Central Splitting of A₂ Discrete Fourier–Weyl Transforms

Abstract

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 $A_2$ 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 $A_2$ 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.

1. Introduction

The purpose of this article is to construct a decomposition of the discrete weight lattice Fourier–Weyl transforms [1,2] associated with the crystallographic reflection group $A_2$ into the corresponding splitting transforms. The decomposition is achieved via the central splitting [3] of a given function that is sampled on points from the triangular fragment of the rescaled $A_2$ weight lattice. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms serve as the first steps for development of fast recursive evaluation algorithms [4].

The discrete Fourier transforms on lattices became possible after the uniform tori discretization of semisimple Lie groups was developed for cosine functions [5,6] and expanded to sine functions [3]. This approach provided the foundation for the Fourier calculus of (anti)symmetric orbit functions that have been developed on the points of the refined weight [7], dual weight [1,8] and dual root lattices [9,10]. The three families of the Weyl orbit C-, S- and E-functions [11,12,13] together with their seven hybrid versions [8,14,15], which constitute the kernels of the discrete Fourier–Weyl transforms, have been formulated and described in full generality. The periodicity, (anti)symmetry and boundary properties of the (anti)symmetric Weyl orbit functions labeled by dominant weights from the weight lattices provide a framework for the generalized discrete cosine and sine transforms pertinent to digital data processing. For the crystallographic reflection group $A_2$, since the fundamental domain of the affine Weyl group has a shape of an equilateral triangle, the unique boundary behavior of the two-variable complex-valued Weyl orbit functions is imposed by the Dirichlet and Neumann boundary conditions [11,12]. As finite sums of the multivariate exponential functions, the Weyl orbit functions are usually sampled on points of the rescaled (dual) weight lattice comprised within the fundamental region and labeled by weights from the magnified fundamental domain [1,8]. The decompositions of these weight lattice transforms to the related root-lattice-based transforms are founded on the idea of the central splitting.

The central splitting of any generic linear combination of the Weyl orbit functions ([3] Section 7) is applied to discretized functions that are expressed as combinations of orbit functions via the weight lattice Fourier–Weyl transforms. Depending on the order of the center of the associated compact Lie group [1,3], a shortcut to a data-decomposition is provided by the central splitting of any function f, that is sampled on the weight lattice points contained in the $A_2$ fundamental region, into a sum of three decomposition functions $f_0$, $f_1$ and $f_2$. This consideration stems from the fact that the Fourier–Weyl transforms of each f-component depend only on one congruence class of the weight lattice labels which characterize the corresponding Weyl orbit functions [3]. Due to the interlaced additional argument symmetry of the decomposition functions $f_0$, $f_1$ and $f_2$ under the action of the extended affine Weyl group, their values are determined on the points of the refined weight lattice comprised within the corresponding kite-shaped fundamental domain. As a result, the reduction of the weight lattice transform is achieved, and the original decomposition of the function f splits into three smaller problems that are tackled faster and more efficiently. Since the root and dual root lattices as well as the weight and dual weight lattices of the root system $A_2$ coincide, the splitting Fourier–Weyl transforms that process the component functions are deduced from the dual-root lattice transforms. In particular, it appears that the crucial exact forms for the $A_2$ splitting transforms are obtained by reversing the roles of the points and labels inherent in the generalized dual root lattice transforms [9,10].

The Fourier–Weyl transforms provide efficient tools for the corresponding Fourier methods [16] and the associated multivariate Chebyshev polynomial methods [15]. For two-variable Weyl orbit functions, the discrete and continuous cosine transforms, together with their continuous interpolations, are formulated in [2,17] and later extended to sine transforms in [18]. Furthermore, the interpolation applications of extended Weyl orbit functions have been recently tested for the discrete armchair honeycomb lattice transforms [19]. In light of the relevance of such applications to digital data processing, in this paper, the developed $A_2$ splitting transforms are examined for their interpolation properties. In general, the discrete and continuous Fourier transforms are fundamental to numerous areas of science and technology involving data processing. Their pertinent applications span various domains, such as signal processing [20,21], pattern recognition [22], encryption [23], image and video compression/decompression [24], magnetic resonance imaging [25], ultrasound imaging [26] and watermarking [27,28]. Ever since the discovery of the fast Fourier transforms [29], attempts to modify and improve the evaluation algorithms have been continuously undertaken [30,31,32]. Since the most ubiquitous recursive algorithms contain as their embedded steps the decompositions directly related to the central splitting idea [4], the developed decomposition directly contributes to formulation of similar recursive evaluation algorithms on the equilateral triangle. Moreover, the decomposition techniques derived for the $A_2$ case represent a foundation for further generalizations of recursive formalisms to analogous 2D and 3D cases.

The paper is organized as follows. In Section 2, pertinent information concerning the $A_2$ root and weight lattices together with the induced crystallographic reflection group $A_2$ and its infinite extensions is recalled. The sets of points and weights of the weight lattice and splitting transforms are introduced. In Section 3, the definition of two families of (anti)symmetric orbit functions and their discrete orthogonality relations are provided, the weight lattice Fourier–Weyl transforms and their splitting versions are presented. In Section 4, the central splitting of the discrete weight lattice transforms is described in detail. The corresponding decomposition of the unitary matrices associated with the normalized discrete Fourier–Weyl transforms is deduced. Comments and follow-up questions are covered in the last section.

2. Splitting Weight and Point Sets

2.1. Root and Weight Lattices

The fundamental concepts and pertinent properties of the simple Lie algebra $A_2$ and its root system are provided in [33,34]. The simple roots ${\alpha }_1$ and ${\alpha }_2$ of $A_2$ form a non-orthogonal $\alpha$-basis in the two-dimensional Euclidean space ${\mathbb{R}}^2$. The geometric properties of the simple roots of $A_2$, such as their lengths and the relative angle between them, are provided by the standard scalar product $\langle ,\rangle$:

$$\langle {\alpha }_1,{\alpha }_1\rangle =\langle {\alpha }_2,{\alpha }_2\rangle =2,\langle {\alpha }_1,{\alpha }_2\rangle =-1.$$

(1)

In addition, it is essential to introduce the basis of fundamental weights that is known as $\omega$-basis. The $\omega$-basis comprises the vectors ${\omega }_1$ and ${\omega }_2$, and the duality between $\alpha$- and $\omega$-bases is provided by the relation

$$\langle {\omega }_i,{\alpha }_j\rangle ={\delta }_{ij},i,j\in \{1,2\}.$$

(2)

The transformation between $\alpha$- and $\omega$-basis is provided by the Cartan matrix as

$$\alpha =C\omega ,\omega =C^{-1}\alpha .$$

(3)

The Cartan matrix of $A_2$ and its inverse have the form

$$C=\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right),C^{-1}=\frac{1}{3}\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right).$$

(4)

The vectors of $\alpha$-basis can be explicitly written in terms of ${\omega }_1$ and ${\omega }_2$ as

$${\alpha }_1=2{\omega }_1-{\omega }_2,{\alpha }_2=-{\omega }_1+2{\omega }_2.$$

For the $A_2$ root system, the notions of the dual weights ${\omega }_k^{\vee }$ [1,10] together with the dual roots ${\alpha }_k^{\vee }$, $k\in \{1,2\}$ coincide with the weights and roots, respectively,

$${\omega }_k^{\vee }={\omega }_k,{\alpha }_k^{\vee }={\alpha }_k.$$

(5)

Using the inverse transform, the vectors of the $\omega$-basis are provided in terms of ${\alpha }_1$ and ${\alpha }_2$ as

$${\omega }_1=\frac{2}{3}{\alpha }_1+\frac{1}{3}{\alpha }_2,{\omega }_2=\frac{1}{3}{\alpha }_1+\frac{2}{3}{\alpha }_2.$$

Similarly to the simple roots, the fundamental weights ${\omega }_1$ and ${\omega }_2$ are characterized by their lengths and the relative angle between them as

$$\langle {\omega }_1,{\omega }_1\rangle =\langle {\omega }_2,{\omega }_2\rangle =\frac{2}{3},\langle {\omega }_1,{\omega }_2\rangle =\frac{1}{3}.$$

(6)

The scalar product of any two vectors $a=a_1{\omega }_1+a_2{\omega }_2$ and $b=b_1{\omega }_1+b_2{\omega }_2$ given in the $\omega$-basis has the form

$$\langle a,b\rangle =\frac{1}{3}(2a_1{b}_1+a_1{b}_2+a_2{b}_1+2a_2{b}_2).$$

(7)

All integer linear combinations of the simple roots ${\alpha }_1$ and ${\alpha }_2$ of $A_2$ form the root lattice $Q\subset {\mathbb{R}}^2$,

$$Q=\mathbb{Z}{\alpha }_1+\mathbb{Z}{\alpha }_2.$$

Subsequently, the weight lattice P is provided by the set of all integer linear combinations of fundamental weights ${\omega }_1$ and ${\omega }_2$ as follows,

$$P=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2.$$

The weight lattice P decomposes into a union of the root lattice Q together with its shifted copies $(Q+{\omega }_i)$, $i\in \{1,2\}$ as

$$P=Q\cup (Q+{\omega }_1)\cup (Q+{\omega }_2).$$

(8)

Hence, the points of the lattice P naturally split into three congruence classes $P_k$, $k\in \{0,1,2\}$ with respect to each component of the union (8) as

$$x=x_1{\omega }_1+x_2{\omega }_2\in P_k,x_1+2x_2=kmod3.$$

(9)

The reflections $r_i$, $i\in \{1,2\}$ across the hyperplanes orthogonal to the simple root ${\alpha }_i$ and passing through the origin are linear maps that for any point $x\in {\mathbb{R}}^2$ are given as

$$r_{i}x=x-\langle x,{\alpha }_i\rangle {\alpha }_i.$$

(10)

The Weyl group W of $A_2$ is generated by the reflections $r_i$. The action of W on any point $x=x_1{\omega }_1+x_2{\omega }_2\in P$ produces the images of x that form the orbit of points equidistant from the origin,

$$Wx=\{(x_1,x_2),(-x_1,x_1+x_2),(x_1+x_2,-x_2),(x_2,-x_1-x_2),(-x_1-x_2,x_1),(-x_1,-x_2)\}.$$

(11)

The lattices Q and P are invariant under the action of the Weyl group,

$$WQ=Q,WP=P.$$

(12)

The determinant c of the Cartan matrix C coincides with the order of the quotient group $P/Q$,

$$c=detC=\left|P/Q\right|=3.$$

(13)

The three representative elements $z_k$, $k\in \{0,1,2\}$ of the quotient group $P/Q$ are according to the decomposition (8) chosen as

$$z_0=0,z_1={\omega }_1,z_2={\omega }_2.$$

(14)

2.2. Splitting Weight Sets

The affine Weyl group $W_Q^{\mathrm{aff}}$ is an infinite extension of the Weyl group W by shift vectors of the root lattice Q,

$$W_Q^{\mathrm{aff}}=Q⋊W.$$

For $q\in Q$ and $w\in W$, the action of an element $T\left(q\right)w$ on any $x\in {\mathbb{R}}^2$ is denoted as

$$T\left(q\right)w·x=wx+q.$$

The affine Weyl group is generated by the reflections $r_i$, $i\in \{1,2\}$ and the affine reflection $r_0$ provided by

$$r_{0}x=r_{\xi }x+\xi ,r_{\xi }x=x-\langle x,\xi \rangle \xi ,$$

(15)

where $r_{\xi }$ represents the reflection across the hyperplane orthogonal to the highest root $\xi ={\alpha }_1+{\alpha }_2$.

