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Open AccessArticle

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

1
Faculty of Mathematics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland
2
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(6), 1018; https://doi.org/10.3390/sym12061018
Received: 21 April 2020 / Revised: 4 June 2020 / Accepted: 5 June 2020 / Published: 16 June 2020
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 . View Full-Text
Keywords: Weyl orbit function; root lattice; congruence class; discrete Fourier transform Weyl orbit function; root lattice; congruence class; discrete Fourier transform
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Czyżycki, T.; Hrivnák, J.; Motlochová, L. Generalized Dual-Root Lattice Transforms of Affine Weyl Groups. Symmetry 2020, 12, 1018.

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