scholarly journals Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices

10.37236/137 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Aisling Kenny
Keyword(s):  
Hyperplane Arrangement ◽  
Reflection Group ◽  
General Construction ◽  
Partition Lattice ◽  
Intersection Lattice ◽  
Affine Hyperplane ◽  
Partition Lattices

For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.

scholarly journals Geometrically Constructed Bases for Homology of Partition Lattices of Types $A$, $B$ and $D$

10.37236/1860 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Anders Björner ◽  
Michelle L. Wachs
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Section ◽  
Hyperplane Arrangements ◽  
Proper Part ◽  
Geometric Method ◽  
Partition Lattice ◽  
General Technique ◽  
Intersection Lattice ◽  
Partition Lattices

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types $A$, $B$ and $D$. This extends and explains the "splitting basis" for the homology of the partition lattice given by M. L. Wachs, thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let ${\cal A}$ be a central and essential hyperplane arrangement in ${\Bbb{R}}^d$. Let $R_1,\dots,R_k$ be the bounded regions of a generic hyperplane section of ${\cal A}$. We show that there are induced polytopal cycles $\rho_{R_i}$ in the homology of the proper part $\overline{L}_{\cal A}$ of the intersection lattice such that $\{\rho_{R_i}\}_{i=1,\dots,k}$ is a basis for $\widetilde{H}_{d-2} (\overline{L}_{\cal A})$. This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types $A$, $B$ and $D$, and to some interpolating arrangements.

scholarly journals From permutahedron to associahedron

2010 ◽  
Vol 53 (2) ◽  
pp. 299-310 ◽  
Author(s):  
Thomas Brady ◽  
Colum Watt
Keyword(s):  
Reflection Group ◽  
Partition Lattice

AbstractFor each finite real reflection group W, we identify a copy of the type-W simplicial generalized associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalized associahedron and the elements of the type-W non-crossing partition lattice that is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for W.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

scholarly journals The Face Semigroup Algebra of a Hyperplane Arrangement

2009 ◽  
Vol 61 (4) ◽  
pp. 904-929 ◽  
Author(s):  
Franco V. Saliola
Keyword(s):  
Algebraic Structure ◽  
Hyperplane Arrangement ◽  
Semigroup Algebra ◽  
Cohomology Algebra ◽  
Koszul Algebra ◽  
Dual Algebra ◽  
Intersection Lattice ◽  
Projective Resolutions ◽  
The Face ◽  
Cartan Invariants

Abstract.This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

scholarly journals ON THE MILNOR MONODROMY OF THE IRREDUCIBLE COMPLEX REFLECTION ARRANGEMENTS

2017 ◽  
Vol 18 (06) ◽  
pp. 1215-1231 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Reflection Group ◽  
Milnor Fiber ◽  
Complex Reflection ◽  
Complex Reflection Group

Using recent results by Măcinic, Papadima and Popescu, and a refinement of an older construction of ours, we determine the monodromy action on $H^{1}(F(G),\mathbb{C})$ , where $F(G)$ denotes the Milnor fiber of a hyperplane arrangement associated to an irreducible complex reflection group $G$ .

scholarly journals Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables

10.37236/1101 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
N. Bergeron ◽  
C. Hohlweg ◽  
M. Rosas ◽  
M. Zabrocki
Keyword(s):  
Symmetric Functions ◽  
Schur Function ◽  
Partition Lattice ◽  
Partition Lattices

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

scholarly journals Bonferroni-Galambos Inequalities for Partition Lattices

10.37236/1838 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Klaus Dohmen ◽  
Peter Tittmann
Keyword(s):  
Network Reliability ◽  
Partition Lattice ◽  
Bonferroni Inequalities ◽  
Uniform Hypergraphs ◽  
Negative Function ◽  
Finite Set ◽  
Partition Lattices

In this paper, we establish a new analogue of the classical Bonferroni inequalities and their improvements by Galambos for sums of type $\sum_{\pi\in {\Bbb P}(U)} (-1)^{|\pi|-1} (|\pi|-1)! f(\pi)$ where $U$ is a finite set, ${\Bbb P}(U)$ is the partition lattice of $U$ and $f:{\Bbb P}(U)\rightarrow{\Bbb R}$ is some suitable non-negative function. Applications of this new analogue are given to counting connected $k$-uniform hypergraphs, network reliability, and cumulants.

scholarly journals Submaximal factorizations of a Coxeter element in complex reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vivien Ripoll
Keyword(s):  
Reflection Group ◽  
Coxeter Element ◽  
Partition Lattice ◽  
Combinatorial Object ◽  
Complex Reflection Groups ◽  
Noncrossing Partition ◽  
Finite Reflection Group ◽  
International Audience ◽  
Cas A ◽  
Noncrossing Partition Lattice

International audience When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

scholarly journals Affine and toric arrangements

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Richard Ehrenborg ◽  
Margaret Readdy ◽  
Michael Slone
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Face Lattice ◽  
Intersection Lattice ◽  
International Audience

International audience We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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