The affine reflections $r_{{\alpha }_i}$ across the hyperplanes orthogonal to the simple roots and passing through ${\alpha }_i/2$, $i\in \{1,2\}$ are given by the formula

$$r_{{\alpha }_i}=r_{i}x+{\alpha }_i.$$

(16)

The fundamental region $F_Q\subset {\mathbb{R}}^2$ of $W_Q^{\mathrm{aff}}$ is represented by an equilateral triangle with the vertices $\{0,{\omega }_1,{\omega }_2\}$. Such a region contains precisely one point of each $W_Q^{\mathrm{aff}}$-orbit,

$$F_Q=\left\{x_1{\omega }_1+x_2{\omega }_2|x_1,x_2\ge 0,x_1+x_2\le 1\right\}.$$

(17)

For any $M\in \mathbb{N}$, the weight sets ${\Lambda }_M$ and ${\tilde{\Lambda }}_M$ are determined by finite fragments of the weight lattice P contained within the magnified fundamental region $M{F}_Q$ and its interior $\mathrm{int}\left(M{F}_Q\right)$, respectively,

$$\begin{array}{l}{\Lambda }_M=P\cap M{F}_Q,\end{array}$$

(18)

$$\begin{array}{l}{\tilde{\Lambda }}_M=P\cap \mathrm{int}\left(M{F}_Q\right).\end{array}$$

(19)

The explicit forms of the weight sets ${\Lambda }_M$ and ${\tilde{\Lambda }}_M$ are as follows,

$$\begin{array}{l}{\Lambda }_M=\left\{{\lambda }_1{\omega }_1+{\lambda }_2{\omega }_2|{\lambda }_0,{\lambda }_1,{\lambda }_2\in {\mathbb{Z}}^{\ge 0},{\lambda }_0+{\lambda }_1+{\lambda }_2=M\right\},\end{array}$$

(20)

$$\begin{array}{l}{\tilde{\Lambda }}_M=\left\{{\lambda }_1{\omega }_1+{\lambda }_2{\omega }_2|{\lambda }_0,{\lambda }_1,{\lambda }_2\in \mathbb{N},{\lambda }_0+{\lambda }_1+{\lambda }_2=M\right\}.\end{array}$$

(21)

The weights in the weight sets (20) are determined by their Kac coordinates,

$$\lambda =[{\lambda }_0,{\lambda }_1,{\lambda }_2]\in {\Lambda }_M.$$

(22)

The splitting weight sets ${\Lambda }_M^{\left(k\right)}$, $k\in \{0,1,2\}$, that form a disjoint decomposition of the weight set ${\Lambda }_M$,

$${\Lambda }_M={\Lambda }_M^{\left(0\right)}\cup {\Lambda }_M^{\left(1\right)}\cup {\Lambda }_M^{\left(2\right)},$$

(23)

are defined by finite fragments of the shifted copies (8) of the root lattice Q comprised within the region $M{F}_Q$,

$${\Lambda }_M^{\left(k\right)}=P_k\cap M{F}_Q.$$

(24)

The interior splitting weight sets ${\tilde{\Lambda }}_M^{\left(k\right)}$, $k\in \{0,1,2\}$, that form a disjoint decomposition of the weight set ${\tilde{\Lambda }}_M$,

$${\tilde{\Lambda }}_M={\tilde{\Lambda }}_M^{\left(0\right)}\cup {\tilde{\Lambda }}_M^{\left(1\right)}\cup {\tilde{\Lambda }}_M^{\left(2\right)},$$

(25)

are contained within the interior $\mathrm{int}\left(M{F}_Q\right)$,

$${\tilde{\Lambda }}_M^{\left(k\right)}=P_k\cap \mathrm{int}\left(M{F}_Q\right).$$

(26)

Taking into account the congruence decomposition relation (9) together with the forms of the weight sets (20) and (21), the weight sets ${\Lambda }_M^{\left(k\right)}$ and ${\tilde{\Lambda }}_M^{\left(k\right)}$ are explicitly described as

$$\begin{array}{l}{\Lambda }_M^{\left(k\right)}=\left\{{\lambda }_1{\omega }_1+{\lambda }_2{\omega }_2|{\lambda }_0,{\lambda }_1,{\lambda }_2\in {\mathbb{Z}}^{\ge 0},{\lambda }_0+{\lambda }_1+{\lambda }_2=M,{\lambda }_1+2{\lambda }_2=k\mathrm{mod}3\right\},\end{array}$$

(27)

$$\begin{array}{l}{\tilde{\Lambda }}_M^{\left(k\right)}=\left\{{\lambda }_1{\omega }_1+{\lambda }_2{\omega }_2|{\lambda }_0,{\lambda }_1,{\lambda }_2\in \mathbb{N},{\lambda }_0+{\lambda }_1+{\lambda }_2=M,{\lambda }_1+2{\lambda }_2=k\mathrm{mod}3\right\}.\end{array}$$

(28)

The numbers of weights in the weight sets ${\Lambda }_M$ and ${\tilde{\Lambda }}_M$ are calculated in [1] as

$$\begin{array}{l}|{\Lambda }_M|=\frac{1}{2}(M^2+3M+2),\end{array}$$

(29)

$$\begin{array}{l}|{\tilde{\Lambda }}_M|=\frac{1}{2}(M^2-3M+2).\end{array}$$

(30)

The numbers of weights in the splitting weight sets ${\Lambda }_M^{\left(k\right)}$, $k\in \{0,1,2\}$ together with their interior versions are determined in the following proposition.

Proposition 1.

The numbers of weights contained in the splitting sets ${\Lambda }_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are determined as

$$|{\Lambda }_M^{\left(0\right)}|=\left\{\begin{array}{cc}\frac{1}{6}(M^2+3M+6)\hfill & M=0\mathrm{mod}3,\hfill \\ \frac{1}{6}(M^2+3M+2)\hfill & otherwise;\hfill \end{array}\right.$$

(31)

$$|{\Lambda }_M^{\left(1\right)}|=|{\Lambda }_M^{\left(2\right)}|=\left\{\begin{array}{cc}\frac{1}{6}(M^2+3M)\hfill & M=0\mathrm{mod}3,\hfill \\ \frac{1}{6}(M^2+3M+2)\hfill & otherwise.\hfill \end{array}\right.$$

(32)

The numbers of weights contained in the interior weight sets ${\tilde{\Lambda }}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are given by

$$|{\tilde{\Lambda }}_M^{\left(0\right)}|=\left\{\begin{array}{cc}\frac{1}{6}(M^2-3M+6)\hfill & M=0\mathrm{mod}3,\hfill \\ \frac{1}{6}(M^2-3M+2)\hfill & otherwise;\hfill \end{array}\right.$$

(33)

$$\left|{\tilde{\Lambda }}_M^{\left(1\right)}\right|=\left|{\tilde{\Lambda }}_M^{\left(2\right)}\right|=\left\{\begin{array}{cc}\frac{1}{6}(M^2-3M)\hfill & M=0\mathrm{mod}3,\hfill \\ \frac{1}{6}(M^2-3M+2)\hfill & otherwise.\hfill \end{array}\right.$$

(34)

Proof. 

Preserving the notation for the point sets $F_{Q^{\vee },M}^{\mathfrak{1}}$ and $F_{Q^{\vee },M}^{{\sigma }^e}$ of $A_2$ from [10], it holds that

$${\Lambda }_M^{\left(0\right)}=M{F}_{Q^{\vee },M}^{\mathfrak{1}},$$

(35)

$${\tilde{\Lambda }}_M^{\left(0\right)}=M{F}_{Q^{\vee },M}^{{\sigma }^e}.$$

(36)

The point sets $F_{Q^{\vee },M}^{\mathfrak{1}}(0,{\omega }_k^{\vee })$ and $F_{Q^{\vee },M}^{{\sigma }^e}(0,{\omega }_k^{\vee })$, $k\in \{1,2\}$ from [9] are related to the current weight sets ${\Lambda }_M^{\left(k\right)}$ and ${\tilde{\Lambda }}_M^{\left(k\right)}$ via the following relations,

$${\Lambda }_M^{\left(k\right)}=M{F}_{Q^{\vee },M}^{\mathfrak{1}}(0,{\omega }_k^{\vee }),$$

(37)

$${\tilde{\Lambda }}_M^{\left(k\right)}=M{F}_{Q^{\vee },M}^{{\sigma }^e}(0,{\omega }_k^{\vee }).$$

(38)

Thus, the current counting formulas are calculated from Theorems 5.4 and 5.5 in [10] and from Theorems 2 and 3 in [9]. □

A discrete function $h_M:{\Lambda }_M\to \mathbb{N}$ is defined for $\lambda \in {\Lambda }_M$ by its values on the Kac coordinates (22) with ${\lambda }_0,{\lambda }_1,{\lambda }_2\ne 0$ as

$$h_M\left(\lambda \right)=\left\{\begin{array}{cc}1,\hfill & \lambda =[{\lambda }_0,{\lambda }_1,{\lambda }_2],\hfill \\ 2,\hfill & \lambda =[0,{\lambda }_1,{\lambda }_2],[{\lambda }_0,0,{\lambda }_2],[{\lambda }_0,{\lambda }_1,0],\hfill \\ 6,\hfill & \lambda =[0,0,{\lambda }_2],[{\lambda }_0,0,0],[0,{\lambda }_1,0].\hfill \end{array}\right.$$

(39)

Since the h-function depends only on the number of zero-valued Kac coordinates of the weight $\lambda \in {\Lambda }_M$, it is invariant under the cyclic permutations of $[{\lambda }_0,{\lambda }_1,{\lambda }_2]$. The weight sets ${\Lambda }_6$, ${\tilde{\Lambda }}_6$ and their decompositions ${\Lambda }_6^{\left(k\right)}$, ${\tilde{\Lambda }}_6^{\left(k\right)}$, $k\in \{0,1,2\}$ are depicted in Figure 1.

2.3. Splitting Point Sets

The extension of the Weyl group W by shift vectors of the weight lattice P yields the extended affine Weyl group $W_P^{\mathrm{aff}}$,

$$W_P^{\mathrm{aff}}=P⋊W.$$

For $p\in P$ and $w\in W$, the action of an element $T\left(p\right)w\in W_P^{\mathrm{aff}}$ on any $x\in {\mathbb{R}}^2$ is defined as

$$T\left(p\right)w·x=wx+p.$$

The abelian group $\Gamma \subset W_P^{\mathrm{aff}}$ is the cyclic group of order 3,

$$\Gamma =\{{\gamma }_0,{\gamma }_1,{\gamma }_2\},$$

(40)

and its elements are expressed by means of the generating reflections and weight lattice translations as follows,

$${\gamma }_0=T\left(0\right)1,{\gamma }_1=T\left({\omega }_1\right)r_1{r}_2,{\gamma }_2=T\left({\omega }_2\right){\left(r_1{r}_2\right)}^2.$$

(41)

The elements of the group $\Gamma$ are equivalently expressed by application of the affine reflections $r_{{\alpha }_1}$, $r_{{\alpha }_2}$ and $r_0$ on the shifted points $x+{\omega }_1$ and $x+{\omega }_2$,

$$\begin{array}{cc}\hfill r_0{r}_{{\alpha }_1}(x+{\omega }_1)& ={\gamma }_{1}x,\hfill \\ \hfill r_0{r}_{{\alpha }_2}(x+{\omega }_2)& ={\gamma }_{2}x.\hfill \end{array}$$

(42)

The points shifted by ${\omega }_1$ and ${\omega }_2$ are brought back to the fundamental region $F_Q$ as shown in Figure 2.

