Four families of Weyl group orbit functions of B_{3} and C_{3}.

# Four families of Weyl group orbit functions of B3 and C3.

Lenka Háková Jiří Hrivnák  and  Jiří Patera
July 9, 2019
###### Abstract.

The properties of the four families of special functions of three real variables, called here , , and functions, are studied. The and functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on finite fragment of lattices of any density and of the symmetry imposed by the weight lattices of and simple Lie algebras/groups. The continuous interpolations, which are induced by the discrete expansions, are exemplified and compared for some model functions.

Centre de Recherches Mathématiques et Département de Mathématiques et de Statistique, Université de Montréal, C. P. 6128 – Centre Ville, Montréal, H3C 3J7, Québec, Canada;
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic;
MIND Research Institute, Irvine, CA 92617, USA

E-mail: hakova@dms.umontreal.ca, jiri.hrivnak@fjfi.cvut.cz, patera@crm.umontreal.ca

## 1. Introduction

Four families of functions, depending on three real variables, called here as , , and functions, are described. Each family is complete within its functional space  and orthogonal when integrated over a finite region of real Euclidean space . Moreover, the functions of each family are discretely orthogonal on a fraction of a lattice in of density of our choice and of symmetry dictated by the simple Lie algebras and .

The definition of these functions for any number of variables and some of their properties are described in [10, 11, 15]. The first two families of functions are also called and functions. They are well known as constituents of irreducible characters of compact simple Lie groups. Indeed, the irreducible characters can be expressed as finite sums of functions with integer coefficients called dominant weight multiplicities. Ratio of two functions appears in the Weyl character formula . Two other families are called and functions. Inspired by the Weyl character formula, we can consider so called hybrid characters, i.e. the ratio of two or functions. It is possible to show that they decompose into the sums of functions with integer coefficients. The explicit formulas for coefficients are in .

It is well known that the functions play an important role in the definition of Jacobi polynomials [3, 12, 13]. One can also remark that the characters and the hybrid characters could be derived as special cases of Jacobi polynomials. However, important insight into the properties of these functions would have been lost in the generality of the approach. For example, their discrete orthogonality appears to be outside of that approach, which is a handicap in the world of ever growing amount of digital data. From our perspective particularly useful are the four families discretized within . The problem of discretization of the and functions, which is now over 20 years old [5, 16, 17, 18], is carried over to the dicretization of the other two families in .

Our motivation for studying these families of functions is guided by ever increasing need to process three-dimensional digital data. Discrete orthogonality of these functions open new efficient possibilities for precisely that. Moreover, the symmetry imposed by the weight lattices of and should be advantageous in describing quantum systems possessing such a symmetry, as well as in some problems of quantum information theory.

There are two undoubtedly interesting extensions of this work. The first one is to combine pairs of functions from the present four families into so called functions, two-variable generalization of the common exponential function are found in . In this way we can find six families of functions, which again are orthogonal as continuous and also as discrete functions over finite extension of .

The second possible extension of this work is to three-variables orthogonal polynomials. Curiously, the extensive literature about the polynomials contains little information about their discretization. This work opens a way to study these problems.

The paper is organized as follows. Section 2 reviews some basic notations concerning the root systems and Weyl groups. We present the short and long fundamental domains of the affine Weyl group of and . In Section 3 we define four families of orbit functions. In Section 4 we describe in detail orbit functions and including their discrete orthogonality and discrete transforms.

## 2. Root systems, affine Weyl groups and fundamental domains

### 2.1. Root systems

Consider a simple Lie algebra of rank three and its ordered set of simple roots . The set forms a basis of the three-dimensional real Euclidean space [8, 9] and satisfies certain specific conditions. There are only two simple Lie algebras of rank three which, with respect to the standard scalar product , have two different lenghts of their simple roots — and . The set is for these two cases decomposed into the set of short simple roots and the set of long simple roots :

 Δ=Δs∪Δl. (1)

The set is usually described by the Coxeter–Dynkin diagram and its corresponding Coxeter matrix or, equivalently, by the Cartan matrix . The vectors called coroots are defined as renormalizations of roots: . In addition to the basis of simple roots and the basis, the following two bases are useful: the weight basis, defined by the relations

 ⟨α∨i,ωj⟩=δij,i,j∈{1,2,3},

and the coweight basis, given by renormalization as

Standardly, the root lattice is the set of all integer linear combinations of the simple roots

 Q=Zα1+Zα2+Zα3

and the coroot lattice is

 Q∨=Zα∨1+Zα∨2+Zα∨3.

