Dual-root lattice discretization

# 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.

Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, CZ-115 19 Prague, Czech Republic

E-mail: jiri.hrivnak@fjfi.cvut.cz, lenka.motlochova@fjfi.cvut.cz

Keywords: Weyl orbit functions, root lattice, discrete Fourier transform, Hartley transform

## 1. Introduction

The purpose of this article is to extend the collection of discrete Fourier transforms of Weyl orbit functions on Weyl group invariant lattices [16, 14, 17]. A finite fragment of a refinement of the classical dual root lattice [2] serves as the starting set of points over which the discrete orthogonality of four types of complex Weyl orbit functions [20, 21, 27] is developed. The entire resulting transform formalism produces a real-valued multidimensional Weyl group invariant generalizations of the one-dimensional discrete Hartley transform [3].

The antisymmetric and symmetric exponential orbit sums over Weyl groups form a standard part of the theory of Lie algebras and their representations [2]. From the viewpoint of the Coxeter groups theory, Weyl groups cover all finite crystallographic reflection groups [18]. Depending on the type of the underlying crystallographic root system, two or four sign homomorphisms exist [27]. Each sign homomorphism determines signs in the exponential sums and thus generates for each Weyl group two or four types of complex special functions. Lattice shift and Weyl group invariance of the resulting Weyl orbit functions generalize periodicity and boundary behaviour of the standard cosine and sine functions of one variable. Investigating Weyl orbit functions as special functions, the results range from generalizations of continuous multivariate Fourier transforms in [20, 21] to generalized Chebyshev polynomial methods [24, 29]. Discrete Fourier methods are comprehensively studied for Weyl orbit functions [16, 14, 17] as well as for their multivariate Chebyshev polynomial generalizations [7, 15, 27, 29]. The refinement of the dual weight lattice intersected with the fundamental domain of the affine Weyl group form a finite point set on which the majority of the discrete Fourier and Chebyshev methods is developed [16, 14, 24, 27]. This choice of the point set generates symmetries of labels of orbit functions governed by the dual affine Weyl group [16]. Choosing as the starting point set the refinement of the weight lattice produced different argument and label symmetries, both controlled by the same affine Weyl group [17].

The dual root and root lattices constitute the last classical Weyl group invariant lattices for which the inherent Fourier methods have not yet been studied. Several apparent relative difficulties, surmounted in the present paper for the dual root lattice, stem from the fact that the label symmetries are in this case determined by the dual extended affine Weyl group. Firstly, even though the structure of the extended affine Weyl group and its dual version is detailed already in [2], relevant rigorous results about their fundamental domains appeared much later in [22]. Moreover, these fundamental domains, essential for selecting the labels of orbit functions in discrete transforms, are determined in [22] only up to their boundaries. A uniform description of the fundamental domains from [22], including a unique layout of the boundary points, is achieved in the present paper by introducing lexicographical ordering on the Kac coordinates [19]. Secondly, the main challenge poses linking the number of weights, found in the fundamental domain of the dual extended affine Weyl group, with the number of points from the refined dual root lattice, lying in the fundamental domain of the affine Weyl group. Both sets form topologically distant finite subsets of the underlying Euclidean space and their common cardinality constitutes a novel invariant characteristic of the crystallographic root systems and corresponding simple Lie algebras. Determining the cardinality of these sets in full generality requires invoking and extending concepts from the theory of invariant polynomials [32, 18]. Common cardinality of the point and weight sets guarantees in turn the existence of both complex-valued Fourier-Weyl and real-valued Hartley-Weyl discrete transforms.