The group $\Gamma$ is isomorphic to the quotient group $P/Q$,

$$\Gamma \cong P/Q.$$

(43)

The fundamental domain $F_P$ of the action of $W_P^{\mathrm{aff}}$ on ${\mathbb{R}}^2$ is the kite-shaped region comprised within the triangle $F_Q$ that contains exactly one point of each $W_P^{\mathrm{aff}}$-orbit,

$$F_P=\left\{x_1{\omega }_1+x_2{\omega }_2\in F_Q|(2x_1+x_2<1,x_1+2x_2<1)\vee (2x_1+x_2=1,x_1\ge x_2)\right\}.$$

(44)

The fundamental domain $F_P$, with the omitted central point of the triangle $F_Q$, forms the domain $F_P^{\prime }$,

$$F_P^{\prime }=\left\{x_1{\omega }_1+x_2{\omega }_2\in F_Q|(2x_1+x_2<1,x_1+2x_2<1)\vee (2x_1+x_2=1,x_1>x_2)\right\}.$$

(45)

For any $M\in N$, the point sets $F_M$ and ${\tilde{F}}_M$ are described by the refined finite fragments of the weight lattice contained within the fundamental region $F_Q$ and its interior, respectively,

$$F_M=\frac{1}{M}P\cap F_Q,$$

(46)

$${\tilde{F}}_M=\frac{1}{M}P\cap \mathrm{int}\left(F_Q\right).$$

(47)

The point sets $F_M$ and ${\tilde{F}}_M$ can be explicitly written as

$$F_M=\left\{\frac{s_1}{M}{\omega }_1+\frac{s_2}{M}{\omega }_2|s_0,s_1,s_2\in {\mathbb{Z}}^{\ge 0},s_0+s_1+s_2=M\right\},$$

(48)

$${\tilde{F}}_M=\left\{\frac{s_1}{M}{\omega }_1+\frac{s_2}{M}{\omega }_2|s_0,s_1,s_2\in \mathbb{N},s_0+s_1+s_2=M\right\}.$$

(49)

The points of the point sets (48) and (49) are conveniently described by their Kac coordinates,

$$s=[s_0,s_1,s_2]\in F_M.$$

(50)

Direct comparison of the expressions for the point and weight sets (20), (21), (48) and (49) yield for the corresponding cardinalities that

$$|F_M|=|{\Lambda }_M|,$$

(51)

$$|{\tilde{F}}_M|=|{\tilde{\Lambda }}_M|.$$

(52)

Note that there is at most one point $s^{\mathrm{fix}}\in F_M$ whose coordinates satisfy the relation $s_0=s_1=s_2=M/3$. Such a point is fixed by the action of $\Gamma$, and it is found in the center of the triangle $F_Q$, if an integer M is divisible by 3.

The splitting point set $F_M^{\left(0\right)}$ is determined by the points from $F_M$ contained within the kite-shaped fundamental domain of the extended affine Weyl group $F_P,$

$$F_M^{\left(0\right)}=F_M\cap F_P$$

(53)

and the splitting point sets $F_M^{\left(1\right)}$ and $F_M^{\left(2\right)}$ are formed by the points from $F_M$ included in the region $F_P^{\prime },$

$$F_M^{\left(1\right)}=F_M^{\left(2\right)}=F_M\cap F_P^{\prime }.$$

(54)

The interior splitting point sets ${\tilde{F}}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are defined similarly as

$${\tilde{F}}_M^{\left(0\right)}={\tilde{F}}_M\cap F_P,$$

(55)

$${\tilde{F}}_M^{\left(1\right)}={\tilde{F}}_M^{\left(2\right)}={\tilde{F}}_M\cap F_P^{\prime }.$$

(56)

Using Formulas (44) and (45) together with the Kac coordinates (50), the point sets $F_M^{\left(k\right)}$ are expressed as

$$\begin{array}{cc}\hfill F_M^{\left(0\right)}=& \left\{[s_0,s_1,s_2]\in F_M|(s_0>s_1,s_0>s_2)\vee (s_0=s_1\ge s_2)\right\},\hfill \\ \hfill F_M^{\left(1\right)}=F_M^{\left(2\right)}=& \left\{[s_0,s_1,s_2]\in F_M|(s_0>s_1,s_0>s_2)\vee (s_0=s_1>s_2)\right\}.\hfill \end{array}$$

(57)

Similarly, the interior point sets ${\tilde{F}}_M^{\left(k\right)}$ are explicitly written as

$$\begin{array}{cc}\hfill {\tilde{F}}_M^{\left(0\right)}=& \left\{[s_0,s_1,s_2]\in {\tilde{F}}_M|(s_0>s_1,s_0>s_2)\vee (s_0=s_1\ge s_2)\right\},\hfill \\ \hfill {\tilde{F}}_M^{\left(1\right)}={\tilde{F}}_M^{\left(2\right)}=& \left\{[s_0,s_1,s_2]\in {\tilde{F}}_M|(s_0>s_1,s_0>s_2)\vee (s_0=s_1>s_2)\right\}.\hfill \end{array}$$

(58)

The cardinalities of the splitting point sets are related to the numbers of elements of the corresponding weight sets in the following proposition.

Proposition 2.

The numbers of points contained in the splitting sets $F_M^{\left(k\right)}$ and ${\tilde{F}}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ coincide with the numbers of weights in the splitting sets ${\Lambda }_M^{\left(k\right)}$ and ${\tilde{\Lambda }}_M^{\left(k\right)}$, respectively,

$$|F_M^{\left(k\right)}|=|{\Lambda }_M^{\left(k\right)}|,$$

(59)

$$\left|{\tilde{F}}_M^{\left(k\right)}\right|=\left|{\tilde{\Lambda }}_M^{\left(k\right)}\right|.$$

(60)

Proof. 

Preserving the notation for the weight sets from [10], it holds that

$$F_M^{\left(0\right)}=\frac{1}{M}{\Lambda }_{P,M}^{\mathfrak{1}},$$

(61)

$${\tilde{F}}_M^{\left(0\right)}=\frac{1}{M}{\Lambda }_{P,M}^{{\sigma }^e},$$

(62)

and the weight sets from [9] are related to the current point sets $F_M^{\left(k\right)}$, ${\tilde{F}}_M^{\left(k\right)}$, $k\in \{1,2\}$ via the relations

$$F_M^{\left(k\right)}=\frac{1}{M}{\Lambda }_{P,M}^{\mathfrak{1}}(0,{\omega }_k^{\vee }),$$

(63)

$${\tilde{F}}_M^{\left(k\right)}=\frac{1}{M}{\Lambda }_{P,M}^{{\sigma }^e}(0,{\omega }_k^{\vee }).$$

(64)

Thus, the current cardinality equalities are obtained from relations (61), (62) and Theorem 5.4 in [10] and from relations (63), (64) together with Theorem 2 in [9]. □

The elements (40) of the abelian group $\Gamma$ preserve the point set $F_M$ and act on any point $s\in F_M$ as cyclic permutations of the Kac coordinates $[s_0,s_1,s_2]$,

$$\begin{array}{cc}\hfill {\gamma }_0[s_0,s_1,s_2]& =[s_0,s_1,s_2],\hfill \\ \hfill {\gamma }_1[s_0,s_1,s_2]& =[s_2,s_0,s_1],\hfill \\ \hfill {\gamma }_2[s_0,s_1,s_2]& =[s_1,s_2,s_0].\hfill \end{array}$$

A discrete function $\epsilon :F_M\to \mathbb{N}$ is defined for $s\in F_M$ by its values on the Kac coordinates (50) with $s_0,s_1,s_2\ne 0$ as

$$\epsilon \left(s\right)=\left\{\begin{array}{cc}6,\hfill & s=[s_0,s_1,s_2],\hfill \\ 3,\hfill & s=[0,s_1,s_2],[s_0,0,s_2],[s_0,s_1,0],\hfill \\ 1,\hfill & s=[0,0,s_2],[s_0,0,0],[0,s_1,0].\hfill \end{array}\right.$$

(65)

Similarly to the h-function (39), the ε-function is invariant under the permutation of the Kac coordinates [s₀,s₁,s₂]. Note also that the h- and ε-functions are related for $\lambda \in {\Lambda }_M$ by the formula

$$h_M\left(\lambda \right)=6{\epsilon }^{-1}\left(\frac{\lambda }{M}\right).$$

(66)

A discrete function $d:F_M\to \mathbb{N}$ depends only on the equality of Kac coordinates $s_0=s_1=s_2$, and it takes one of the following two values,

$$d\left(s\right)=\left\{\begin{array}{cc}3,\hfill & s_0=s_1=s_2,\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(67)

The point sets $F_6$, ${\tilde{F}}_6$ together with their subsets $F_6^{\left(k\right)}$ and ${\tilde{F}}_6^{\left(k\right)}$, $k\in \{0,1,2\}$ are depicted in Figure 1.

3. Weight Lattice Fourier–Weyl Transforms

3.1. C- and S-Functions

The Weyl orbit functions and their pertinent properties have been extensively studied in several papers, see for instance [11,12,13]. The orbit functions of $A_2$ can be written as sums of multivariate exponential functions $e^{2\pi \mathrm{i}\langle b,x\rangle }$, $b\in {\mathbb{R}}^2$ of the variable $x\in {\mathbb{R}}^2$. For any label $b\in {\mathbb{R}}^2$ and a point $x\in {\mathbb{R}}^2$, consider the two families of complex-valued smooth functions,

$${\Phi }_b\left(x\right)=\sum _{w\in W}{e}^{2\pi \mathrm{i}\langle wb,x\rangle },$$

(68)

$${\phi }_b\left(x\right)=\sum _{w\in W}det\left(w\right)e^{2\pi \mathrm{i}\langle wb,x\rangle }.$$

(69)

The functions (68) and (69), known as the C- and S-functions, correspond to the classical univariate cosine and sine functions. For the C-functions, all terms have positive sign; hence, they are referred to as symmetric orbit functions. The signs of terms composing S-functions depend on $det\left(w\right)$, and they are called anti-symmetric orbit functions.

The duality of the Weyl orbit functions [11,12] is expressed as

$${\Phi }_b\left(x\right)={\Phi }_x\left(b\right),{\phi }_b\left(x\right)={\phi }_x\left(b\right),$$

(70)

and, for any real-valued parameter $t\in \mathbb{R}$, the scaling symmetry is determined by

$${\Phi }_b\left(tx\right)={\Phi }_{tb}\left(x\right),{\phi }_b\left(tx\right)={\phi }_{tb}\left(x\right).$$

(71)

Considering the scalar product (7) and expression for the Weyl orbit (11), the C- and S-functions can be written for the label $b=b_1{\omega }_1+b_2{\omega }_2$ and the point $x=x_1{\omega }_1+x_2{\omega }_2$ in the $\omega$-basis explicitly as

$$\begin{array}{cc}\hfill {\Phi }_b\left(x\right)=& e^{\frac{2}{3}\pi \mathrm{i}\left((2b_1+b_2)x_1+(b_1+2b_2)x_2\right)}+e^{\frac{2}{3}\pi \mathrm{i}((-b_1+b_2)x_1+(b_1+2b_2)x_2)}+e^{\frac{2}{3}\pi \mathrm{i}((-b_1-2b_2)x_1+(b_1-b_2)x_2)}\hfill \\ \hfill & +e^{\frac{2}{3}\pi \mathrm{i}((-b_1-2b_2)x_1+(-2b_1-b_2)x_2)}+e^{\frac{2}{3}\pi \mathrm{i}((-b_1+b_2)x_1+(-2b_1-b_2)x_2)}+e^{\frac{2}{3}\pi \mathrm{i}((2b_1+b_2)x_1+(b_1-b_2)x_2)},\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\phi }_b\left(x\right)=& e^{\frac{2}{3}\pi \mathrm{i}((2b_1+b_2)x_1+(b_1+2b_2)x_2)}-e^{\frac{2}{3}\pi \mathrm{i}((-b_1+b_2)x_1+(b_1+2b_2)x_2)}+e^{\frac{2}{3}\pi \mathrm{i}((-b_1-2b_2)x_1+(b_1-b_2)x_2)}\hfill \\ \hfill & -e^{\frac{2}{3}\pi \mathrm{i}((-b_1-2b_2)x_1+(-2b_1-b_2)x_2)}+e^{\frac{2}{3}\pi \mathrm{i}((-b_1+b_2)x_1+(-2b_1-b_2)x_2)}-e^{\frac{2}{3}\pi \mathrm{i}((2b_1+b_2)x_1+(b_1-b_2)x_2)}.\hfill \end{array}$$