The weight lattice and the coweight lattice are given standardly as

 P=Zω1+Zω2+Zω3,P∨=Zω∨1+Zω∨2+Zω∨3.

Two important subsets of the weight lattice are the cone of dominant weights and the cone of strictly dominant weights :

 P+=Z≥0ω1+Z≥0ω2+Z≥0ω3 ⊃ P++=Nω1+Nω2+Nω3.

The decomposition (1) induces two subsets of which are crucial for description of the orbit functions. The first excludes points from which are orthogonal to short roots,

 P+s={ω∈P+|(∀α∈Δs)(⟨ω,α⟩>0)} (2)

and the second excludes points orthogonal to long roots

 P+l={ω∈P+|(∀α∈Δl)(⟨ω,α⟩>0)}. (3)

### 2.2. Affine Weyl groups

The reflection , , which fixes the plane orthogonal to and passing through the origin, is explicitly written for as . Weyl group is a finite group generated by reflections . The system of vectors obtained by the action of on the set of simple roots forms a root system which contains its unique highest root . The marks are the coefficients of the highest root in basis, .

The affine reflection with respect to this highest root is given by

 r0x=rξx+2ξ⟨ξ,ξ⟩,rξx=x−2⟨x,ξ⟩⟨ξ,ξ⟩ξ,x∈R3.

The affine Weyl group is generated by reflections from the set . The decomposition (1) induces a decomposition of the generator set ,

 R=Rs∪Rl (4)

where the subsets and are given by

 Rs ={rα|rα∈Δs} Rl ={rα|rα∈Δl}∪{r0}.

The affine Weyl group consists of orthogonal transformations from and of shifts by vectors from the coroot lattice . The fundamental domain of the action of on is a tetrahedron with vertices .

The set of reflections corresponding to dual roots also generates the Weyl group . The system of vectors is a root system and contains the highest dual root . The dual marks are the coefficients of the dual highest root in basis, .

The dual affine reflection with respect to the highest dual root is given by

 r∨0x=rηx+2η⟨η,η⟩,rηx=x−2⟨x,η⟩⟨η,η⟩η,x∈R3.

The dual affine Weyl group is generated by reflections from the set , see . The decomposition (1) also induces a decomposition of the generator set ,

 R∨={r∨0,r1,r2,r3}=Rs∨∪Rl∨ (5)

where the subsets and are given by

 Rs∨ ={rα|rα∈Δs}∪{r∨0} Rl∨ ={rα|rα∈Δl}.

The dual affine group consists of orthogonal transformations from and of shifts by vectors from the root lattice . The dual fundamental domain of the action of on is a tetrahedron with vertices .

### 2.3. Short and long fundamental domains

The boundary of the fundamental domain consists of points stabilized by the generators from . Two types of the boundaries of are determined by the decomposition (4) — those points which are stabilized by are collected in the short boundary ,

 Hs={a∈F|(∃r∈Rs)(ra=a)}

and the points stabilized by in the long boundary ,

 Hl={a∈F|(∃r∈Rl)(ra=a)}.

The points from which do not lie on the short boundary form the short fundamental domain ,

 Fs=F∖Hs

and the points which do not lie on the long boundary form the long fundamental domain ,

 Fl=F∖Hl.

Similarly, the boundary of the dual fundamental domain consists of points stabilized by the generators from and two types of the boundaries of are determined by the decomposition (5). The points which are stabilized by are collected in the short dual boundary ,

 Hs∨={a∈F∨|(∃r∈Rs∨)(ra=a)}

and the points stabilized by in the long dual boundary ,

 Hl∨={a∈F∨|(∃r∈Rl∨)(ra=a)}.

The points from which do not lie on the short dual boundary form the short dual fundamental domain ,

 Fs∨=F∨∖Hs∨

and the points not on the long dual boundary form the long dual fundamental domain ,

 Fl∨=F∨∖Hl∨.