Both complex and real types of developed Fourier-like transforms significantly enhance the collection of available Weyl group invariant discrete transforms [16, 14, 24, 27]. The topology of the point sets of the dual root lattice discretization and their relative position with respect to the fundamental domain of the affine Weyl group substantially diverge from the locations of the original dual weight lattice points. Moreover, fundamentally novel options are generated by combining both dual root and dual weight discretizations to produce discrete transforms on generalized and composed grids. These options encompass transforms on the refined dual weight lattice with points from the dual root lattice omitted, such as two-variable transforms on the honeycomb lattice. Assuming Weyl symmetric or antisymmetric boundary conditions in mechanical graphene eigenvibrations model [5] or in quantum field lattice models [8] potentially yields novel Weyl invariant solutions and dispersion relations in terms of four types of extended Weyl orbit functions. Real-valued multivariate Hartley-Weyl versions of the transforms augment the application potential of discrete Hartley transforms in pattern recognition [4], geophysics [23], signal processing [30, 31], optics [25] and measurement [33]. The unitary matrices of the derived transforms also permit the construction of novel analogues of the Kac-Peterson matrices from the conformal field theory [17].

The paper is organized as follows. In Section 2, the necessary facts concerning root systems and invariant lattices of Weyl groups are recalled. Section 3 contains a description of infinite extensions of Weyl groups. In Section 4, fundamental domains of infinite extensions of Weyl groups and the corresponding invariant polynomials are detailed. Section 5 is devoted to the study of finite sets of points and weights. The identical cardinality of these sets is proven and explicit counting formulas for the cardinalities are listed in full generality. Section 6 describes Weyl and Hartley orbit functions together with their discrete orthogonality and discrete Fourier transforms. Comments and follow-up questions are contained in the last section.

## 2. Invariant lattices of Weyl groups

### 2.1. Root systems and Weyl groups

The notation used in this article is based on papers [16, 14]. The purpose of this section is to extend this notation and recall other pertinent details [2]. Each simple Lie algebra from the classical four series , , , and from the five exceptional cases determines its set of simple roots [2, 34]. For the cases of simple Lie algebras with two different root-lengths, the set is disjointly decomposed into of short simple roots and of long simple roots,

 Δ=Δs∪Δl.

The set forms a non-orthogonal basis of the Euclidean space with the standard scalar product . To every simple root corresponds a reflection given by the formula

 ria≡a−2⟨a,αi⟩⟨αi,αi⟩αi,a∈Rn.

Reflections generate an irreducible Weyl group

 W≡⟨r1,…,rn⟩

and in turn generates the entire root system . The set of simple roots induces a partial ordering on such that for it holds if and only if with for all . There exists a unique root , highest with respect to this ordering, of the form

 ξ=m1α1+⋯+mnαn.

To each simple root relates the dual simple root given by

 α∨i=2αi⟨αi,αi⟩.

The set of dual simple roots generates the entire dual root system . The dual root system contains the highest dual root with respect to the ordering induced by of the form

 η=m∨1α∨1+⋯+m∨nα∨n.

The expansion coefficients of the highest root and of the highest dual root , named the marks and the dual marks, respectively, are listed in Table 1 in [16]. Setting additionally , the Coxeter number is given by

 m=n∑i=0mi=n∑i=0m∨i. (1)

The dual marks with unitary values determine an important index set ,

 J≡{1≤i≤n|m∨i=1}.

Note that is empty for the simple Lie algebras and .

Minimal number generators , necessary to generate an element is called the length of . There is a unique longest element , also called the opposite involution, and for its length it holds that

 l(w0)=|Π|2.

For root systems with even Coxeter numbers are the opposite involutions of the form

 w0=(r1…rn)m2.

An important parabolic subgroup of the Weyl group is obtained by omitting from the set of generators of

 Wi≡⟨r1,…,ri−1,ri+1,…,rn⟩.

The subgroup forms the Weyl group corresponding to the root system . The opposite involution in with respect to the root system is denoted by .

### 2.2. Invariant lattices

Four classical Weyl group invariant lattices [2] comprise the root lattice, the dual weight lattice, the dual root lattice and the weight lattice. The root lattice is the -span of the set of simple roots ,

 Q=Zα1+⋯+Zαn.

