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 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 .

A Euclidean interpretation of Dynkin diagrams and its relation to root systems

1987 ◽  
Vol 22 (1) ◽  
Author(s):  
Daniel Drucker ◽  
Daniel Frohardt
Keyword(s):  
Root Systems ◽  
Dynkin Diagrams

Triple root systems, rational quivers and examples of linear free divisors

2018 ◽  
Vol 29 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Kazunori Nakamoto ◽  
Ayşe Sharland ◽  
Meral Tosun
Keyword(s):  
Comparison Theorem ◽  
Triple Point ◽  
Root Systems ◽  
Rational Singularities ◽  
Dynkin Diagrams ◽  
Free Divisors ◽  
Resolution Graphs

The dual resolution graphs of rational triple point (RTP) singularities can be seen as a generalization of Dynkin diagrams. In this work, we study the triple root systems corresponding to those diagrams. We determine the number of roots for each RTP singularity, and show that for each root we obtain a linear free divisor. Furthermore, we deduce that linear free divisors defined by rational triple quivers with roots in the corresponding triple root systems satisfy the global logarithmic comparison theorem. We also discuss a generalization of these results to the class of rational singularities with almost reduced Artin cycle.

scholarly journals Twisted quadratic foldings of root systems

2021 ◽  
Vol 33 (1) ◽  
pp. 65-84
Author(s):  
M. Lanini ◽  
K. Zainoulline
Keyword(s):  
Root System ◽  
Equivariant Cohomology ◽  
Quaternion Algebra ◽  
Root Systems ◽  
Group Cohomology ◽  
Characteristic Classes ◽  
Reflection Groups ◽  
Content Type ◽  
Dynkin Diagrams ◽  
Projective Homogeneous Varieties

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

scholarly journals Central Splitting of A2 Discrete Fourier–Weyl Transforms

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1828 ◽  
Author(s):  
Jiří Hrivnák ◽  
Mariia Myronova ◽  
Jiří Patera
Keyword(s):  
Reflection Group ◽  
Unitary Matrix ◽  
Weyl Transform ◽  
Matrix Decompositions ◽  
Weight Lattice ◽  
Affine Weyl Group ◽  
Weyl Transforms ◽  
Original Weight ◽  
Unitary Transform ◽  
Point Set

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

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.

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