### 2.4. The Lie algebra B3

For practical purposes, the most convenient way of specifying the root system and the bases , and is to evaluate their coordinates in a fixed orthonormal basis. With respect to the standard orthonormal basis of , these four bases of are of the form

 α1 =(1,−1,0), ω1 =(1,0,0), α∨1 =(1,−1,0), ω∨1 =(1,0,0), α2 =(0,1,−1), ω2 =(1,1,0), α∨2 =(0,1,−1), ω∨2 =(1,1,0), α3 =(0,0,1), ω3 =(12,12,12), α∨3 =(0,0,2), ω∨3 =(1,1,1).

In this setting it holds that and , which means that , are the long roots and is the short root of — the decomposition (1) is

 Δs={α3},Δl={α1,α2}.

The short and long subsets (2), (3) of the grid are of the form

 P+s=Z≥0ω1+Z≥0ω2+Nω3,P+l=Nω1+Nω2+Z≥0ω3.

The highest root and the dual highest root are given as

 ξ=α1+2α2+2α3,η=2α∨1+2α∨2+α∨3

which determine the fundamental domain and the dual fundamental domain explicitly as

 F ={y1ω∨1+y2ω∨2+y3ω∨3|y0,y1,y2,y3∈R≥0,y0+y1+2y2+2y3=1}, F∨ ={z1ω1+z2ω2+z3ω3|z0,z1,z2,z3∈R≥0,z0+2z1+2z2+z3=1}.

The induced decompositions (4), (5) are of the form

 Rs ={r3}, Rl ={r0,r1,r2}, Rs∨ ={r∨0,r3}, Rl∨ ={r1,r2},

which give the short and the long fundamental domains explicitly

 Fs ={ys1ω∨1+ys2ω∨2+ys3ω∨3|ys0,ys1,ys2∈R≥0,ys3∈R>0,ys0+ys1+2ys2+2ys3=1}, Fl ={yl1ω∨1+yl2ω∨2+yl3ω∨3|yl0,yl1,yl2∈R>0,yl3∈R≥0,yl0+yl1+2yl2+2yl3=1},

together with their dual versions

 Fs∨ ={zs1ω1+zs2ω2+zs3ω3|zs0,zs3∈R>0,zs1,zs2∈R≥0,zs0+2zs1+2zs2+zs3=1}, Fl∨ ={zl1ω1+zl2ω2+zl3ω3|zl0,zl3∈R≥0,zl1,zl2∈R>0,zl0+2zl1+2zl2+zl3=1}.

The and bases, together with the fundamental domains , and of are depicted in Figure 1. Figure 1. The α and ω−bases and the fundamental domain F of B3. The tetrahedron F without the grey back face, which depicts Hs, is the short fundamental domain Fs; the tetrahedron F without the three unmarked faces is the long fundamental domain Fl.

### 2.5. The Lie algebra C3

With respect to the standard orthonormal basis of , the four bases of are of the form

 α1 =1√2(1,−1,0), ω1 =1√2(1,0,0), α∨1 =√2(1,−1,0), ω∨1 =(√2,0,0), α2 =1√2(0,1,−1), ω2 =1√2(1,1,0), α∨2 =√2(0,1,−1), ω∨2 =√2(1,1,0), α3 =(0,0,√2), ω3 =1√2(1,1,1), α∨3 =(0,0,√2), ω∨3 =1√2(1,1,1).

In this setting it holds that and , which means that , are the short roots and is the long root of — the decomposition (1) is

 Δs={α1,α2},Δl={α3}.

The short and long subsets (2), (3) of the grid are of the form

 P+s=Nω1+Nω2+Z≥0ω3,P+l=Z≥0ω1+Z≥0ω2+Nω3.

The highest root and the dual highest root are given as

 ξ=2α1+2α2+α3,η=α∨1+2α∨2+2α∨3

which determine the fundamental domain and the dual fundamental domain explicitly as

 F ={y1ω∨1+y2ω∨2+y3ω∨3|y0,y1,y2,y3∈R≥0,y0+2y1+2y2+y3=1}, F∨ ={z1ω1+z2ω2+z3ω3|z0,z1,z2,z3∈R≥0,z0+z1+2z2+2z3=1}.

