Advances in Mathematics, ISSN 0001-8708, 09/2018, Volume 335, pp. 466 - 518

We establish general counting formulas and bijections for deformations of the braid arrangement...

Bijection | Regions | Enumeration | Coxeter arrangements | MATHEMATICS | WEYL GROUPS | NUMBERS | HYPERPLANE ARRANGEMENTS | SHI ARRANGEMENT | Combinatorics | Mathematics

Bijection | Regions | Enumeration | Coxeter arrangements | MATHEMATICS | WEYL GROUPS | NUMBERS | HYPERPLANE ARRANGEMENTS | SHI ARRANGEMENT | Combinatorics | Mathematics

Journal Article

Advances in Applied Mathematics, ISSN 0196-8858, 06/2019, Volume 107, pp. 32 - 73

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three...

Simplicial arrangements | Supersolvable arrangements | Hyperplane arrangements | Coxeter graph | Root system | Reflection arrangements | MATHEMATICS, APPLIED | Algebra | Mathematics - Combinatorics

Simplicial arrangements | Supersolvable arrangements | Hyperplane arrangements | Coxeter graph | Root system | Reflection arrangements | MATHEMATICS, APPLIED | Algebra | Mathematics - Combinatorics

Journal Article

Journal of Combinatorial Theory, Series A, ISSN 0097-3165, 02/2017, Volume 146, pp. 169 - 183

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman...

Ish arrangement | Shi arrangement | Supersolvable arrangement | Fiber-type arrangement | Hyperplane arrangement | Coxeter arrangement | Free arrangement | MATHEMATICS | HYPERPLANE ARRANGEMENTS | Mathematics - Combinatorics

Ish arrangement | Shi arrangement | Supersolvable arrangement | Fiber-type arrangement | Hyperplane arrangement | Coxeter arrangement | Free arrangement | MATHEMATICS | HYPERPLANE ARRANGEMENTS | Mathematics - Combinatorics

Journal Article

Journal of Fixed Point Theory and Applications, ISSN 1661-7738, 03/2019, Volume 21, Issue 1, p. 1

We compute the total cohomology of the complement of the toric arrangement associated with the root system An as a representation of the corresponding Weyl group via fixed point theory of a twisted action of the group...

Arrangements | Fixed points | Cohomology | Root systems | Weyl groups | COMPLEMENTS | MATHEMATICS | MATHEMATICS, APPLIED | COXETER GROUP-ACTIONS | REPRESENTATION | COMBINATORICS

Arrangements | Fixed points | Cohomology | Root systems | Weyl groups | COMPLEMENTS | MATHEMATICS | MATHEMATICS, APPLIED | COXETER GROUP-ACTIONS | REPRESENTATION | COMBINATORICS

Journal Article

Discrete and Computational Geometry, ISSN 0179-5376, 01/2019, Volume 61, Issue 1, pp. 185 - 197

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2019, Volume 581, pp. 405 - 412

We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino...

Coxeter groups | Rotation groups | Reflection groups | Subspace arrangements | MATHEMATICS | MATHEMATICS, APPLIED | COHEN-MACAULAY | INVARIANTS | FINITE-GROUPS | DEPTH | Rotation | Reflection

Coxeter groups | Rotation groups | Reflection groups | Subspace arrangements | MATHEMATICS | MATHEMATICS, APPLIED | COHEN-MACAULAY | INVARIANTS | FINITE-GROUPS | DEPTH | Rotation | Reflection

Journal Article

Communications in Mathematical Physics, ISSN 0010-3616, 6/2014, Volume 328, Issue 3, pp. 1117 - 1157

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function...

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | HUYGENS PRINCIPLE | QUASI-INVARIANTS | PHYSICS, MATHEMATICAL | COMMUTATIVE RINGS | COXETER GROUPS | Algebra

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | HUYGENS PRINCIPLE | QUASI-INVARIANTS | PHYSICS, MATHEMATICAL | COMMUTATIVE RINGS | COXETER GROUPS | Algebra

Journal Article

Archiv der Mathematik, ISSN 0003-889X, 4/2019, Volume 112, Issue 4, pp. 347 - 359

.... This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement...

52B30 | 14N20 | Poset of regions of a real arrangement | Restriction of a Coxeter arrangement | Factorization of rank-generating function | Mathematics, general | Mathematics | 52C35 | Coxeter arrangement | 20F55 | MATHEMATICS

52B30 | 14N20 | Poset of regions of a real arrangement | Restriction of a Coxeter arrangement | Factorization of rank-generating function | Mathematics, general | Mathematics | 52C35 | Coxeter arrangement | 20F55 | MATHEMATICS

Journal Article

Discrete & Computational Geometry, ISSN 0179-5376, 1/2019, Volume 61, Issue 1, pp. 185 - 197

Let G be a simple graph on $$ \ell $$ ℓ vertices $$ \{1, \dots , \ell \} $$ { 1 , ⋯ , ℓ } with edge set $$ E_{G} $$ E G . The graphical arrangement...

