrangements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Thomas McConville
Keyword(s):  
Homotopy Type ◽  
Coxeter Groups ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Homotopy Equivalent ◽  
Hasse Diagram ◽  
Arrangement Of Hyperplanes ◽  
International Audience ◽  
Supersolvable Arrangement ◽  
Reduced Words

International audience We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau's Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its intervals are either homotopy equivalent to spheres or are contractible. One may view this as a analogue of a result of Edelman and Walker on the homotopy type of intervals of a poset of chambers of a hyperplane arrangement.

scholarly journals Multi-cluster complexes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Jean-Philippe Labbé ◽  
Christian Stump
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Simplicial Complexes ◽  
Cluster Complex ◽  
Geometric Properties ◽  
Cluster Complexes ◽  
Combinatorial Description ◽  
Compatibility Relation ◽  
Positive Roots ◽  
International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

scholarly journals Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Nantel Bergeron ◽  
Cesar Ceballos ◽  
Jean-Philippe Labbé
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Nous Obtenons ◽  
Open Case ◽  
International Audience ◽  
Paper Work ◽  
Subword Complex

International audience We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$. Nous construisons des éventails simpliciaux complets ayant la combinatoire des complexes de sous-mots de type $A_n$ pour $n\leq 3$. Par conséquent, nous obtenons des constructions d’éventails des multi-associaèdres $\Delta_{2k+4,k}$, dont les facettes correspondent aux $k$-triangulations d’un $(2k+4)$-gone. Cette construction confirme l’existence d’éventails ayant la combinatoire du multi-associaèdres pour une famille dont la polytopalité n’est pas confirmée. Les techniques utilisées fonctionnent pour tous les groupes de Coxeter et nous espérons qu’elles seront utiles afin de construire des éventails réalisant les complexes de sous-mots en général. En particulier, nous présentons des éventails pour deux complexes de sous-mots de type $A_4$ dont l’existence était inconnue: les multi-associaèdres $\Delta_{9,2}$ et $\Delta_{11,3}$.

scholarly journals Hyperplane Arrangements and Diagonal Harmonics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Symmetric Functions ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Combinatorial Interpretation ◽  
Nabla Operator ◽  
Affine Weyl Group ◽  
Diagonal Harmonics ◽  
International Audience

International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

The Homotopy Type of the Symmetric Products of Bouquets of Circles

2003 ◽  
Vol 14 (05) ◽  
pp. 489-497 ◽  
Author(s):  
Boon W. Ong
Keyword(s):  
Homotopy Type ◽  
Hyperplane Arrangement ◽  
Homotopy Equivalent ◽  
Symmetric Product ◽  
Symmetric Products

A wedge of k circles is homotopy equivalent to the plane with k points removed. By looking at [Formula: see text]∖{w1, …, wk} instead of ⋁k S1, we could view the symmetric product of a wedge of k circles as [Formula: see text], the complement space [3] of a hyperplane arrangement, [Formula: see text]. Applying a theorem of Hattori [2], we see that if k > n, the symmetric product of ⋁k S1 is, up to homotopy, the union of n-dimensional subtorii in a k-torus.

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

scholarly journals Criteria for components of a function space to be homotopy equivalent

2008 ◽  
Vol 145 (1) ◽  
pp. 95-106 ◽  
Author(s):  
GREGORY LUPTON ◽  
SAMUEL BRUCE SMITH
Keyword(s):  
Function Space ◽  
Homotopy Type ◽  
Homotopy Equivalent ◽  
General Method

AbstractWe give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

scholarly journals How to get the weak order out of a digraph ?

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Francois Viard
Keyword(s):  
Wreath Product ◽  
Symmetric Function ◽  
Coxeter Groups ◽  
Symmetric Functions ◽  
Weak Order ◽  
Möbius Function ◽  
Mobius Function ◽  
Acyclic Digraph ◽  
International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, ainsi qu’une variante de l’ordre faible sur les produits en couronne &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$<sub>$n-1$</sub> où l’on obtient les séries de Stanley classiques.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

scholarly journals Special involutions and bulky parabolic subgroups in finite Coxeter groups

2005 ◽  
Vol 79 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Götz Pfeiffer ◽  
Gerhard Röhrle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Hyperplane Arrangement ◽  
Conjugacy Classes ◽  
Parabolic Subgroups ◽  
Finite Coxeter Group

AbstractThe conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

scholarly journals The Frobenius Complex

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Eric Clark ◽  
Richard Ehrenborg
Keyword(s):  
Homotopy Equivalent ◽  
Order Complex ◽  
Frobenius Problem ◽  
Nous Montrons ◽  
International Audience ◽  
Frobenius Complex

International audience Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers $\mathbb{Z}$, that is, for a sub-semigroup $\Lambda$ of the non-negative integers $(\mathbb{N},+)$, we define the order by $n \leq_{\Lambda} m$ if $m-n \in \Lambda$. When $\Lambda$ is generated by two relatively prime integers $a$ and $b$, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when $\Lambda$ is generated by the integers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, the order complex is homotopy equivalent to a wedge of spheres. Motivé par le problème de Frobenius classique, nous introduisons l'ensemble partiellement ordonné de Frobenius sur les entiers $\mathbb{Z}$, c.à.d. que pour un sous-semigroupe $\Lambda$ de les entiers non-négatifs $(\mathbb{N},+)$ nous définissons l'ordre par $n \leq_{\Lambda} m$ si $m-n \in \Lambda$. Quand le $\Lambda$ est engendré par deux nombres $a$ et $b$, relativement premiers entre eux, nous montrons que le complexe des chaînes d'un intervalle quelconque dans l'ensemble partiellement ordonné de Frobenius est soit contractible soit homotopiquement équivalent à une sphère. Nous montrons aussi que dans le cas où $\Lambda$ est engendré par les entiers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, le complexe des chaînes a le type de homotopie d'un bouquet de sphères.

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