The induced decompositions (4), (5) are of the form

 Rs ={r1,r2}, Rl ={r0,r3}, Rs∨ ={r∨0,r1,r2}, Rl∨ ={r3},

which give the short and the long fundamental domains explicitly

 Fs ={ys1ω∨1+ys2ω∨2+ys3ω∨3|ys0,ys3∈R≥0,ys1,ys2∈R>0,ys0+2ys1+2ys2+ys3=1}, Fl ={yl1ω∨1+yl2ω∨2+yl3ω∨3|yl0,yl3∈R>0,yl1,yl2∈R≥0,yl0+2yl1+2yl2+yl3=1},

together with their dual versions

 Fs∨ ={zs1ω1+zs2ω2+zs3ω3|zs0,zs1,zs2∈R>0,zs3∈R≥0,zs0+zs1+2zs2+2zs3=1}, Fl∨ ={zl1ω1+zl2ω2+zl3ω3|zl0,zl1,zl2∈R≥0,zl3∈R>0,zl0+zl1+2zl2+2zl3=1}.

The and bases, together with the fundamental domains , and of are depicted in Figure 2. Figure 2. The α and ω−bases and the fundamental domain F of C3. The tetrahedron F without the two grey faces, which depict Hs, is the short fundamental domain Fs; the tetrahedron F without the two unmarked faces is the long fundamental domain Fl.

## 3. Orbit functions

### 3.1. Orbits and stabilizers

Considering any , the stabilizer of is the set and its order is denoted by ,

 dλ≡|StabW(λ)|. (6)

For calculation of continuous orthogonality of various types of orbit functions, the number of elements in the Weyl group and the volume of the fundamental domain are needed. Their product is denoted by and it holds that (see e.g. )

 K≡|W||F|={2for B32√2for C3. (7)

The orbits and the stabilizers on the torus are needed for the discrete calculus of orbit functions. An arbitrarily chosen natural number controls the density of the grids appearing in this calculus . The discrete calculus of orbit functions is performed over the finite group . The finite complement set of weights is taken as the quotient group . For any , its orbit by the action of is given by and its order is denoted by ,

 ε(x)≡|Wx|. (8)

For any , its stabilizer by the action of is given by

 Stab∨(λ)={w∈W|wλ=λ}.

and its order is denoted by

 h∨λ≡|Stab∨(λ)|. (9)

Moreover, for calculation of discrete orthogonality of various types of orbit functions, the determinant of the Cartan matrix is needed. The product is denoted by and it holds that (see e.g. )

 k≡|W|detC=96,for B3,C3. (10)

### 3.2. Four types of orbit functions

The Weyl group can also be abstractly defined by the presentation of a Coxeter group

 r2i=1,(rirj)mij=1,i,j=1,2,3, (11)

where integers denote elements of the Coxeter matrix. The Coxeter matrices of and are of the form

 M(B3)=⎛⎜⎝132314241⎞⎟⎠,M(C3)=⎛⎜⎝142413231⎞⎟⎠. (12)

Crucial tool for defining the orbit functions are ’sign’ homomorphisms . A sign homomorphism can be defined by prescribing its values on the generators of . An admissible mapping has to satisfy the presentation condition (11)

 σ(ri)2=1,(σ(ri)σ(rj))mij=1,i,j=1,2,3. (13)

Two obvious choices and for every lead to the standard homomorphisms and with values given for any as

 1(w) =1, σe(w) =detw.

It turns out that for root systems with two different lengths of roots there are two other available choices . This can also be directly seen for the cases of and — the non-diagonal elements of the Coxeter matrices (12) are even except for the elements corresponding simultaneously to two short roots or two long roots. Therefore, if we set one value of on all the short roots and, independently, another value for all the long roots, the admissibility condition (13) is still satisfied. Consequently, there are two more sign homomorphisms, denoted by and ,

 σs(rα) ={1,α∈Δl−1,α∈Δs σl(rα) ={1,α∈Δs−1,α∈Δl.