(73)

The C- and S-functions are (anti)symmetric with respect to the action of any Weyl group element $w\in W,$

$${\Phi }_b\left(wx\right)={\Phi }_b\left(x\right),{\phi }_b\left(wx\right)=det\left(w\right){\phi }_b\left(x\right).$$

(74)

For the discretized labels from the weight lattice $\lambda \in P$, the C- and S-functions are invariant under the translations by any root-lattice vector $q\in Q,$

$${\Phi }_{\lambda }(x+q)={\Phi }_{\lambda }\left(x\right),{\phi }_{\lambda }(x+q)={\phi }_{\lambda }\left(x\right).$$

(75)

Therefore, both families of the orbit functions ${\Phi }_{\lambda }$ and ${\phi }_{\lambda }$, $\lambda \in P$ are (anti)symmetric with respect to the affine Weyl group $W_Q^{\mathrm{aff}}$. Hence, the C- and S-functions are usually restricted to the corresponding fundamental domain (17). The normal derivative of the C-functions and values of the S-functions vanish on the boundary of $F_Q$. Symmetries of the orbit functions under the action of the $\Gamma$-group are formulated in the following proposition.

Proposition 3.

For any $\lambda \in P_k$, ${\gamma }_j\in \Gamma$, $j,k\in \{0,1,2\}$ and $x\in {\mathbb{R}}^2$, it holds that

$${\Phi }_{\lambda }\left({\gamma }_{j}x\right)=e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }{\Phi }_{\lambda }\left(x\right),$$

(76)

$${\phi }_{\lambda }\left({\gamma }_{j}x\right)=e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }{\phi }_{\lambda }\left(x\right).$$

(77)

Proof. 

Taking into account identity (5), the representative elements $z_j$, $j\in \{0,1,2\}$ constitute according to [9] admissible shifts of the root lattice of $A_2$. Thus, the invariance property ([9] Equation (48)) is specialized for any $\lambda \in P$ and $w\in W$ as

$$e^{2\pi \mathrm{i}\langle w\lambda ,z_j\rangle }=e^{2\pi \mathrm{i}\langle \lambda ,z_j\rangle }.$$

(78)

Moreover, for any weight $\lambda =z_k+q\in P_k$ with $q\in Q$, the $\mathbb{Z}$-duality relation (2) of the weight and root lattices guarantees that

$$e^{2\pi \mathrm{i}\langle \lambda ,z_j\rangle }=e^{2\pi \mathrm{i}\langle z_k+q,z_j\rangle }=e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }.$$

(79)

The action of the elements of the $\Gamma$-group of $A_2$ are for any $x\in {\mathbb{R}}^2$ from defining relation (41) given as

$${\gamma }_{j}x=w_{j}x+z_j,$$

(80)

while it holds that

$$det\left(w_j\right)=1.$$

(81)

The symmetry properties of the C-functions under the action of the $\Gamma$-group are calculated for $\lambda \in P_k$ using (78)–(80) as

$${\Phi }_{\lambda }\left({\gamma }_{j}x\right)=\sum _{w\in W}{e}^{2\pi \mathrm{i}\langle w\lambda ,w_{j}x+z_j\rangle }=\sum _{w\in W}{e}^{2\pi \mathrm{i}\langle w\lambda ,w_{j}x\rangle }{e}^{2\pi \mathrm{i}\langle w\lambda ,z_j\rangle }=e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }{\Phi }_{\lambda }\left(x\right).$$

In addition, utilizing relation (81), the resulting symmetry properties of the S-functions are obtained via the calculation

$$\begin{array}{cc}\hfill {\phi }_{\lambda }\left({\gamma }_{j}x\right)& =\sum _{w\in W}det\left(w\right)e^{2\pi \mathrm{i}\langle w\lambda ,w_{j}x+z_j\rangle }=\sum _{w\in W}det\left(w\right)e^{2\pi \mathrm{i}\langle w\lambda ,w_{j}x\rangle }{e}^{2\pi \mathrm{i}\langle w\lambda ,z_j\rangle }\hfill \\ \hfill & =det\left(w_j\right)e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }{\phi }_{\lambda }\left(x\right).\hfill \end{array}$$

□

3.2. Discrete Orthogonality

The orthogonality relations of C- and S-functions whenever they are integrated over the fundamental region are described in [35]. Here we recall the discrete orthogonality relations of both families of functions, as well as define the discrete orthogonality of C- and S-functions summed over finite point sets comprised within the kite-shaped region $F_P$. Employing the ε-function (65), the scalar product of two functions $f,g:F_M\to \mathbb{C}$ on a refined fragment of the weight lattice (46) contained within the fundamental domain $F_Q$ is provided by the formula

$${\langle f,g\rangle }_{F_M}=\sum _{s\in F_M}\epsilon \left(s\right)f\left(s\right)\overline{g\left(s\right)}.$$

(82)

The Hilbert space ${\mathcal{H}}_M$ is the space of complex-valued functions $f:F_M\to \mathbb{C}$ equipped with the weighted scalar product (82).

Since the points of the interior point set ${\tilde{F}}_M$ retain their Kac coordinates non-zero, the discrete ε-function takes according to (65) the constant value,

$$\epsilon (s)=6,s\in {\tilde{F}}_M.$$

(83)

Thus, the scalar product of two complex-valued functions $f,g:{\tilde{F}}_M\to \mathbb{C}$ on the interior point set ${\tilde{F}}_M$ is given as

$${\langle f,g\rangle }_{{\tilde{F}}_M}=6\sum _{s\in {\tilde{F}}_M}f\left(s\right)\overline{g\left(s\right)}.$$

(84)

The Hilbert space ${\tilde{\mathcal{H}}}_M$ is the space of complex-valued functions $f:{\tilde{F}}_M\to \mathbb{C}$ equipped with the scalar product (84).

For any weights $\lambda ,{\lambda }^{\prime }\in {\Lambda }_M$, the discrete orthogonality relations of the C-functions (68) with respect to the scalar product (82) are of the form [1],

$${\langle {\Phi }_{\lambda },{\Phi }_{{\lambda }^{\prime }}\rangle }_{F_M}=18M^2{h}_M\left(\lambda \right){\delta }_{\lambda ,{\lambda }^{\prime }},$$

(85)

and for any interior weights $\lambda ,{\lambda }^{\prime }\in {\tilde{\Lambda }}_M$, the discrete orthogonality relations of the S-functions (69) are given as

$${\langle {\phi }_{\lambda },{\phi }_{{\lambda }^{\prime }}\rangle }_{{\tilde{F}}_M}=18M^2{\delta }_{\lambda ,{\lambda }^{\prime }}.$$

(86)

The scalar product of two functions $f,g:F_M^{\left(k\right)}\to \mathbb{C}$, $k\in \{0,1,2\}$ on the refined fragments of the weight lattice is defined as

$${\langle f,g\rangle }_{F_M^{\left(k\right)}}=\sum _{s\in F_M^{\left(k\right)}}\epsilon \left(s\right)d^{-1}\left(s\right)f\left(s\right)\overline{g\left(s\right)}.$$

(87)

The Hilbert spaces ${\mathcal{H}}_M^{\left(k\right)}$ are the spaces of complex-valued functions $f:F_M^{\left(k\right)}\to \mathbb{C}$ equipped with the weighted scalar product (87).

Taking into account the interior ε-function values (83), the interior scalar product of two functions $f:{\tilde{F}}_M^{\left(k\right)}\to \mathbb{C},k\in \{0,1,2\}$ is given by

$${\langle f,g\rangle }_{{\tilde{F}}_M^{\left(k\right)}}=6\sum _{s\in {\tilde{F}}_M^{\left(k\right)}}{d}^{-1}\left(s\right)f\left(s\right)\overline{g\left(s\right)}.$$

(88)

Discrete orthogonality relations with respect to the scalar products (87) and (88) of the C- and S-functions are formulated in the following proposition.

Proposition 4.

The discrete orthogonality relations of C-functions labeled by any weights $\lambda ,{\lambda }^{\prime }\in {\Lambda }_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are of the form

$${\langle {\Phi }_{\lambda },{\Phi }_{{\lambda }^{\prime }}\rangle }_{F_M^{\left(k\right)}}=6M^2{h}_M\left(\lambda \right){\delta }_{\lambda ,{\lambda }^{\prime }}.$$

(89)

The discrete orthogonality relations of S-functions labeled by any weights $\lambda ,{\lambda }^{\prime }\in {\tilde{\Lambda }}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are determined as

$${\langle {\phi }_{\lambda },{\phi }_{{\lambda }^{\prime }}\rangle }_{{\tilde{F}}_M^{\left(k\right)}}=6M^2{\delta }_{\lambda ,{\lambda }^{\prime }}.$$

(90)

Proof. 

The discrete orthogonality relations of the $A_2$ orbit C-functions ([9] Theorem 4) and the corresponding Plancherel formulas ([9] Equation (172)) lead for points $a,a^{\prime }\in F_{Q^{\vee },M}^{\mathfrak{1}}(0,{\omega }_k^{\vee })$, $k\in \{1,2\}$ to the following relations,

$$\sum _{\mu \in {\Lambda }_{P,M}^{\mathfrak{1}}(0,{\omega }_k^{\vee })}{h}_M^{-1}\left(\mu \right)d^{-1}\left(\frac{\mu }{M}\right){\Phi }_{\mu }\left(a\right)\overline{{\Phi }_{\mu }\left(a^{\prime }\right)}=6M^2{\epsilon }^{-1}\left(a\right){\delta }_{a,a^{\prime }}.$$

(91)

Denoting $Ma=\lambda$, $M{a}^{\prime }={\lambda }^{\prime }$ and $\mu =Ms$, it follows from weight and point set relations (37) and (63) that $\lambda ,{\lambda }^{\prime }\in {\Lambda }_M^{\left(k\right)}$ and $s\in F_M^{\left(k\right)}$, $k\in \{1,2\}$. Thus, discrete orthogonality relation (91) is rewritten as

$$\sum _{s\in F_M^{\left(k\right)}}{h}_M^{-1}\left(Ms\right)d^{-1}\left(s\right){\Phi }_{Ms}\left(\frac{\lambda }{M}\right)\overline{{\Phi }_{Ms}\left(\frac{{\lambda }^{\prime }}{M}\right)}=6M^2{\epsilon }^{-1}\left(\frac{\lambda }{M}\right){\delta }_{\lambda ,{\lambda }^{\prime }}.$$

(92)

Utilizing the duality and scaling symmetry of orbit functions (70) and (71) together with the relation between h- and ε-functions (66), the discrete orthogonality (92) is reformulated as

$$\sum _{s\in F_M^{\left(k\right)}}\epsilon \left(s\right)d^{-1}\left(s\right){\Phi }_{\lambda }\left(s\right)\overline{{\Phi }_{{\lambda }^{\prime }}\left(s\right)}=6M^2{h}_M\left(\lambda \right){\delta }_{\lambda ,{\lambda }^{\prime }}.$$

(93)