The dual weight lattice is dual to the root lattice ,

 P∨={ω∨∈Rn|⟨ω∨,αi⟩∈Z,∀αi∈Δ}=Zω∨1+⋯+Zω∨n,

with the dual fundamental weights given by

 ⟨ω∨i,αj⟩=δij.

The dual root lattice is the -span of the set of dual simple roots

 Q∨=Zα∨1+⋯+Zα∨n.

The weight lattice is dual to the dual root lattice ,

 P={ω∈Rn|⟨ω,α∨i⟩∈Z,∀α∨i∈Δ∨}=Zω1+⋯+Zωn, (2)

with the fundamental weights given by

 ⟨ωi,α∨j⟩=δij.

The weight lattice is partitioned into components and this decomposition consists of the root lattice and its shifted copies,

 P=Q∪⋃i∈J(ωi+Q).

The Cartan matrix with entries given by

 Cij=⟨αi,α∨j⟩ (3)

relates the simple roots and fundamental weights as well as their dual versions via formulas

 ωi=n∑j=1C−1ijαj,ω∨i=n∑j=1(CTij)−1α∨j. (4)

The determinant of the Cartan matrix determines the index of connection of the root system and the order of the quotient group ,

 c=detC=|P/Q|=|J|+1.

### 2.3. Sign homomorphisms

Recall from [14] that a homomorphism is called a sign homomorphism. The identity and the determinant sign homomorphisms, which exist for any Weyl group are given on the generating reflections as

 1(ri) = 1, σe(ri) = −1.

For the root systems with two lengths of roots, the short and long sign homomorphisms and are defined as

 σs(ri) ={−1if αi∈Δs,1otherwise, σl(ri) ={−1if αi∈Δl,1otherwise.

To each sign homomorphism is attached a vector defined by

 ϱ1i≡0 i=1,…,n, (5) ϱσei≡1 i=1,…,n, ϱσsi≡1 αi∈Δs, ϱσsi≡0 αi∈Δl, ϱσli≡0 αi∈Δs, ϱσli≡1 αi∈Δl.

The vector becomes the standard vector defined as half-sum of the positive roots. Zero coordinates of the vectors are for convenience defined by

 ϱ10≡0,ϱσe0≡1,ϱσs0≡1,ϱσl0≡0.

Furthermore, to each sign homomorphism is associated a generalized Coxeter number by defining relation

 mσ=n∑i=0m∨iϱσi. (6)

Note that and the number coincides with the standard Coxeter number. The short and long Coxeter numbers are tabulated in [14].

## 3. Extensions of Weyl groups

### 3.1. Affine Weyl groups

The affine Weyl group is a semidirect product of the group of translations and

 Waff=Q∨⋊W. (7)

For and , any element acts on as

 T(q∨)w⋅a≡wa+q∨,a∈Rn.

The standard retraction homomorphism of the semidirect product (7) is given by

 ψ(T(q∨)w)=w.

The fundamental domain of is a simplex explicitly given by

 F={a1ω∨1+⋯+anω∨n|a0+m1a1+⋯+mnan=1,ai≥0,i=0,…,n}.

The stabilizer is a subgroup of stabilizing

 StabWaff(a)={waff∈Waff∣waffa=a},

and the function is defined by

 ε(a)=|W|∣∣StabWaff(a)∣∣. (8)

The stabilizers and are conjugated and therefore the function is invariant

 ε(a)=ε(waffa),waff∈Waff. (9)

The standard action of on the torus generates for its isotropy groups and orbits of orders

 ˜ε(x)≡|Wx|,x∈Rn/Q∨.

The following three properties from Proposition 2.2 in [16] of the action of on the torus are essential,

1. For any , there exists and such that

 x=wx′. (10)
2. If and , , then

 x′=x=wx. (11)
3. If , i.e. , , then and

 Stab(x)≅StabWaff(a). (12)

Relation (12) grants that for , it holds that

 ε(a)=˜ε(x). (13)

Note that instead of , the symbol is used for , in [16, 14]. The algorithm for calculation of the coefficients is described in [16, §3.7].

