# Orthogonal Polynomials of Compact Simple Lie Groups

Authors: Maryna Nesterenko; Jiří Patera; Agnieszka Tereszkiewicz
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Recursive algebraic construction of two infinite families of polynomials inn variables is
proposed as a uniform method applicable to every semisimple Lie group of rank n. Its result
recognizes Chebyshev polynomials of the first and second kind as the special case of the
simple group of type A1. The obtained not Laurent-type polynomials are equivalent to the
partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for
the Lie groups of types A1, A2, A3, C2, C3, G2, and B3 together with lowest polynomials.


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## Body

## 1. Introduction

The majority of special functions and orthogonal polynomials introduced during the last decade are associated with Lie groups or their generalizations. In particular, special functions of mathematical physics are in fact matrix elements of representations of Lie groups [1] and recent multivariate generalizations of classical hypergeometric orthogonal polynomials are based on root systems of simple Lie groups/algebras [2–9]. In this connection a number of elegant results in theory of these polynomials, such as explicit (determinantal) computation of polynomials [10–12] and Pieri formulas [13, 14], were obtained, see also [15–17] and references therein.Our primary objective at this stage is to establish a constructive method for finding orthogonal multivariate polynomials related to orbit functions of simple Lie groups of rankn, indeed, for actually seeing them. As far as we deal with the functions invariant/skew-invariant under the action of the corresponding Weyl group the obtained polynomials appear as building blocks in all multivariate polynomials associated with root systems. Unlike Gram-Schmidt type orthogonalization of the monomial basis with respect to Haar measure [3, 7, 8] or determinantal construction of polynomials [10–12] we make profit from decomposition of products of Weyl group orbits and from basic properties of the characters of irreducible finite dimensional representations.Our method is purely algebraic and we propose three different ways to transform aC- or S-orbit function into a polynomial. The first one substitutes for each multivariable exponential term in an orbit function a monomial of as many variables. In 1D this results in Chebyshev polynomials written as Laurent polynomials with symmetrically placed positive and negative powers of the variable; and in the case of A2 our results coincide with those from [18].The second transformation, the “truly trigonometric” form, is based on the fact that, for many simple Lie algebras (see the list in (3.3) below), each C and S-orbit function consists of pairs of exponential terms that add up to either cosine or sine. Hence such a function is a sum of trigonometric terms (or a polynomial of one-dimensional Chebyshev polynomials). For the Chebyshev polynomials we obtain in this way their trigonometric form. Note from (3.3) that this method does not apply to the groups An for n>1.But this paper focuses on polynomials obtained by the third substitution of variables, mimicking Weyl's method for the construction of finite-dimensional representations fromn fundamental representations. Thus the C polynomials have n variables that are the C-orbit functions, one for each fundamental weight ωj. This approach results in a simple recursive construction that allows one to represent any orbit function/monomial symmetric function in non-Laurent polynomial form.In addition to the general approach and associated tools we present a lot of explicit and practically useful data and discussions, namely, in AppendixA  we compare the classical Chebyshev polynomials (Dickson polynomials) and orbit functions of A1 with their recursion relations. Suitably normalized, the Chebyshev polynomials of the first and second kind coincide with the C and S polynomials. A table of the polynomials of each kind is presented. Appendices B, C, and D contain, respectively, the recursion relations for polynomials of the Lie algebras A2, C2, and G2. In Appendix E recursion relations for A3, B3, and C3 polynomials of both kinds are listed together with useful tools for solving these recursion relations.


