Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element

Jean Michel

doi:10.1090/proc12753

Solution of tetrahedron equation and cluster algebras

Journal of High Energy Physics ◽

10.1007/jhep05(2021)103 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

P. Gavrylenko ◽

M. Semenyakin ◽

Y. Zenkevich

Keyword(s):

Integrable System ◽

Cluster Algebra ◽

Newton Polygon ◽

Cluster Algebras ◽

Weyl Groups ◽

Newton Polygons ◽

Tetrahedron Equation ◽

Affine Weyl Groups ◽

New Formalism ◽

Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽

10.3390/sym13061060 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1060

Author(s):

Enrico Celeghini ◽

Manuel Gadella ◽

Mariano A. del del Olmo

Keyword(s):

Hilbert Space ◽

Fourier Transform ◽

Heisenberg Group ◽

Real Line ◽

Hermite Functions ◽

Unitary Representations ◽

Weyl Groups ◽

The Real ◽

Symmetry Properties ◽

The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

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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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