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 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 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.

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

scholarly journals On the Monodromy of Milnor Fibers of Hyperplane Arrangements

2014 ◽  
Vol 57 (4) ◽  
pp. 697-707 ◽  
Author(s):  
Pauline Bailet
Keyword(s):  
Cohomology Group ◽  
Hyperplane Arrangement ◽  
General Setting ◽  
Hyperplane Arrangements ◽  
Milnor Fiber ◽  
First Cohomology Group

AbstractWe describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

scholarly journals Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

2012 ◽  
Vol 206 ◽  
pp. 75-97 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monodromy Operator ◽  
Low Dimensions ◽  
Milnor Fiber ◽  
Necessary And Sufficient ◽  
Complex Hyperplane

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

scholarly journals Face monoid actions and tropical hyperplane arrangements

2017 ◽  
Vol 27 (08) ◽  
pp. 1001-1025
Author(s):  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Oriented Matroid ◽  
Definition Of ◽  
The Way

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropical hyperplane arrangement, in terms of partial bijections which attain permanents of submatrices of a matrix which naturally encodes the arrangement.

scholarly journals Circle-valued Morse theory for complex hyperplane arrangements

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Toshitake Kohno ◽  
Andrei Pajitnov
Keyword(s):  
Morse Theory ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Complex Hyperplane

AbstractLet 𝒜 be an essential complex hyperplane arrangement in

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 Freeness and multirestriction of hyperplane arrangements

2012 ◽  
Vol 148 (3) ◽  
pp. 799-806 ◽  
Author(s):  
Mathias Schulze
Keyword(s):  
Characteristic Polynomial ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements

AbstractGeneralizing a result of Yoshinaga in dimension three, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

scholarly journals Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Hyperplane Arrangement ◽  
Schubert Variety ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Poincaré Polynomial ◽  
Generalized Flag Manifolds ◽  
Nous Montrons ◽  
International Audience

International audience We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

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