## 2. Preliminaries and Conventions

This section serves to fix notations and terminology, additional details can be found for example in [19–26].Letℝn be the Euclidean space spanned by the simple roots of a simple Lie group G. The basis of the simple roots and the basis of fundamental weights are hereafter referred to as the α-basis and ω-basis, respectively. Bases dual to α- and ω-bases are denoted by α̌- and ω̌-bases. In addition we use {e1,…,en}, the orthonormal basis of ℝn. The root lattice Q and the weight lattice P of G are formed by all integer linear combinations of the α-basis and ω-basis, respectively. In P we define the cone of dominant weights P+ and its subset of strictly dominant weights P++.HereafterW=W(G) is the Weyl group of size |Wλ|, and Wλ is the orbit containing the (dominant) point λ∈P+⊂ℝn. The fundamental region F(G)⊂ℝn is the convex hull of the vertices {0,(ω1/q1),…,(ωn/qn)}, where qj, j=1,n¯ are comarks of the highest root.Definition 2.1.
TheC-function Cλ(x) is defined as
(2.1)Cλ(x)∶=∑μ∈Wλ(G)e2πi〈μ,x〉,x∈Rn,λ∈P+.
Occasionally it is useful to scale upCλ of nongeneric λ by the stabilizer of λ in W.Definition 2.2.
TheS-function Sλ(x) is defined as
(2.2)Sλ(x)∶=∑μ∈Wλ(G)(-1)p(μ)e2πi〈μ,x〉,x∈Rn,λ∈P++,
where p(μ) is the number of elementary reflections necessary to obtain μ from λ.In this paper, we always suppose thatλ,μ∈P are given in ω-basis and x∈ℝn is given in α̌-basis, hence the orbit functions have the following forms: (2.3)Cλ(x)=∑μ∈Wλe2πi∑j=1nμjxj=∑μ∈Wλ∏j=1ne2πiμjxj,Sλ(x)=∑μ∈Wλ(-1)p(μ)e2πi∑j=1nμjxj=∑μ∈Wλ(-1)p(μ)∏j=1ne2πiμjxj.There is a fundamental relation between theC- and S-orbit functions for simple Lie group G of any type and rank, called the Weyl character formula: (2.4)χλ(x)=Sλ+ρ(x)Sρ(x)=∑μmμλCμ(x),x∈Rn,λ,μ∈P+,ρ=∑k=1nωk.
The positive integer mμλ is the Kostka number [27, 28].The rank of the underlying semisimple Lie group/algebra is the number of variables of the orbit functions.C and S functions are continuous and have continuous derivatives; they are, respectively, symmetric and antisymmetric with respect to the (n-1)-dimensional boundary of F [23–25]. Moreover, any pair of orbit functions from the same family is orthogonal on the corresponding fundamental region [20], these families of functions are complete, and Cλ(x)- and Sλ(x)-orbit functions are eigenfunctions of the n-dimensional Laplace operator.


