Dual-Root Lattice Discretization of Weyl Orbit Functions

Abstract

Four types of discrete transforms of Weyl orbit functions on the finite point sets are developed. The point sets are formed by intersections of the dual-root lattices with the fundamental domains of the affine Weyl groups. The finite sets of weights, labelling the orbit functions, obey symmetries of the dual extended affine Weyl groups. Fundamental domains of the dual extended affine Weyl groups are detailed in full generality. Identical cardinality of the point and weight sets is proved and explicit counting formulas for these cardinalities are derived. Discrete orthogonality of complex-valued Weyl and real-valued Hartley orbit functions over the point sets is established and the corresponding discrete Fourier-Weyl and Hartley-Weyl transforms are formulated.

Discrete cosine and sine transforms generalized to honeycomb lattice

“…The numbers of weights in the interior weight sets Λ Q,M and Λ P,M are proven in [12,15] to coincide with the number of points in interiors F P,M and F Q,M , respectively,…”

“…The introduced extended Weyl and Hartley-Weyl orbit functions, obeying the three non-linear conditions, represent a unique novel discretely orthogonal parametric systems of functions over the subtractively constructed honeycomb point sets. The counting formulas for the numbers of elements in the point and label sets in [11] and [12] guarantee existence of both Weyl and Hartley-Weyl discrete Fourier transforms over the honeycomb point sets.…”

“…Note that the points from F P,M are described by the coordinates from (10) as and the set F Q,M ⊂ F P,M contains only such points from F P,M , which satisfy the additional condition s 1 + 2s 2 = 0 mod 3. The number of points in the point sets F P,M and F Q,M are calculated in [12,15] as…”

“…The purpose of this article is to generalize the discrete cosine and sine transforms [2,31] to a finite fragment of the honeycomb lattice. Two-variable (anti)symmetric complex-valued Weyl and the realvalued Hartley-Weyl orbit functions [11,12,21,22] are modified by six extension coefficients sequences and the resulting family of the extended functions contains the set of the discretely orthogonal honeycomb orbit functions. The triangularly shaped fragment of the honeycomb lattice forms the set of nodes of the honeycomb Fourier-Weyl and Hartley-Weyl discrete transforms.…”

“…Discrete orthogonality and transforms of the Weyl orbit functions over a fragment of the refined weight lattice are formulated in connection with the conformal field theory in [17]. Discrete orthogonality and transforms of the Hartley-Weyl orbit functions over a fragment of the refined dual weight lattice are obtained in a more general theoretical setting in [11] and explicit formulations of discrete orthogonality and transforms of both Weyl and Hartley-Weyl functions over a fragment of the refined dual root lattice are achieved in [12]. For the crystallographic root system A 2 , the dual weight and weight lattices as well as the dual root and root lattices coincide.…”