MAT-free Reflection Arrangements

Michael Cuntz; Paul Mücksch

doi:10.37236/8820

MAT-free Reflection Arrangements

The Electronic Journal of Combinatorics ◽

10.37236/8820 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Michael Cuntz ◽

Paul Mücksch

Keyword(s):

Addition Theorem ◽

Hyperplane Arrangements ◽

Complex Reflection ◽

Multiple Addition

We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization of the Multiple Addition Theorem by Abe and Terao. We give classifications of the irreducible complex reflection arrangements which are MAT-free respectively MAT2-free. Furthermore, we ask some questions concerning relations to other classes of free arrangements.

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Multiple addition theorem for discrete and continuous nonlinear problems

Journal of Physics A General Physics ◽

10.1088/0305-4470/34/48/318 ◽

2001 ◽

Vol 34 (48) ◽

pp. 10549-10558 ◽

Cited By ~ 2

Author(s):

J A Zagrodzinski ◽

T Nikiciuk

Keyword(s):

Nonlinear Problems ◽

Addition Theorem ◽

Multiple Addition

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Multiple addition, deletion and restriction theorems for hyperplane arrangements

Proceedings of the American Mathematical Society ◽

10.1090/proc/14592 ◽

2019 ◽

Vol 147 (11) ◽

pp. 4835-4845 ◽

Cited By ~ 1

Author(s):

Takuro Abe ◽

Hiroaki Terao

Keyword(s):

Hyperplane Arrangements ◽

Restriction Theorems ◽

Multiple Addition

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Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Discrete & Computational Geometry ◽

10.1007/pl00009404 ◽

1999 ◽

Vol 21 (1) ◽

pp. 117-130 ◽

Cited By ~ 1

Author(s):

C. A. Athanasiadis

Keyword(s):

Hyperplane Arrangements ◽

Monotone Path

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Mathematical equations for potentiometric analysis by the multiple addition method

Journal of Analytical Chemistry ◽

10.1134/s1061934806060049 ◽

2006 ◽

Vol 61 (6) ◽

pp. 540-543 ◽

Cited By ~ 4

Author(s):

B. M. Mar’yanov ◽

E. V. Pozdnyakov

Keyword(s):

Addition Method ◽

Potentiometric Analysis ◽

Mathematical Equations ◽

Multiple Addition

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On polytopes and generalizations of the KLT relations

Journal of High Energy Physics ◽

10.1007/jhep12(2020)057 ◽

2020 ◽

Vol 2020 (12) ◽

Cited By ~ 1

Author(s):

Nikhil Kalyanapuram

Keyword(s):

Scattering Amplitudes ◽

Large Class ◽

Moduli Space ◽

Differential Forms ◽

Natural Generalization ◽

Intersection Theory ◽

Hyperplane Arrangements ◽

Number Of Solutions ◽

Scalar Theories ◽

Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

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The Tutte polynomial of symmetric hyperplane arrangements

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500042 ◽

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.

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Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Nagoya Mathematical Journal ◽

10.1017/s0027763000009880 ◽

2010 ◽

Vol 197 ◽

pp. 175-212

Author(s):

Maria Chlouveraki

Keyword(s):

Infinite Series ◽

Hecke Algebras ◽

Weyl Groups ◽

Reflection Groups ◽

Complex Reflection ◽

Complex Reflection Groups ◽

Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

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