Each of the sign homomorphisms , , and induces a family of complex orbit functions. The functions in each family are labeled by the weights and their general form is

 ψσλ(a)=∑w∈Wσ(w)e2πi⟨wλ,a⟩,a∈R3. (14)

Among the basic general properties of all these families of orbit functions are their invariance with respect to shifts from

 ψσλ(a+q∨)=ψσλ(a) (15)

and their invariance or antiinvariance with respect to action of elements from

 ψσλ(wa) =σ(w)ψσλ(a) (16) ψσwλ(a) =σ(w)ψσλ(a). (17)

### 3.3. C− and S−functions

Choosing , so-called functions  are obtained from (14). Following , these functions are here denoted by the symbol . The invariance (15), (16) with respect to the affine Weyl group allows to consider only on the fundamental domain . Similarly, the invariance (17) restricts the weights to the set . Thus, we have

 Φλ(x)=∑w∈We2πi⟨wλ,x⟩,x∈F,λ∈P+.

For the detailed review of functions see  and references therein. Continuous and discrete orthogonality of functions are studied for all rank two cases in detail in [20, 21]. The discretization properties of functions on a finite fragment of the grid is described in full generality in .

Choosing , well–known functions  are obtained from (14). Following [5, 11], these functions are here denoted by the symbol . The invariance (15) together with the antiinvariance (16) allows to consider only on the interior of the fundamental domain . Similarly, the antiinvariance (17) restricts the weights to the set . Thus, we have

 φλ(x)=∑w∈W(detw)e2πi⟨wλ,x⟩,x∈F∘,λ∈P++.

For the detailed review of functions see  and references therein. Continuous and discrete orthogonality of functions are studied for all rank two cases in detail in . The discretization properties of functions on a finite fragment of the grid is described in full generality in .

## 4. Ss− and Sl−functions

### 4.1. Ss−functions

Choosing , so–called functions  are obtained from (14). Following , these functions are here denoted by the symbol . The antiinvariance (16) with respect to the short reflections together with shift invariance (15) imply zero values of functions on the boundary ,

 φsλ(a′)=0,a′∈Hs.

Therefore, the functions are considered on the fundamental domain only. Similarly, the antiinvariance (17) restricts the weights to the set . Thus, we have

 φsλ(x)=∑w∈Wσs(w)e2πi⟨wλ,x⟩,x∈Fs,λ∈P+s.

#### 4.1.1. Continuous orthogonality and Ss−transforms

For any two weights the corresponding functions are orthogonal on

 ∫Fsφsλ(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯φsλ′(x)dx=Kdλδλλ′ (18)

where , are given by (6), (7), respectively. The functions determine symmetrized Fourier series expansions,

 f(x)=∑λ∈P+scsλφsλ(x),where csλ=1Kdλ∫Fsf(x)¯¯¯¯¯¯¯¯¯¯¯¯¯φsλ(x)dx.

#### 4.1.2. Discrete orthogonality and discrete Ss−transforms

The finite set of points is given by

 FsM=1MP∨/Q∨∩Fs

and the corresponding finite set of weights as

 ΛsM=P/MQ∩MFs∨.

Then, for , the following discrete orthogonality relations hold,

 ∑x∈FsMε(x)φsλ(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯φsλ′(x)=kM3h∨λδλλ′ (19)

where , and are given by (8), (9) and (10), respectively. The discrete symmetrized function expansion is given by

 f(x)=∑λ∈ΛsMcsλφsλ(x),where csλ=1kM3h∨λ∑x∈FsMε(x)f(x)¯¯¯¯¯¯¯¯¯¯¯¯¯φsλ(x). (20)

#### 4.1.3. Ss−functions of B3

For a point with coordinates in basis and a weight with coordinates in basis of , the coressponding functions are explicitly evaluated as

 φsλ (x,y,z)=2i{sin(2π(ax+by+cz))+sin(2π(−ax+(a+b)y+cz)) +sin(2π((a+b)x−by+(2b+c)z))−sin(2π(ax+(b+c)y−cz)) +sin(2π(bx−(a+b)y+(2a+2b+c)z))−sin(2π(−ax+(a+b+c)y−cz)) +sin(2π(−(a+b)x+ay+(2b+c)z))−sin(2π((a+b)x+(b+c)y−(2b+c)z)) −sin(2π((a+b+c)x−(b+c)y+(2b+c)z))+sin(2π(−bx−ay−(2a+2b+c)z)) −sin(2