The remaining case for $k=0$ and the orthogonality relations of the S-functions over the interior sets are shown similarly via Theorem 6.4 in [10] together with relations (35), (61) and interior set expressions (36), (38), (62) and (64). □

3.3. Splitting Transforms

Based on the discrete orthogonality relations of the two families of C- and S-functions (85) and (86), the discrete Fourier analysis is applied in the context of the Weyl group symmetry. Taking into account cardinality Formulas (51) and (52), the C- and S-functions constitute orthogonal bases of the Hilbert spaces ${\mathcal{H}}_M$ and ${\tilde{\mathcal{H}}}_M$, respectively. For a given complex-valued function $f:F_Q\to \mathbb{C}$, there exist two interpolating functions $\mathrm{I}{\left[f\right]}_M:{\mathbb{R}}^2\to \mathbb{C}$ and $\tilde{\mathrm{I}}{\left[f\right]}_M:{\mathbb{R}}^2\to \mathbb{C}$. The interpolating functions $\mathrm{I}{\left[f\right]}_M$ and $\tilde{\mathrm{I}}{\left[f\right]}_M$ are constructed as linear combinations of the Weyl orbit functions,

$$\mathrm{I}{\left[f\right]}_M\left(x\right)=\sum _{\lambda \in {\Lambda }_M}{c}_{\lambda }{\left[f\right]}_M{\Phi }_{\lambda }\left(x\right),$$

(94)

$$\tilde{\mathrm{I}}{\left[f\right]}_M\left(x\right)=\sum _{\lambda \in {\tilde{\Lambda }}_M}{\tilde{c}}_{\lambda }{\left[f\right]}_M{\phi }_{\lambda }\left(x\right),$$

(95)

which coincide with the function f on the interpolation nodes $F_M$ and ${\tilde{F}}_M$, respectively,

$$\mathrm{I}{\left[f\right]}_M\left(s\right)=f\left(s\right),s\in F_M,$$

(96)

$$\tilde{\mathrm{I}}{\left[f\right]}_M\left(s\right)=f\left(s\right),s\in {\tilde{F}}_M.$$

(97)

Due to discrete orthogonality relations (85) and (86), the frequency spectrum coefficients $c_{\lambda }{\left[f\right]}_M$, $\lambda \in {\Lambda }_M$ and ${\tilde{c}}_{\lambda }{\left[f\right]}_M$, $\lambda \in {\tilde{\Lambda }}_M$ are uniquely determined via the weight lattice Fourier–Weyl C- and S-transforms [1] of $A_2$,

$$c_{\lambda }{\left[f\right]}_M=\frac{{\langle f,{\Phi }_{\lambda }\rangle }_{F_M}}{{\langle {\Phi }_{\lambda },{\Phi }_{\lambda }\rangle }_{F_M}}={\left(18M^2{h}_M\left(\lambda \right)\right)}^{-1}\sum _{s\in F_M}\epsilon \left(s\right)f\left(s\right)\overline{{\Phi }_{\lambda }\left(s\right)},$$

(98)

$${\tilde{c}}_{\lambda }{\left[f\right]}_M=\frac{{\langle f,{\phi }_{\lambda }\rangle }_{{\tilde{F}}_M}}{{\langle {\phi }_{\lambda },{\phi }_{\lambda }\rangle }_{{\tilde{F}}_M}}={\left(3M^2\right)}^{-1}\sum _{s\in {\tilde{F}}_M}f\left(s\right)\overline{{\phi }_{\lambda }\left(s\right)}.$$

(99)

Formulation of the weight lattice splitting Fourier–Weyl C- and S-transforms on the Hilbert spaces ${\mathcal{H}}_M^{\left(k\right)}$ and ${\tilde{\mathcal{H}}}_M^{\left(k\right)}$ is founded on orbit function orthogonal bases constructed in the following proposition.

Proposition 5.

For each $k\in \{0,1,2\}$, the C-functions ${\Phi }_{\lambda }$, $\lambda \in {\Lambda }_M^{\left(k\right)}$ form an orthogonal basis of the Hilbert space ${\mathcal{H}}_M^{\left(k\right)}$, and the S-functions ${\phi }_{\lambda }$, $\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}$ form an orthogonal basis of the Hilbert space ${\tilde{\mathcal{H}}}_M^{\left(k\right)}$.

Proof. 

Discrete orthogonality relations of the C- and S-functions in Proposition 4 guarantee that the functions ${\Phi }_{\lambda }$, $\lambda \in {\Lambda }_M^{\left(k\right)}$ and ${\phi }_{\lambda }$, $\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}$ are linearly independent in the spaces ${\mathcal{H}}_M^{\left(k\right)}$ and ${\tilde{\mathcal{H}}}_M^{\left(k\right)}$, respectively. The dimensions of the functional Hilbert spaces ${\mathcal{H}}_M^{\left(k\right)}$ and ${\tilde{\mathcal{H}}}_M^{\left(k\right)}$ coincide with the cardinalities of the underlying point sets $F_M^{\left(k\right)}$ and ${\tilde{F}}_M^{\left(k\right)}$, and Proposition 2 provides that

$$\begin{array}{cc}\hfill dim{\mathcal{H}}_M^{\left(k\right)}& =\left|F_M^{\left(k\right)}\right|=\left|{\Lambda }_M^{\left(k\right)}\right|,\hfill \\ \hfill dim{\tilde{\mathcal{H}}}_M^{\left(k\right)}& =\left|{\tilde{F}}_M^{\left(k\right)}\right|=\left|{\tilde{\Lambda }}_M^{\left(k\right)}\right|.\hfill \end{array}$$

□

For a given complex-valued function $f:F_P\to \mathbb{C}$, utilizing the orthogonal bases of the discretized Weyl orbit functions, there exist six interpolating functions $\mathrm{I}{\left[f\right]}_M^{\left(k\right)}:{\mathbb{R}}^2\to \mathbb{C}$ and $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(k\right)}:{\mathbb{R}}^2\to \mathbb{C}$, $k\in \{0,1,2\}$. The interpolating functions $\mathrm{I}{\left[f\right]}_M^{\left(k\right)}$ and $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(k\right)}$ are constructed as linear combinations of the Weyl orbit functions,

$$\mathrm{I}{\left[f\right]}_M^{\left(k\right)}\left(x\right)=\sum _{\lambda \in {\Lambda }_M^{\left(k\right)}}{c}_{\lambda }{\left[f\right]}_M^{\left(k\right)}{\Phi }_{\lambda }\left(x\right),$$

(100)

$$\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(k\right)}\left(x\right)=\sum _{\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}}{\tilde{c}}_{\lambda }{\left[f\right]}_M^{\left(k\right)}{\phi }_{\lambda }\left(x\right),$$

(101)

that coincide with the function f on the interpolation nodes $F_M^{\left(k\right)}$ and ${\tilde{F}}_M^{\left(k\right)}$, respectively,

$$\mathrm{I}{\left[f\right]}_M^{\left(k\right)}\left(s\right)=f\left(s\right),s\in F_M^{\left(k\right)},$$

(102)

$$\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(k\right)}\left(s\right)=f\left(s\right),s\in {\tilde{F}}_M^{\left(k\right)}.$$

(103)

Obtained as the standard Fourier coefficients from Propositions 4 and 5, the frequency spectrum coefficients $c_{\lambda }{\left[f\right]}_M^{\left(k\right)}$, $\lambda \in {\Lambda }_M^{\left(k\right)}$ and ${\tilde{c}}_{\lambda }{\left[f\right]}_M^{\left(k\right)}$, $\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}$ are uniquely determined as

$$c_{\lambda }{\left[f\right]}_M^{\left(k\right)}=\frac{{\langle f,{\Phi }_{\lambda }\rangle }_{F_M^{\left(k\right)}}}{{\langle {\Phi }_{\lambda },{\Phi }_{\lambda }\rangle }_{F_M^{\left(k\right)}}}={\left(6M^2{h}_M\left(\lambda \right)\right)}^{-1}\sum _{s\in F_M^{\left(k\right)}}\epsilon \left(s\right)d^{-1}\left(s\right)f\left(s\right)\overline{{\Phi }_{\lambda }\left(s\right)},$$

(104)

$${\tilde{c}}_{\lambda }{\left[f\right]}_M^{\left(k\right)}=\frac{{\langle f,{\phi }_{\lambda }\rangle }_{{\tilde{F}}_M^{\left(k\right)}}}{{\langle {\phi }_{\lambda },{\phi }_{\lambda }\rangle }_{{\tilde{F}}_M^{\left(k\right)}}}=M^{-2}\sum _{s\in {\tilde{F}}_M^{\left(k\right)}}{d}^{-1}\left(s\right)f\left(s\right)\overline{{\phi }_{\lambda }\left(s\right)}.$$

(105)

Frequency spectrum coefficients Formulas (104) and (105) constitute the forward weight lattice splitting Fourier–Weyl C- and S-transforms, respectively. Interpolation properties of the splitting types of the Fourier–Weyl transforms are tested in the following example.

Example 1 (Interpolation by Splitting Transforms).

As a model function for the interpolation tests of the splitting transforms, the following real-valued function f on the kite-shaped fundamental domain of the extended affine Weyl group $F_P$ is introduced for any point $x=x_1{\omega }_1+x_2{\omega }_2$ in the ω-basis,

$$f\left(x\right)=0.4e^{-\frac{1}{{\sigma }^2}\left({\left(x_1-\frac{1}{6}\right)}^2+\frac{1}{3}{\left(x_1+2x_2-\frac{1}{2}\right)}^2\right)}.$$

(106)

The $3D$ graph and contour plot of the model function f, with $\sigma =0.065$ chosen as a fixed value, are shown in Figure 3.

The function f is interpolated by the anti(symmetric) interpolating functions (100) and (101) with the frequency spectrum coefficients computed from the weight lattice splitting Fourier–Weyl transforms (104) and (105), respectively. The symmetric interpolating functions $\mathrm{I}{\left[f\right]}_M^{\left(0\right)}$ and $\mathrm{I}{\left[f\right]}_M^{\left(1\right)}$ are for $M=10,14$ and 18 plotted in Figure 4 and Figure 5.

The antisymmetric interpolating functions $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(0\right)}$ and $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(1\right)}$ are for $M=10,14$ and 18 plotted in Figure 6 and Figure 7.

The integral error estimates of both types of interpolations are presented in

Table 1. The integral error estimates of the interpolations $\mathrm{I}{\left[f\right]}_M^{\left(0\right)}$, $\mathrm{I}{\left[f\right]}_M^{\left(1\right)}$, $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(0\right)}$ and $\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(1\right)}$ are tabulated for $M=10,12,14,16$ and 18.

M	10	12	14	16	18
${\int }_{F_P}{\left|f-\mathrm{I}{\left[f\right]}_M^{\left(0\right)}\right|}^2$	$1377\times {10}^{-7}$	$373\times {10}^{-7}$	$123\times {10}^{-7}$	$39\times {10}^{-7}$	$4\times {10}^{-7}$
${\int }_{F_P}{\left|f-\mathrm{I}{\left[f\right]}_M^{\left(1\right)}\right|}^2$	$1530\times {10}^{-7}$	$649\times {10}^{-7}$	$117\times {10}^{-7}$	$52\times {10}^{-7}$	$9\times {10}^{-7}$
${\int }_{F_P}{\left|f-\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(0\right)}\right|}^2$	$1814\times {10}^{-7}$	$763\times {10}^{-7}$	$266\times {10}^{-7}$	$69\times {10}^{-7}$	$9\times {10}^{-7}$
${\int }_{F_P}{\left|f-\tilde{\mathrm{I}}{\left[f\right]}_M^{\left(1\right)}\right|}^2$	$1520\times {10}^{-7}$	$753\times {10}^{-7}$	$255\times {10}^{-7}$	$48\times {10}^{-7}$	$9\times {10}^{-7}$

.