Computational Mathematics and Numerical Analysis | 05C22 | Chordal graph | Mathematics | Hyperplane arrangement | 52C35 | Coxeter arrangement | Free arrangement | Graphical arrangement | 32S22 | Ish arrangement | Supersolvable arrangement | 13N15 | Combinatorics | Vertex-weighted graph | Multiarrangement | 05C15 | 20F55

Computational Mathematics and Numerical Analysis | 05C22 | Chordal graph | Mathematics | Hyperplane arrangement | 52C35 | Coxeter arrangement | Free arrangement | Graphical arrangement | 32S22 | Ish arrangement | Supersolvable arrangement | 13N15 | Combinatorics | Vertex-weighted graph | Multiarrangement | 05C15 | 20F55

Journal Article

Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova, ISSN 0041-8994, 2017, Volume 138, pp. 147 - 191

We introduce the notion of a Tits arrangement on a convex open cone as a special case of (in finite...

Simplicial arrangement | Coxeter group | Tits cone | NICHOLS ALGEBRA | MATHEMATICS | MATHEMATICS, APPLIED | RANK 2 | WEYL GROUPOIDS

Simplicial arrangement | Coxeter group | Tits cone | NICHOLS ALGEBRA | MATHEMATICS | MATHEMATICS, APPLIED | RANK 2 | WEYL GROUPOIDS

Journal Article

Journal of Algebra, ISSN 0021-8693, 09/2020, Volume 558, pp. 336 - 349

We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W...

Coxeter groups | Orlik-Solomon algebras | Hyperplane complements | Arrangements of hyperplanes | MATHEMATICS | INVOLUTIONS | PARABOLIC SUBGROUPS | COMPUTATIONS

Coxeter groups | Orlik-Solomon algebras | Hyperplane complements | Arrangements of hyperplanes | MATHEMATICS | INVOLUTIONS | PARABOLIC SUBGROUPS | COMPUTATIONS

Journal Article

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, ISSN 0138-4821, 11/2016, Volume 57, Issue 4, pp. 891 - 897

Let $$\mathcal {A}$$ A be a finite real linear hyperplane arrangement in three dimensions...

Geometry | Algebra | Finite Coxeter systems | Hyperplane arrangements | 52Cxx | Convex and Discrete Geometry | Algebraic Geometry | Mathematics | Spherical geometry | 51F15 | 97G60 | 20F55

Geometry | Algebra | Finite Coxeter systems | Hyperplane arrangements | 52Cxx | Convex and Discrete Geometry | Algebraic Geometry | Mathematics | Spherical geometry | 51F15 | 97G60 | 20F55

Journal Article

Journal of Algebra, ISSN 0021-8693, 01/2015, Volume 422, pp. 89 - 104

In his affirmative answer to the Edelman–Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules...

Shi arrangements | Free arrangements | Weyl arrangements | Root systems | Arrangements of hyperplanes | MATHEMATICS | FREENESS | COXETER ARRANGEMENTS

Shi arrangements | Free arrangements | Weyl arrangements | Root systems | Arrangements of hyperplanes | MATHEMATICS | FREENESS | COXETER ARRANGEMENTS

Journal Article

Advances in Mathematics, ISSN 0001-8708, 2012, Volume 229, Issue 1, pp. 691 - 709

Using the classification of finite Weyl groupoids we prove that crystallographic arrangements, a large subclass of the class of simplicial arrangements which was recently defined, are hereditarily inductively free...

Reflection | Arrangement of hyperplanes | Inductively free | Coxeter | MATHEMATICS

Reflection | Arrangement of hyperplanes | Inductively free | Coxeter | MATHEMATICS

Journal Article

International Journal of Algebra and Computation, ISSN 0218-1967, 12/2017, Volume 27, Issue 8, pp. 1001 - 1025

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical...

tropical convexity | tropical matrix permanents | Hyperplane face monoids | tropical oriented matroids | MATHEMATICS | SEMIGROUP ALGEBRA | LEFT-REGULAR BAND | COXETER GROUP | RANDOM-WALKS | COMPLEXES | MACKEY FORMULA

tropical convexity | tropical matrix permanents | Hyperplane face monoids | tropical oriented matroids | MATHEMATICS | SEMIGROUP ALGEBRA | LEFT-REGULAR BAND | COXETER GROUP | RANDOM-WALKS | COMPLEXES | MACKEY FORMULA

Journal Article

Advances in Mathematics, ISSN 0001-8708, 09/2016, Volume 300, pp. 788 - 834

We discuss arrangements of equal minors of totally positive matrices. More precisely, we investigate the structure of equalities and inequalities between the minors...

