Generalized Dual-Root Lattice Transforms of Affine Weyl Groups
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 lattice...
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MDPI AG
2020-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/6/1018 |
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| author | Tomasz Czyżycki Jiří Hrivnák Lenka Motlochová |
| author_facet | Tomasz Czyżycki Jiří Hrivnák Lenka Motlochová |
| author_sort | Tomasz Czyżycki |
| collection | DOAJ |
| description | 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-03-10T19:08:04Z |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T19:08:04Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
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| series | Symmetry |
| spelling | doaj.art-51f7cdfc614a4cbf9fd680c036fdcebc2023-11-20T04:00:26ZengMDPI AGSymmetry2073-89942020-06-01126101810.3390/sym12061018Generalized Dual-Root Lattice Transforms of Affine Weyl GroupsTomasz Czyżycki0Jiří Hrivnák1Lenka Motlochová2Faculty of Mathematics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandDepartment of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech RepublicDepartment of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech RepublicDiscrete 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula>.https://www.mdpi.com/2073-8994/12/6/1018Weyl orbit functionroot latticecongruence classdiscrete Fourier transform |
| spellingShingle | Tomasz Czyżycki Jiří Hrivnák Lenka Motlochová Generalized Dual-Root Lattice Transforms of Affine Weyl Groups Symmetry Weyl orbit function root lattice congruence class discrete Fourier transform |
| title | Generalized Dual-Root Lattice Transforms of Affine Weyl Groups |
| title_full | Generalized Dual-Root Lattice Transforms of Affine Weyl Groups |
| title_fullStr | Generalized Dual-Root Lattice Transforms of Affine Weyl Groups |
| title_full_unstemmed | Generalized Dual-Root Lattice Transforms of Affine Weyl Groups |
| title_short | Generalized Dual-Root Lattice Transforms of Affine Weyl Groups |
| title_sort | generalized dual root lattice transforms of affine weyl groups |
| topic | Weyl orbit function root lattice congruence class discrete Fourier transform |
| url | https://www.mdpi.com/2073-8994/12/6/1018 |
| work_keys_str_mv | AT tomaszczyzycki generalizeddualrootlatticetransformsofaffineweylgroups AT jirihrivnak generalizeddualrootlatticetransformsofaffineweylgroups AT lenkamotlochova generalizeddualrootlatticetransformsofaffineweylgroups |