4. Central Splitting

4.1. Central Splitting of Discrete Transforms

The center of the compact simple Lie group $SU\left(3\right)$, associated with the root system $A_2$, is isomorphic to both groups $\Gamma$ and $P/Q$. The central splitting of a function $f:F_Q\to \mathbb{C}$ represents the functional decomposition [3] of the form

$$f=f_0+f_1+f_2,$$

(107)

where each component $f_k:F_Q\to \mathbb{C}$, $k\in \{0,1,2\}$ is determined by

$$f_k\left(x\right)=\frac{1}{3}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_k,z_j\rangle }f\left({\gamma }_{j}x\right).$$

(108)

The current defining formula of the central splitting (108) is a specialization of relation ([3] Equation (27)) with incorporated action of the $\Gamma$-group elements from relations (41) and (42). The exponential coefficients $e^{-2\pi \mathrm{i}\langle z_k,z_j\rangle }$, $j,k\in \{0,1,2\}$ are calculated from relation (6) as elements of the group of the third roots of unity $U_3$,

$$U_3=\{1,e^{\frac{2\pi \mathrm{i}}{3}},e^{\frac{-2\pi \mathrm{i}}{3}}\}.$$

(109)

Direct calculations from defining relation (108) of the central splitting yields the following symmetry property of the component functions under the action of the $\Gamma$-group for $j\in \{0,1,2\}$ and $x\in {\mathbb{R}}^2$,

$$f_k\left({\gamma }_{j}x\right)=e^{2\pi \mathrm{i}\langle z_k,z_j\rangle }{f}_k\left(x\right).$$

Based on the decomposition of the central components into the sum of orbit functions labeled by the weights from the corresponding congruence class from ([3] Section 7), the central splitting is utilized to decompose the weight lattice Fourier–Weyl transform into the smaller splitting transforms.

Theorem 1 (Central Splitting of Weight Lattice Transforms).

The spectral coefficients $c_{\lambda }{\left[f\right]}_M$, $\lambda \in {\Lambda }_M^{\left(k\right)}$, $k\in \{0,1,2\}$ of the weight lattice Fourier–Weyl C-transforms (98), corresponding to the function $f:F_M\to \mathbb{C}$, coincide with the spectral coefficients of the splitting Fourier–Weyl C-transforms (104) of the central components $f_k:F_M^{\left(k\right)}\to \mathbb{C}$,

$$c_{\lambda }{\left[f\right]}_M=c_{\lambda }{\left[f_k\right]}_M^{\left(k\right)},\lambda \in {\Lambda }_M^{\left(k\right)}.$$

(110)

The spectral coefficients ${\tilde{c}}_{\lambda }{\left[f\right]}_M$, $\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}$ of the weight lattice Fourier–Weyl S-transforms (99), corresponding to the function $f:{\tilde{F}}_M\to \mathbb{C}$, coincide with the spectral coefficients of the splitting Fourier–Weyl S-transforms (105) of the central components $f_k:{\tilde{F}}_M^{\left(k\right)}\to \mathbb{C}$,

$${\tilde{c}}_{\lambda }{\left[f\right]}_M={\tilde{c}}_{\lambda }{\left[f_k\right]}_M^{\left(k\right)},\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)}.$$

(111)

Proof. 

The weight lattice Fourier–Weyl C-transform of the discretized function $f:F_M\to \mathbb{C}$ provides from relations (94) and (96) the following expansion,

$$f\left(s\right)=\sum _{\lambda \in {\Lambda }_M}{c}_{\lambda }{\left[f\right]}_M{\Phi }_{\lambda }\left(s\right),s\in F_M.$$

(112)

The splitting Fourier–Weyl C-transform of the discretized central component $f_k:F_M^{\left(k\right)}\to \mathbb{C}$ provides from relations (100) and (102) the following expansion,

$$f_k\left(s\right)=\sum _{\lambda \in {\Lambda }_M^{\left(k\right)}}{c}_{\lambda }{\left[f_k\right]}_M^{\left(k\right)}{\Phi }_{\lambda }\left(s\right),s\in F_M^{\left(k\right)}.$$

(113)

Substituting into the defining relation of the central splitting (108) the expansion (112) and taking into account the symmetry property (76) and the disjoint decomposition (23) yield for $s\in F_M^{\left(k\right)}$ that

$$\begin{array}{cc}\hfill f_k\left(s\right)& =\frac{1}{3}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_k,z_j\rangle }f\left({\gamma }_{j}s\right)=\frac{1}{3}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_k,z_j\rangle }\sum _{\lambda \in {\Lambda }_M}{c}_{\lambda }{\left[f\right]}_M{\Phi }_{\lambda }\left({\gamma }_{j}s\right)\hfill \\ \hfill & =\frac{1}{3}\sum _{j,l=0}^2{e}^{2\pi \mathrm{i}\langle z_l-z_k,z_j\rangle }\sum _{\lambda \in {\Lambda }_M^{\left(l\right)}}{c}_{\lambda }{\left[f\right]}_M{\Phi }_{\lambda }\left(s\right).\hfill \end{array}$$

(114)

Recall from ([1] Corollary 5.2) that for the classes of weights $\lambda =z_l+Q$, ${\lambda }^{\prime }=z_k+Q$, $k,l\in \{0,1,2\}$ of the root system $A_2$, the orthogonality relations of the multivariate exponential functions are specialized to the form

$$\sum _{j=0}^2{e}^{2\pi \mathrm{i}\langle z_l-z_k,z_j\rangle }=3{\delta }_{kl}.$$

(115)

Using the orthogonality relation of the multivariate exponential functions (115) in expression (114) provides the final form of the expasion of the discretized central component,

$$f_k\left(s\right)=\sum _{\lambda \in {\Lambda }_M^{\left(k\right)}}{c}_{\lambda }{\left[f\right]}_M{\Phi }_{\lambda }\left(s\right),s\in F_M^{\left(k\right)}.$$

(116)

Comparing the resulting expression (116) to the original expansion of the central component (113) provides the statement (110). Since, for $A_2$, the crucial symmetry property of the S-functions (77) under the action of the $\Gamma$-group is of the same form as its C-functions counterpart, the version of the statement (111) is obtained similarly for S-functions via the comparison of the discrete transforms (95), (97) and (101), (103), together with the disjoint decomposition (25). □

4.2. Decompositions of Unitary Transform Matrices

The Fourier–Weyl transforms on a discrete set of points $F_M$ are carried out by predetermined square matrices that multiply any given column-vector of data [36]. The currently constructed unitary transform matrices correspond to the normalized versions of the weight lattice Fourier–Weyl transforms. Any arbitrary fixed orderings of the weight sets ${\Lambda }_M^{\left(k\right)}$ and ${\tilde{\Lambda }}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ induce uniquely from the decompositions (23) and (25) the orderings of the weight sets ${\Lambda }_M$ and ${\tilde{\Lambda }}_M$. The fixed orderings of the splitting point sets $F_M^{\left(k\right)}$ and ${\tilde{F}}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ can be chosen independently on the ordering of the point sets $F_M$ and ${\tilde{F}}_M$.

Using formulas for the spectrum coefficients (98) and (99), the unitary matrices ${\mathbb{I}}_M$ and ${\tilde{\mathbb{I}}}_M$ of the normalized weight lattice Fourier–Weyl C- and S-transforms are given by their entries as follows,

$${\left({\mathbb{I}}_M\right)}_{\lambda s}=\sqrt{\frac{\epsilon \left(s\right)}{18M^2{h}_M\left(\lambda \right)}}\overline{{\Phi }_{\lambda }\left(s\right)},\lambda \in {\Lambda }_M,s\in F_M,$$

(117)

$${\left({\tilde{\mathbb{I}}}_M\right)}_{\lambda s}=\frac{1}{\sqrt{3M^2}}\overline{{\phi }_{\lambda }\left(s\right)},\lambda \in {\tilde{\Lambda }}_M,s\in {\tilde{F}}_M.$$

(118)

The unitary matrices ${\mathbb{I}}_M^{\left(k\right)}$ and ${\tilde{\mathbb{I}}}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ of the normalized splitting Fourier–Weyl C- and S-transforms are constructed using the Formulas (104) and (105),

$${\left({\mathbb{I}}_M^{\left(k\right)}\right)}_{\lambda s}=\sqrt{\frac{\epsilon \left(s\right)}{6M^{2}d\left(s\right)h_M\left(\lambda \right)}}\overline{{\Phi }_{\lambda }\left(s\right)},\lambda \in {\Lambda }_M^{\left(k\right)},s\in F_M^{\left(k\right)},$$

(119)

$${\left({\tilde{\mathbb{I}}}_M^{\left(k\right)}\right)}_{\lambda s}=\frac{1}{\sqrt{M^{2}d\left(s\right)}}\overline{{\phi }_{\lambda }\left(s\right)},\lambda \in {\tilde{\Lambda }}_M^{\left(k\right)},s\in {\tilde{F}}_M^{\left(k\right)}.$$

(120)

The unitary transform matrices ${\mathbb{T}}_M$ and ${\tilde{\mathbb{T}}}_M$, that realize the normalized central splitting (107), are represented by the following block matrices

$${\mathbb{T}}_M=\left(\begin{array}{c}{\mathbb{T}}_M^{\left(0\right)}\\ {\mathbb{T}}_M^{\left(1\right)}\\ {\mathbb{T}}_M^{\left(2\right)}\end{array}\right),{\tilde{\mathbb{T}}}_M=\left(\begin{array}{c}{\tilde{\mathbb{T}}}_M^{\left(0\right)}\\ {\tilde{\mathbb{T}}}_M^{\left(1\right)}\\ {\tilde{\mathbb{T}}}_M^{\left(2\right)}\end{array}\right),$$

(121)

where the block components ${\mathbb{T}}_M^{\left(k\right)}$ and ${\tilde{\mathbb{T}}}_M^{\left(k\right)}$, $k\in \{0,1,2\}$ are determined by their entries as

$${\left({\mathbb{T}}_M^{\left(k\right)}\right)}_{s{s}^{\prime }}=\frac{1}{3}\sqrt{\frac{3}{d\left(s\right)}}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_{j}s,s^{\prime }},s\in F_M^{\left(k\right)},s^{\prime }\in F_M,$$

(122)

$${\left({\tilde{\mathbb{T}}}_M^{\left(k\right)}\right)}_{s{s}^{\prime }}=\frac{1}{3}\sqrt{\frac{3}{d\left(s\right)}}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_{j}s,s^{\prime }},s\in {\tilde{F}}_M^{\left(k\right)},s^{\prime }\in {\tilde{F}}_M.$$

(123)

The unitary transform matrices of the normalized weight lattice Fourier–Weyl transforms are decomposed into the normalized central splitting matrices and splitting transform matrices in the following theorem.

Theorem 2 (Decompositions of Transform Matrices).

The following matrix equalities hold for any $M\in \mathbb{N}$,

$${\mathbb{I}}_M=\left({\mathbb{I}}_M^{\left(0\right)}\oplus {\mathbb{I}}_M^{\left(1\right)}\oplus {\mathbb{I}}_M^{\left(2\right)}\right){\mathbb{T}}_M,$$

(124)

$${\tilde{\mathbb{I}}}_M=\left({\tilde{\mathbb{I}}}_M^{\left(0\right)}\oplus {\tilde{\mathbb{I}}}_M^{\left(1\right)}\oplus {\tilde{\mathbb{I}}}_M^{\left(2\right)}\right){\tilde{\mathbb{T}}}_M.$$

(125)

Proof. 