Gröbner bases | The Eulerian numbers | Alcoved polytopes | Triangulations and thrackles | Mutation distance | The affine Coxeter arrangement | Schur positivity | Matrix completion problem | honeycombs | Cluster algebras and plabic graphs | The positive Grassmannian | Chain reactions of mutations | Minors and Plücker coordinates | Totally positive matrices | Arrangements of equal minors | The Laurent phenomenon | Weakly separated and sorted sets | hypersimplices | Minors and Plucker coordinates | Mutation distance, honeycombs | Grobner bases,Schur positivity | MATHEMATICS | Alcoved polytopes, hypersimplices | MATRICES | CLUSTER ALGEBRAS | Algebra

Gröbner bases | The Eulerian numbers | Alcoved polytopes | Triangulations and thrackles | Mutation distance | The affine Coxeter arrangement | Schur positivity | Matrix completion problem | honeycombs | Cluster algebras and plabic graphs | The positive Grassmannian | Chain reactions of mutations | Minors and Plücker coordinates | Totally positive matrices | Arrangements of equal minors | The Laurent phenomenon | Weakly separated and sorted sets | hypersimplices | Minors and Plucker coordinates | Mutation distance, honeycombs | Grobner bases,Schur positivity | MATHEMATICS | Alcoved polytopes, hypersimplices | MATRICES | CLUSTER ALGEBRAS | Algebra

Journal Article

Annals of Combinatorics, ISSN 0218-0006, 12/2016, Volume 20, Issue 4, pp. 719 - 735

The reflection arrangement of a Coxeter group is a well-known instance of a free hyperplane arrangement...

14N20 | braid arrangement | free arrangement | Mathematics | 52C35 | Coxeter arrangement | inductively free multiarrangement | Combinatorics | 20F55 | MATHEMATICS, APPLIED | inductively free multi-arrangement | MULTIPLICITIES | COXETER ARRANGEMENTS

14N20 | braid arrangement | free arrangement | Mathematics | 52C35 | Coxeter arrangement | inductively free multiarrangement | Combinatorics | 20F55 | MATHEMATICS, APPLIED | inductively free multi-arrangement | MULTIPLICITIES | COXETER ARRANGEMENTS

Journal Article

Advances in Mathematics, ISSN 0001-8708, 11/2015, Volume 285, pp. 709 - 736

We show that an element w of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement I associated to the inversion set of w is inductively free, and the product (d1+1)⋯(dl+1...

Schubert varieties | Coxeter groups | Hyperplane arrangements | Inductive freeness | MATHEMATICS | ORDER | BRUHAT INTERVALS

Schubert varieties | Coxeter groups | Hyperplane arrangements | Inductive freeness | MATHEMATICS | ORDER | BRUHAT INTERVALS

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 11/2011, Volume 363, Issue 11, pp. 6063 - 6083

In this paper, we study k-equal arrangements for finite real reflection groups. When k=2 \mathbb{C} Coxeter arrangement is isomorphic to the pure Artin group of type W...

Homomorphisms | Homotopy theory | Algebra | Braiding | Hyperplanes | Mathematical lattices | Partially ordered sets | Mathematics | Combinatorics | Vertices | MATHEMATICS | COMPLEX | FUNDAMENTAL GROUP | Subspace arrangement | Eilenberg-MacLane space | discrete homotopy theory | right-angled Coxeter group | HOMOTOPY-THEORY

Homomorphisms | Homotopy theory | Algebra | Braiding | Hyperplanes | Mathematical lattices | Partially ordered sets | Mathematics | Combinatorics | Vertices | MATHEMATICS | COMPLEX | FUNDAMENTAL GROUP | Subspace arrangement | Eilenberg-MacLane space | discrete homotopy theory | right-angled Coxeter group | HOMOTOPY-THEORY

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 8/2009, Volume 80, Issue 1, pp. 121 - 134

We define specific multiplicities on the braid arrangement by using signed graphs...

MATHEMATICS | HYPERPLANE ARRANGEMENTS | DEFORMATIONS | COXETER ARRANGEMENTS

MATHEMATICS | HYPERPLANE ARRANGEMENTS | DEFORMATIONS | COXETER ARRANGEMENTS

Journal Article

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