scholarly journals Affine Permutations of Type $A$

10.37236/1276 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Anders Björner ◽  
Francesco Brenti
Keyword(s):  
Coxeter Group ◽  
Type A ◽  
Bruhat Order ◽  
Weak Order ◽  
Combinatorial Properties ◽  
Affine Permutations

We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group $\tilde{A}_{n}$.

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 On the Topology of the Cambrian Semilattices

10.37236/2910 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Myrto Kallipoliti ◽  
Henri Mühle
Keyword(s):  
Coxeter Group ◽  
Closed Interval ◽  
Open Interval ◽  
Weak Order ◽  
Edge Labeling

For an arbitrary Coxeter group $W$, Reading and Speyer defined Cambrian semilattices $\mathcal{C}_{\gamma}$ as sub-semilattices of the weak order on $W$ induced by so-called $\gamma$-sortable elements. In this article, we define an edge-labeling of $\mathcal{C}_{\gamma}$, and show that this is an EL-labeling for every closed interval of $\mathcal{C}_{\gamma}$. In addition, we use our labeling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Reading.

scholarly journals Symplectic Keys and Demazure Atoms in Type C

10.37236/9235 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
João Miguel Santos
Keyword(s):  
Type A ◽  
Bruhat Order ◽  
Young Tableaux ◽  
Hyperoctahedral Group ◽  
Similar Role ◽  
Type C ◽  
Left And Right ◽  
Demazure Atoms ◽  
The Right

We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

scholarly journals Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

2020 ◽  
Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Combinatorial Properties ◽  
Coxeter Matroid ◽  
Finite Coxeter Group ◽  
Bruhat Intervals

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

scholarly journals A result on the Bruhat order of a Coxeter group

1990 ◽  
Vol 128 (2) ◽  
pp. 510-516 ◽  
Author(s):  
Shi Jianyi
Keyword(s):  
Coxeter Group ◽  
Bruhat Order

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

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.

The unit group of a partial Burnside ring of a reducible Coxeter group of type A

2019 ◽  
Vol 48 (2) ◽  
pp. 345-356
Author(s):  
Fumihito ODA ◽  
Masahiro WAKATAKE
Keyword(s):  
Coxeter Group ◽  
Type A ◽  
Unit Group ◽  
Burnside Ring
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