Performing the multiplication of the block matrices in statement (124) provides the matrix block form

$${\mathbb{I}}_M=\left(\begin{array}{c}{\mathbb{I}}_M^{\left(0\right)}{\mathbb{T}}_M^{\left(0\right)}\\ {\mathbb{I}}_M^{\left(1\right)}{\mathbb{T}}_M^{\left(1\right)}\\ {\mathbb{I}}_M^{\left(2\right)}{\mathbb{T}}_M^{\left(2\right)}\end{array}\right).$$

(126)

Since the ordering of the weights labeling the rows of ${\mathbb{I}}_M$ is induced by the decomposition (23), the matrix equality (126) is reformulated via the corresponding entries for $k\in \{0,1,2\}$ as

$${\left({\mathbb{I}}_M\right)}_{\lambda s^{\prime }}=\sum _{s\in F_M^{\left(k\right)}}{\left({\mathbb{I}}_M^{\left(k\right)}\right)}_{\lambda s}{\left({\mathbb{T}}_M^{\left(k\right)}\right)}_{s{s}^{\prime }},\lambda \in {\Lambda }_M^{\left(k\right)},s^{\prime }\in F_M.$$

(127)

Direct calculations from defining relation (122), symmetry property (76) and ε-function $\Gamma$-invariance lead to the invariance of the following products for $r_l\in \Gamma ,l\in \{0,1,2\}$,

$$\sqrt{\epsilon ({\gamma }_{l}s)}\overline{{\Phi }_{\lambda }\left({\gamma }_{l}s\right)}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_j{\gamma }_{l}s,s^{\prime }}=\sqrt{\epsilon \left(s\right)}\overline{{\Phi }_{\lambda }\left(s\right)}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_{j}s,s^{\prime }}.$$

(128)

Taking into account the $\Gamma$-invariance from expression (128) provides the following simplification of the matrix multiplication (127) for $\lambda \in {\Lambda }_M^{\left(k\right)}$ and $s^{\prime }\in F_M$,

$$\begin{array}{cc}\hfill \sum _{s\in F_M^{\left(k\right)}}{\left({\mathbb{I}}_M^{\left(k\right)}\right)}_{\lambda s}{\left({\mathbb{T}}_M^{\left(k\right)}\right)}_{s{s}^{\prime }}& =\sum _{s\in F_M^{\left(k\right)}}3d^{-1}\left(s\right)\sqrt{\frac{\epsilon \left(s\right)}{162M^2{h}_M\left(\lambda \right)}}\overline{{\Phi }_{\lambda }\left(s\right)}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_{j}s,s^{\prime }}\hfill \\ \hfill & =\sum _{s\in F_M}\sqrt{\frac{\epsilon \left(s\right)}{162M^2{h}_M\left(\lambda \right)}}\overline{{\Phi }_{\lambda }\left(s\right)}\sum _{j=0}^2{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }{\delta }_{{\gamma }_{j}s,s^{\prime }}.\hfill \end{array}$$

(129)

Employing again the symmetry property (76) together with $\Gamma$-invariance of both ε-function and point set $F_M$ yields from relation (129) the desired result,

$$\begin{array}{cc}\hfill \sum _{s\in F_M^{\left(k\right)}}{\left({\mathbb{I}}_M^{\left(k\right)}\right)}_{\lambda s}{\left({\mathbb{T}}_M^{\left(k\right)}\right)}_{s{s}^{\prime }}& ={\left(162M^2{h}_M\left(\lambda \right)\right)}^{-\frac{1}{2}}\sum _{j=0}^2\sqrt{\epsilon \left({\gamma }_j^{-1}{s}^{\prime }\right)}\overline{{\Phi }_{\lambda }\left({\gamma }_j^{-1}{s}^{\prime }\right)}{e}^{-2\pi \mathrm{i}\langle z_j,z_k\rangle }\hfill \\ \hfill & =\sqrt{\frac{\epsilon \left(s^{\prime }\right)}{18M^2{h}_M\left(\lambda \right)}}\overline{{\Phi }_{\lambda }\left(s^{\prime }\right)}={\left({\mathbb{I}}_M\right)}_{\lambda s^{\prime }}.\hfill \end{array}$$

The symmetry property of the S-functions (77) allows to obtain the S-transform matrix relation (125) by performing analogous steps. □

4.3. Decompositions of Transform Matrices ${\mathbb{I}}_3$ and ${\tilde{\mathbb{I}}}_6$

The weights of the splitting weight sets ${\Lambda }_3^{\left(k\right)}$, $k\in \{0,1,2\}$ are calculated from expression (27) and ordered as

$$\begin{array}{cc}\hfill {\Lambda }_3^{\left(0\right)}& =\left\{\right[3,0,0],[1,1,1],[0,3,0],[0,0,3\left]\right\},\hfill \\ \hfill {\Lambda }_3^{\left(1\right)}& =\left\{\right[2,1,0],[1,0,2],[0,2,1\left]\right\},\hfill \\ \hfill {\Lambda }_3^{\left(2\right)}& =\left\{\right[2,0,1],[1,2,0],[0,1,2\left]\right\}.\hfill \end{array}$$

(130)

The ordering of the entire weight set ${\Lambda }_3$ is induced by the decomposition (23),

$${\Lambda }_3={\Lambda }_3^{\left(0\right)}\cup {\Lambda }_3^{\left(1\right)}\cup {\Lambda }_3^{\left(2\right)},$$

where the weights of each splitting weight set are ordered as in the lists of weights (130). The points of the point set $F_3$ are calculated in Kac coordinates from relation (48), and ordered as follows,

$$F_3=\left\{[3,0,0],[2,1,0],[2,0,1],[1,2,0],[1,1,1],[1,0,2],[0,3,0],[0,2,1],[0,1,2],[0,0,3]\right\}.$$

The unitary transform matrix ${\mathbb{I}}_3$ comprises rows indexed by the ordered weight set ${\Lambda }_3$ and columns indexed by the point set $F_3$. Utilizing the C-function formula (72), h- and ε-function expressions (39) and (65), the transform matrix ${\mathbb{I}}_3$ is calculated from definition (117) as

$${\mathbb{I}}_3=\left(\begin{array}{cccccccccc}\frac{1}{3\sqrt{3}}& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& \frac{\sqrt{2}}{3}& \frac{1}{3}& \frac{1}{3\sqrt{3}}& \frac{1}{3}& \frac{1}{3}& \frac{1}{3\sqrt{3}}\\ \frac{\sqrt{2}}{3}& 0& 0& 0& -\frac{1}{\sqrt{3}}& 0& \frac{\sqrt{2}}{3}& 0& 0& \frac{\sqrt{2}}{3}\\ \frac{1}{3\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}{e}^{-\frac{2\mathrm{i}\pi }{3}}& \frac{\sqrt{2}}{3}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3\sqrt{3}}\\ \frac{1}{3\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}{e}^{\frac{2\mathrm{i}\pi }{3}}& \frac{\sqrt{2}}{3}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\mathrm{i}\pi }{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3\sqrt{3}}\\ \frac{1}{3}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{\frac{4\mathrm{i}\pi }{9}}a}{6\sqrt{3}}& 0& \frac{e^{-\frac{4\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}\\ \frac{1}{3}& \frac{e^{-\frac{4\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{\frac{4\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& 0& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{e^{\frac{2\mathrm{i}\pi }{9}}b}{6\sqrt{3}}& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}\\ \frac{1}{3}& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& 0& \frac{e^{\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& \frac{e^{-\frac{4\mathrm{i}\pi }{9}}b}{6\sqrt{3}}& \frac{e^{\frac{4\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}\\ \frac{1}{3}& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{-\frac{4\mathrm{i}\pi }{9}}b}{6\sqrt{3}}& 0& \frac{e^{\frac{4\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}\\ \frac{1}{3}& \frac{e^{\frac{4\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{-\frac{4\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& 0& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}\\ \frac{1}{3}& -\frac{1}{3}\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{1}{3}\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& 0& \frac{e^{-\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{1}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& \frac{e^{\frac{4\mathrm{i}\pi }{9}}a}{6\sqrt{3}}& \frac{e^{-\frac{4\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{1}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}\end{array}\right),$$

where $a=3+\sqrt{3}\mathrm{i}$ and $b=3-\sqrt{3}\mathrm{i}$.

Each unitary splitting matrix ${\mathbb{I}}_3^{\left(k\right)}$, $k\in \{0,1,2\}$ contains rows indexed by the corresponding weight set in (130) and columns indexed by the point sets $F_3^{\left(k\right)}$ obtained from expression (57) as

$$\begin{array}{cc}\hfill F_3^{\left(0\right)}& =\left\{\right[3,0,0],[2,1,0],[2,0,1],[1,1,1\left]\right\},\hfill \\ \hfill F_3^{\left(1\right)}=F_3^{\left(2\right)}& =\left\{\right[3,0,0],[2,1,0],[2,0,1\left]\right\}.\hfill \end{array}$$

Utilizing the d-function values from (67), the splitting transform matrices ${\mathbb{I}}_3^{\left(0\right)}$, ${\mathbb{I}}_3^{\left(1\right)}$ and ${\mathbb{I}}_3^{\left(2\right)}$ are calculated from definition (119) as

$$\begin{array}{cc}\hfill {\mathbb{I}}_3^{\left(0\right)}& =\left(\begin{array}{cccc}\frac{1}{3}& \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{3}}& \frac{\sqrt{2}}{3}\\ \sqrt{\frac{2}{3}}& 0& 0& -\frac{1}{\sqrt{3}}\\ \frac{1}{3}& \frac{e^{\frac{2\pi \mathrm{i}}{3}}}{\sqrt{3}}& \frac{e^{-\frac{2\pi \mathrm{i}}{3}}}{\sqrt{3}}& \frac{\sqrt{2}}{3}\\ \frac{1}{3}& \frac{e^{-\frac{2\pi \mathrm{i}}{3}}}{\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{3}}}{\sqrt{3}}& \frac{\sqrt{2}}{3}\end{array}\right),\hfill \\ \hfill {\mathbb{I}}_3^{\left(1\right)}& =\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}& \frac{\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}& -\frac{\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}& \frac{1}{6}b{e}^{-\frac{4\pi \mathrm{i}}{9}}& \frac{1}{6}a{e}^{\frac{4\pi \mathrm{i}}{9}}\\ \frac{1}{\sqrt{3}}& \frac{\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}& -\frac{\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}\end{array}\right),\hfill \\ \hfill {\mathbb{I}}_3^{\left(2\right)}& =\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}& \frac{1}{6}a{e}^{-\frac{2\pi \mathrm{i}}{9}}& \frac{1}{6}b{e}^{\frac{2\pi \mathrm{i}}{9}}\\ \frac{1}{\sqrt{3}}& \frac{1}{6}a{e}^{\frac{4\pi \mathrm{i}}{9}}& \frac{1}{6}b{e}^{-\frac{4\pi \mathrm{i}}{9}}\\ \frac{1}{\sqrt{3}}& -\frac{\mathrm{i}{e}^{-\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}& \frac{\mathrm{i}{e}^{\frac{2\pi \mathrm{i}}{9}}}{\sqrt{3}}\end{array}\right).\hfill \end{array}$$