To each sign homomorphism is associated a subset of the form

 Fσ≡{a∈F∣σ∘ψ(StabWaff(a))={1}}. (14)

The sets are detailed in [16, 14] and described via non-negative symbols defined by

 a10,a1i≥0 if αi∈Δ, aσe0,aσei>0 if αi∈Δ, aσs0,aσsi≥0 if αi∈Δl, aσsi>0 if αi∈Δs, aσl0,aσli>0 if αi∈Δl, aσli≥0 if αi∈Δs.

The explicit formulas for are thus of the form

 Fσ={aσ1ω∨1+⋯+aσnω∨n|aσ0+m1aσ1+⋯+mnaσn=1}. (15)

Note that and .

### 3.2. Dual affine Weyl groups

The dual affine Weyl group is a semidirect product of the group of translations in the root lattice and

 WaffQ≡Q⋊W. (16)

For and , any element acts on as

 T(q)w⋅b≡wb+q,b∈Rn. (17)

The fundamental domain of the dual affine Weyl group , denoted by in [16, 14], is explicitly given by

 FQ≡{b1ω1+⋯+bnωn|b0+m∨1b1+⋯+m∨nbn=1,bi≥0,i=0,…,n}. (18)

By identifying each with its Kac coordinates from (18),

 b≡[b0,b1,…,bn],

the lexicographic ordering on is introduced in the following way. An element is lexicographically higher than ,

 b>lexb′, (19)

if and only if for the first where differs from .

The standard dual retraction homomorphism of the semidirect product (16) is given by

 ˆψ(T(q)w)=w. (20)

Similar to (14), four subsets of are introduced by

 FσQ≡{b∈FQ∣∣σ∘ˆψ(StabWaffQ(b))={1}}.

The domains are explicitly described by

 FσQ={bσ1ω1+⋯+bσnωn|bσ0+m∨1bσ1+⋯+m∨nbσn=1} (21)

with the symbols satisfying

 b10,b1i≥0 if αi∈Δ, (22) bσe0,bσei>0 if αi∈Δ, bσsi≥0 if αi∈Δl, bσs0,bσsi>0 if αi∈Δs, bσli>0 if αi∈Δl, bσl0,bσli≥0 if αi∈Δs.

### 3.3. Extended dual affine Weyl groups

The extended dual affine Weyl group is a semidirect product of the group of translations and the Weyl group

 WaffP≡P⋊W. (23)

For and , extending the action (17) on to elements yields

 T(p)w⋅b≡wb+p,b∈Rn.

The extended dual retraction homomorphism of the semidirect product (23) is a natural extension of (20) given by

 ˆψ(T(p)w)=w.

Introducing the subgroup of all elements of leaving the fundamental domain of invariant

 Γ={γ∈WaffP|γ⋅FQ=FQ} (24)

allows to represent as a semidirect product

 WaffP=WaffQ⋊Γ. (25)

Explicit structure of the abelian group is detailed in [2, Ch.VI,§2] as

 Missing or unrecognized delimiter for \right (26)

The action of on assigns to each fixed a bijection on given on Kac coordinates by

 γ⋅b=γ⋅[b0,…,bn]=[bπγ(0),…,bπγ(n)] (27)

where denotes a permutation of the index set . These permutations of the Kac coordinates , which determine the group , are specified for every simple Lie algebra in Table 1. Direct analysis of the permutations in Table 1 on the vector (5) yields for its coordinates that

 ϱσπγ(i)=ϱσi,i∈{0,1,…,n}. (28)

Considering a resolution factor , an important class of subgroups of is given by

 ΓM={1,γM,i∈WaffP|i∈J},γM,i≡T(Mωi)wiw0,|ΓM|=c. (29)

For each , the subgroup is isomorphic to by an isomorphism

 ΓM∋γM↦γ∈Γ (30)

defined by assigning to . The action of on is directly related to action of by

 γM⋅b=M(γ⋅bM). (31)

Consequently, acts naturally on the magnified fundamental domain . Setting for the magnified Kac coordinates , this action is described by

 γM⋅b=γM⋅[b0,…,bn]=[bπγ(0),…,bπγ(n)]. (32)

To each element is thus assigned the same permutation of from Table 1 as to the corresponding in (27).

