scholarly journals Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1018 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Weyl Groups ◽  
Root Lattice ◽  
Affine Weyl Groups ◽  
Fundamental Domains ◽  
Weyl Transforms ◽  
Orthogonality Relations ◽  
Discrete Transforms ◽  
Unitary Transform ◽  
Dual Affine ◽  
Root Lattices

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

scholarly journals Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 61
Author(s):  
Adam Brus ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Case Analysis ◽  
Root Systems ◽  
Trigonometric Functions ◽  
Root Lattice ◽  
Direct Evaluation ◽  
Trigonometric Transforms ◽  
Weyl Transforms ◽  
Dynkin Diagrams ◽  
Discrete Transforms ◽  
Unitary Transform

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems A1 and Cn are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms.

scholarly journals Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 938 ◽  
Author(s):  
Adam Brus ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Three Dimensional ◽  
Weight Functions ◽  
Recursive Algorithms ◽  
Interpolation Methods ◽  
Orthogonality Relations ◽  
Discrete Transforms ◽  
Unitary Transform ◽  
Sine Functions

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

scholarly journals Solution of tetrahedron equation and cluster algebras

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.

scholarly journals Bijective projections on parabolic quotients of affine Weyl groups

2014 ◽  
Vol 41 (4) ◽  
pp. 911-948 ◽  
Author(s):  
Elizabeth Beazley ◽  
Margaret Nichols ◽  
Min Hae Park ◽  
XiaoLin Shi ◽  
Alexander Youcis
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

2011 ◽  
Vol 39 (2) ◽  
pp. 730-749 ◽  
Author(s):  
Saeid Azam ◽  
Valiollah Shahsanaei
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups

scholarly journals Poincaré duality and Langlands duality for extended affine Weyl groups

2018 ◽  
Vol 3 (3) ◽  
pp. 491-522
Author(s):  
Graham Niblo ◽  
Roger Plymen ◽  
Nick Wright
Keyword(s):  
Weyl Groups ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Affine Weyl Groups ◽  
Langlands Duality

scholarly journals Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

2019 ◽  
Vol 351 ◽  
pp. 897-946 ◽  
Author(s):  
Boris Dubrovin ◽  
Ian A.B. Strachan ◽  
Youjin Zhang ◽  
Dafeng Zuo
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups ◽  
Frobenius Manifolds
Sign in / Sign up

Export Citation Format

Share Document