The rows of the unitary central splitting matrix ${\mathbb{T}}_3$ are labeled by the union of the splitting point sets $F_3^{\left(0\right)}\cup F_3^{\left(1\right)}\cup F_3^{\left(2\right)}$, and the columns are labeled by the elements of the point set $F_3$. Thus, the transform matrix ${\mathbb{T}}_3$ is calculated from definitions (121) and (122) as

$${\mathbb{T}}_3=\left(\begin{array}{cccccccccc}\frac{\sqrt{3}}{3}& 0& 0& 0& 0& 0& \frac{\sqrt{3}}{3}& 0& 0& \frac{\sqrt{3}}{3}\\ 0& \frac{\sqrt{3}}{3}& 0& 0& 0& \frac{\sqrt{3}}{3}& 0& \frac{\sqrt{3}}{3}& 0& 0\\ 0& 0& \frac{\sqrt{3}}{3}& \frac{\sqrt{3}}{3}& 0& 0& 0& 0& \frac{\sqrt{3}}{3}& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ \frac{\sqrt{3}}{3}& 0& 0& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& 0& 0& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}\\ 0& \frac{\sqrt{3}}{3}& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& 0& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& 0& 0\\ 0& 0& \frac{\sqrt{3}}{3}& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& 0& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& 0\\ \frac{\sqrt{3}}{3}& 0& 0& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& 0& 0& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}\\ 0& \frac{\sqrt{3}}{3}& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& 0& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& 0& 0\\ 0& 0& \frac{\sqrt{3}}{3}& \frac{\sqrt{3}}{3}{e}^{-\frac{2\pi \mathrm{i}}{3}}& 0& 0& 0& 0& \frac{\sqrt{3}}{3}{e}^{\frac{2\pi \mathrm{i}}{3}}& 0\end{array}\right).$$

(131)

The weights of the splitting weight sets ${\tilde{\Lambda }}_6^{\left(k\right)}$, $k\in \{0,1,2\}$ are calculated from expression (28) and ordered as

$$\begin{array}{cc}\hfill {\tilde{\Lambda }}_6^{\left(0\right)}& =\left\{\right[4,1,1],[2,2,2],[1,4,1],[1,1,4\left]\right\},\hfill \\ \hfill {\tilde{\Lambda }}_6^{\left(1\right)}& =\left\{\right[3,2,1],[2,1,3],[1,3,2\left]\right\},\hfill \\ \hfill {\tilde{\Lambda }}_6^{\left(2\right)}& =\left\{\right[3,1,2],[2,3,1],[1,2,3\left]\right\}.\hfill \end{array}$$

(132)

The ordering of the entire weight set ${\tilde{\Lambda }}_6$ is induced by the decomposition (25),

$${\tilde{\Lambda }}_6={\tilde{\Lambda }}_6^{\left(0\right)}\cup {\tilde{\Lambda }}_6^{\left(1\right)}\cup {\tilde{\Lambda }}_6^{\left(2\right)},$$

where the weights of each splitting weight set are ordered as in the sets of weights (132).

The points of the point set ${\tilde{F}}_6$ are calculated in Kac coordinates from relation (49) and ordered as follows,

$${\tilde{F}}_6=\{[4,1,1],[3,2,1],[3,1,2],[2,3,1],[2,2,2],[2,1,3],[1,4,1],[1,3,2,],[1,2,3],[1,1,4]\}.$$

Utilizing the S-function formula (73), the transform matrix ${\tilde{\mathbb{I}}}_6$ is calculated from definition (118) as

$${\tilde{\mathbb{I}}}_6=\left(\begin{array}{cccccccccc}\frac{\mathrm{i}}{6}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{2}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{6}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{3}& \frac{\mathrm{i}}{6}\\ \frac{\mathrm{i}}{2}& 0& 0& 0& -\frac{\mathrm{i}}{2}& 0& \frac{\mathrm{i}}{2}& 0& 0& \frac{\mathrm{i}}{2}\\ \frac{\mathrm{i}}{6}& -\frac{b}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{2}& -\frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{6}& -\frac{b}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{6}\\ \frac{\mathrm{i}}{6}& \frac{b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{2}& \frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{6}& \frac{b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& \frac{\mathrm{i}}{6}\\ \frac{\mathrm{i}}{3}& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& 0& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}\\ \frac{\mathrm{i}}{3}& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& 0& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}\\ \frac{\mathrm{i}}{3}& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& 0& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}\\ \frac{\mathrm{i}}{3}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& 0& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}\\ \frac{\mathrm{i}}{3}& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& 0& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}\\ \frac{\mathrm{i}}{3}& \frac{e^{-\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{e^{\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& -\frac{e^{-\frac{2\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& 0& \frac{e^{\frac{2\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{b}{6\sqrt{3}}& -\frac{e^{-\frac{\pi \mathrm{i}}{9}}a}{6\sqrt{3}}& \frac{e^{\frac{\pi \mathrm{i}}{9}}b}{6\sqrt{3}}& -\frac{b}{6\sqrt{3}}\end{array}\right),$$

where $a=3+\sqrt{3}\mathrm{i}$ and $b=3-\sqrt{3}\mathrm{i}$.

Each unitary splitting matrix ${\tilde{\mathbb{I}}}_6^{\left(k\right)}$, $k\in \{0,1,2\}$ contains rows indexed by the corresponding interior weight set in (132) and columns indexed by the interior point sets ${\tilde{F}}_6^{\left(k\right)}$ obtained from expression (58) as

$$\begin{array}{cc}\hfill {\tilde{F}}_6^{\left(0\right)}& =\left\{\right[4,1,1],[3,2,1],[3,1,2],[2,2,2\left]\right\},\hfill \\ \hfill {\tilde{F}}_6^{\left(1\right)}=F_6^{\left(2\right)}& =\left\{\right[4,1,1],[3,2,1],[3,1,2\left]\right\}.\hfill \end{array}$$

The splitting transform matrices ${\tilde{\mathbb{I}}}_6^{\left(0\right)}$, ${\tilde{\mathbb{I}}}_6^{\left(1\right)}$ and ${\tilde{\mathbb{I}}}_6^{\left(2\right)}$ are calculated from definition (120) as

$$\begin{array}{cc}\hfill {\tilde{\mathbb{I}}}_6^{\left(0\right)}& =\left(\begin{array}{cccc}\frac{\mathrm{i}}{2\sqrt{3}}& \frac{\mathrm{i}}{\sqrt{3}}& \frac{\mathrm{i}}{\sqrt{3}}& \frac{\mathrm{i}}{2}\\ \frac{i\sqrt{3}}{2}& 0& 0& -\frac{\mathrm{i}}{2}\\ \frac{\mathrm{i}}{2\sqrt{3}}& -\frac{1}{6}b& \frac{1}{6}b& \frac{\mathrm{i}}{2}\\ \frac{\mathrm{i}}{2\sqrt{3}}& \frac{1}{6}b& -\frac{1}{6}b& \frac{\mathrm{i}}{2}\end{array}\right),\hfill \\ \hfill {\tilde{\mathbb{I}}}_6^{\left(1\right)}& =\left(\begin{array}{ccc}\frac{\mathrm{i}}{\sqrt{3}}& -\frac{1}{6}b{e}^{-\frac{2\mathrm{i}\pi }{9}}& \frac{1}{6}a{e}^{\frac{2\mathrm{i}\pi }{9}}\\ \frac{\mathrm{i}}{\sqrt{3}}& \frac{1}{6}b{e}^{\frac{\mathrm{i}\pi }{9}}& -\frac{1}{6}a{e}^{-\frac{\mathrm{i}\pi }{9}}\\ \frac{\mathrm{i}}{\sqrt{3}}& -\frac{1}{6}a{e}^{\frac{\mathrm{i}\pi }{9}}& \frac{1}{6}b{e}^{-\frac{\mathrm{i}\pi }{9}}\end{array}\right),\hfill \\ \hfill {\tilde{\mathbb{I}}}_6^{\left(2\right)}& =\left(\begin{array}{ccc}\frac{\mathrm{i}}{\sqrt{3}}& \frac{1}{6}a{e}^{\frac{2\mathrm{i}\pi }{9}}& -\frac{1}{6}b{e}^{-\frac{2\mathrm{i}\pi }{9}}\\ \frac{\mathrm{i}}{\sqrt{3}}& -\frac{1}{6}a{e}^{-\frac{\mathrm{i}\pi }{9}}& \frac{1}{6}b{e}^{\frac{\mathrm{i}\pi }{9}}\\ \frac{\mathrm{i}}{\sqrt{3}}& \frac{1}{6}b{e}^{-\frac{\mathrm{i}\pi }{9}}& -\frac{1}{6}a{e}^{\frac{\mathrm{i}\pi }{9}}\end{array}\right).\hfill \end{array}$$

Using the Formulas (121) and (123) to construct the unitary central splitting matrix ${\tilde{\mathbb{T}}}_6$, the rows are labeled by the union of the point sets ${\tilde{F}}_6^{\left(0\right)}\cup {\tilde{F}}_6^{\left(1\right)}\cup {\tilde{F}}_6^{\left(2\right)}$, and the columns are labeled by the elements of the interior point set ${\tilde{F}}_6$. Since the ordering of the interior point sets for $M=6$ is chosen to be compatible with the ordering of the C-transform point sets for $M=3$, the transform matrix ${\tilde{\mathbb{T}}}_6$ coincides with the splitting matrix ${\mathbb{T}}_3$,

$${\tilde{\mathbb{T}}}_6={\mathbb{T}}_3.$$

5. Concluding Remarks

• The decompositions of the weight lattice Fourier–Weyl transforms into the splitting transforms are considered as a point of departure in exploring the fast bivariate discrete Fourier transforms involving the (anti)symmetric orbit functions of the Weyl group $A_2$. The central splitting method offers an advantageous approach to computational efficiency, using the reduction of the initial weight lattice Fourier–Weyl transform into three smaller-weight lattice splitting transforms. Even though the recursive method to determine the further splitting of currently developed two-variable cosine and sine transforms has not been determined yet, the demonstrated reduced weight transform is considered as a stepping stone towards the fast discrete transforms. Moreover, analogously to the interpolation tests conducted for cosine and sine discrete transforms [2,17,18], the developed discrete splitting transforms also manifest excellent interpolation properties.
• For the crystallographic reflection group $A_1$, repeatedly using the central splitting of one-variable discrete cosine and sine transforms produces the standard versions of the fast split-radix transforms [4]. In this case, the possibility of rearranging the splitting point sets into the original format governed by the affine Weyl group ensures the central splitting method’s recursive behavior. The decompositions of the transform matrices, similar to the formulated unitary matrix decompositions in Theorem 2, have been rigorously proven for the one-dimensional sine and cosine transforms in [30]. However, in the case of the $A_2$ group, since a further splitting of the points in the kite-shaped domain has not been formulated yet, the recursive central splitting method remains an unsolved problem.
• Given the importance of multi-dimensional digital data processing [16,37,38,39,40], a central-splitting mechanism could be potentially developed for other compact simple Lie groups with non-trivial elements of the center, such as $A_n$ with its center provided by a cyclic group of $n+1$ elements, $B_n$, $C_n$ and $E_7$ with the center given by a cyclic group of order 2, $D_n$ whose center contains 4 elements, $E_6$ that equivalently to $A_2$ has 3 elements of the center. Such an approach would be considered as a first step to a general multidimensional fast transform. Since a similar behavior of the central splitting in the case of the finite reflection group $A_n$ is expected, the extension of the developed Fourier–Weyl transforms to higher-dimensional cases should be treated independently.
• Another family of orbit functions, known as E-functions, is obtained by symmetrizing multivariate exponential terms over even subgroups of a considered Weyl group. Such functions are developed in [13], and their corresponding Fourier–Weyl transforms together with continuous interpolations are examined in detail [41]. There is one type of the E-functions for the root systems with the roots of one length [3]. For the root systems with two lengths of simple roots, the six types of E-functions, together with their even complex-valued dual weight lattice Fourier–Weyl transforms, are formulated in [14]. Hence, instead of the discrete transforms based on the C- and S-functions, the central splitting of the transforms developed by means of the E-functions represents an open problem. Furthermore, the Fourier–Weyl transforms with their kernel represented by the combinations of different types of Weyl orbit functions have not been previously explored.