Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Abstract

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 .

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

“…Recall from [4,18] that the admissible shifts ν ∨ of the dual root lattice Q ∨ form the trivial zero shift together with the n−th dual fundamental weight ω ∨ n and the admissible shifts of the weight lattice P are the trivial zero shift and one half of the n−th fundamental weight ω n /2,…”

“…The purpose of this article is to develop an explicit link between the multivariate discrete (anti)symmetric trigonometric transforms [1,2] and generalized dual-root lattice Fourier-Weyl transforms [3,4]. The established correspondence [5] between the (anti)symmetric trigonometric functions and Weyl orbit functions induced by the crystallographic root systems A 1 and C n [6,7] is utilized to interlace the kernels, point and label sets of the discrete transforms.…”

“…Formulated on the Weyl group invariant lattices, the discrete Fourier-Weyl transforms constitute multivariate generalizations of the classical discrete trigonometric transforms that utilize as their kernels the Weyl orbit functions. The standard lattice form of the Fourier-Weyl transforms, realized on the refined dual weight [14,15], weight [16,17], and dual root lattices [3], admits further extensions via admissible shifts of the (dual) root and weight lattices [4,18]. It appears that only after taking into account recent Fourier-Weyl transforms with the rescaled shifted dual root lattices point sets and the shifted weight lattices label sets [3,4], the entire family of 32 multivariate discrete (anti)symmetric trigonometric transforms permits embedding into the Fourier-Weyl formalism of the crystallographic root systems C n .…”

“…The standard lattice form of the Fourier-Weyl transforms, realized on the refined dual weight [14,15], weight [16,17], and dual root lattices [3], admits further extensions via admissible shifts of the (dual) root and weight lattices [4,18]. It appears that only after taking into account recent Fourier-Weyl transforms with the rescaled shifted dual root lattices point sets and the shifted weight lattices label sets [3,4], the entire family of 32 multivariate discrete (anti)symmetric trigonometric transforms permits embedding into the Fourier-Weyl formalism of the crystallographic root systems C n . Since both (anti)symmetric trigonometric and Fourier-Weyl transforms attain uniform characterization by their complementary unitary transform matrices [4,10], coupling together the ordered label and point sets as well as the weight and normalization functions produces the exact one-to-one correspondence between the unitary matrices of the induced discrete transforms.…”

“…Intrinsic in the established one-to-one correspondence, the exact evaluation techniques for the C n transforms devise a foundation for similar explicit expositions of the Fourier-Weyl transforms related to the remaining infinite series of the crystallographic root systems [4,10]. Furthermore, the actualized coincidence between the (anti)symmetric trigonometric and Fourier-Weyl transforms offers tools for connection and transfer of the discrete Fourier methods [11,19].…”