## 4. Fundamental domains and invariant polynomials

### 4.1. Stabilizers

The stabilizer is a subgroup of stabilizing and the related counting discrete function is for any defined by

 hP,M(b)=∣∣∣StabWaffP(bM)∣∣∣. (33)

Characterizing the structure of in the following proposition subsequently allows the calculation of the function .

###### Proposition 4.1.

For any it holds that

 StabWaffP(b)=StabWaffQ(b)⋊StabΓ(b). (34)
###### Proof.

The semidirect decomposition (25), where is normal in , directly guarantees that is a normal subgroup of . Moreover, for any stabilized by both and is Conversely, for any exists from (25) a unique decomposition with and . Invariance (24) of under the action of implies that . Since the fundamental domain contains only one point from each orbit, the relation

 wP⋅b=wQ⋅(γ⋅b)=b

forces . ∎

The isomorphism (30) and relation (31) imply for the stabilizers that

 StabΓM(b)≃StabΓ(bM). (35)

The resulting counting formula for is deduced for from relations (34) and (35) as

 hP,M(b)=∣∣∣StabWaffQ(bM)∣∣∣⋅∣∣StabΓM(b)∣∣.

The calculation procedure for is described in [16, §3.7]. The orders of stabilizers are directly derived from the explicit form of permutations given by (27) in Table 1.

Note that since and from (26) and (29) differ only in the translation part, their retractions coincide

 ˆψ(γM)=ˆψ(γ). (36)

Consequently, the retraction of the stabilizers are also identical

 ˆψ(StabΓM(b))=ˆψ(StabΓ(bM)). (37)

### 4.2. Fundamental domains

Significant results concerning the structure of the fundamental domain of the group are achieved in [22]. Firstly, from the semidirect decomposition (25) follows that the fundamental domain coincides with a fundamental domain of the group acting on . Secondly, the interior of is determined as [22]

 int(FP)={b∈int(FQ)|⟨η+α∨i,b⟩<1,i∈J}.

It is also asserted that the extended interior of defined by

 ent(FP)≡{b∈FQ|⟨η+α∨i,b⟩<1,i∈J} (38)

forms a subset of , .

In order to uniquely determine the remaining boundary points from , note that the defining relation in (38)

 bi+b1m∨1+⋯+m∨nbn=⟨η+α∨i,b⟩<1=b0+m∨1b1+⋯+m∨nbn

implies that . Taking into account explicit forms of permutations of from Table 1 and the lexicographic ordering (19), the inequality grants that the point is the lexicographically highest among the points lying in its orbit. Consequently, the fundamental domain is taken as such subset of which contains the lexicographically highest point from each orbit of . This resulting form of the fundamental domain , which contains exactly one point from each orbit of is summarized in the following theorem.

###### Theorem 4.2.

The set defined by

 FP={b∈FQ|b=max>lexΓb} (39)

forms a fundamental domain of the extended dual affine Weyl group

Four crucial subsets of are for each sign homomorphism defined by

 FσP≡{b∈FP|σ∘ˆψ(StabWaffP(b))={1}}. (40)

Recall also from [22] that the stabilizer of any interior point is trivial, therefore it holds that . The form of the sets is simplified in the following proposition.

###### Proposition 4.3.

The sets , defined by (40), are of the form

 FσP={b∈FP∩FσQ|σ∘ˆψ(StabΓ(b))={1}}. (41)
###### Proof.

Since both stabilizers and are subgroups of , it holds for any point that

 σ∘ˆψ(StabWaffQ(