## 3. Multivariate Orthogonal Polynomials Corresponding to Orbit Functions

In this section we consider several transformations(3.1)T:Rn⟶Cn;(x1,…,xn)⟼(X1,…,Xn),
that represent the C(x)- and S(x)-orbit functions in polynomial form.It directly follows from the orthogonality of the orbit functions that such polynomials are orthogonal on the domainF̃ with the weight function det-1(D(X)/D(x)), where F̃ is the image of the fundamental region F under the transformation 𝔗.(i) The first type of transformation is rather straightforward(3.2)Xj∶=e2πixj,xj∈R,j=1,2,…,n.Polynomial summands are products∏j=1nXjμj, where μj∈ℤ are components of the orbit points relative to a suitable basis. Under this transformation orbit function, Cλ(x) and Sλ(x), given by (2.3), become Laurent polynomials in n variables Xj, where j={1,2,…,n}.The exponential substitution polynomials are complex-valued in general, admit negative powers, and have all their coefficients equal to one inC polynomials, and 1 or −1 in S polynomials.(ii) TheW-orbits of the Lie groups
(3.3)A1,Bn(n≥3),Cn(n≥2),D2n(n≥2),E7,E8,F4,G2
have an additional property ±μ∈Wλ(L) for all μ∈Wλ(L), then the pair of corresponding terms of the function of Wλ(L) can be combined so that Cλ(x) and Sλ(x) become linear combinations of cosines and sines:
(3.4)e2πi〈μ,x〉+e-2πi〈μ,x〉=2cos⁡(2π〈μ,x〉)∈Cλ(x),e2πi〈μ,x〉-e-2πi〈μ,x〉=2isin⁡(2π〈μ,x〉)∈Sλ(x),
that admits a truly trigonometric substitution of variables. Note that in the case of A1, this is precisely the trigonometric substitution made for Chebyshev polynomials of the first and second kind.Remark 3.1.
Chebyshev polynomials of one variable play crucial role for the orbit functions of the above-mentioned Lie groups as far as they allow us to calculate the polynomial coefficients explicitly.
Really, as far as we suppose thatλ are given in ω-basis and x is given in α̌-basis, then, using common trigonometric identities, cosines and sines can be expressed through the cosines and sines of 2πkxi, k∈ℕ, i=1,…,n. What immediately represents our orbit functions as polynomials of Chebyshev polynomials of the first Tk(xi) and second Uk(xi) kind with well-known formulas for coefficients.(iii) The third transformation, which we propose here, works uniformly for simple Lie algebras of all types. We chooseC functions of the n fundamental weights as the n new variables:
(3.5)Xj∶=Xj(x)∶=Cωj(x),j=1,2,…,n,x∈Rn
completed by one more variable, the lowest S function,
(3.6)S∶=Sρ(x),x∈Rn,ρ=(1,1,…,1)=∑m=1nωm.
The recursive construction of C-polynomials begins by multiplying the variables Xj and C-functions and decomposing their products into sums of C-polynomials. A judicious choice of the sequence of products allows one to find ever higher degree C-polynomials.First,generic recursion relations are found as the decomposition of products XjC(a1,a2,…,an) with “sufficiently large” a1,a2,…,an (i.e., all C functions in the decomposition should correspond to generic points). Then the rest of necessary recursions (“additional”) are constructed. An efficient way to find the decompositions is to work with products of Weyl group orbits, rather than with orbit functions. Their decomposition has been studied, and many examples have been described in [29]. Note that these recursion relations are always linear and the corresponding matrix is triangular. The procedure is exemplified in Appendices A–E for Simple Lie groups of ranks 1, 2, and 3.Results of the recursive procedures can be summarized as follows (see [30] for the proof).Proposition 3.2.
Any irreducibleC-function and any character χλ of a simple Lie group G can be represented as a polynomial of C-functions of the fundamental weights ω1,…,ωn, that is, a polynomial in the variables X1,X2,…,Xn.The recursive construction ofS-polynomials starts by multiplying the variables S and Xj and decomposing their products into sums of S polynomials. However, the higher the rank of the underlying Lie algebra, the recursive procedure for S polynomials becomes more laborious, what caused by the presence of negative terms in S polynomials. Fortunately, there is an alternative to the recursive procedure. Once the C polynomials have been calculated, they can be used in Weyl character formula for finding S polynomials as sums of C polynomials multiplied by the variable S. In practice, polynomials Sλ/S should be used instead of Sλ.Remark 3.3.
There are two easy and practical checks on recursion relations applicable to all simple Lie algebras. The first one is the equality of numbers of exponential terms inS- or C-functions on both sides of a recursion relation (the numbers of exponential terms are calculated using the sizes of Weyl group orbits). The second check is the equality of congruence numbers.Remark 3.4.
Polynomial forms ofC and S functions introduced in this section are partial cases of the Macdonald symmetric polynomials.
AllC- and S-orthogonal polynomials (and, therefore, the Macdonald polynomials) inherit from orbit functions important discretization properties. A uniform discretization of these polynomials follows from their invariance with respect to the affine Weyl group of G and from the well-established discretization of the fundamental region F(G) [20]. One more advantage is the cubature formula introduced in [31].For the application reason in AppendicesA–E we present recurrence relations and lowest polynomials for the simple Lie groups A1, A2, C2, G2, A3, B3, and C3. All cases contain both generic and additional recursions or, instead of cumbersome additional recursions, we present all their solutions in form of lowest polynomials. The skipped explicit formulas are available in [30, 32].The content of AppendicesA–E is also motivated by the fact that calculation of additional recurrences is not suitable for complete computer automatization. However, as soon as additional recurrences (or their solutions) were obtained, all other calculations concerning polynomials and their applications become very algorithmic and can easily be done by computer algebra packages for Lie theory.


## 4. Conclusion

There is an alternative way to our construction of the polynomials in all but in theAn cases. The crucial substitution (3.5) can be replaced by (4.1)Xk∶=χωk(x),k=1,2,…,n.
In (4.1) the variables are characters of irreducible representations with highest weights given as the fundamental weights, while in (3.5) the variables are C functions of the fundamental weights. Only for An the two coincide, Cωk(x)=χωk(x) for all k=1,…,n and for all x∈ℝn. Already for the rank two cases other than A2 there is a difference. Indeed, (4.1) reads as follows: (4.2)C2:X1=χω1(x)=Cω1(x),X2=χω2(x)=Cω2(x)+2,G2:X1=χω1(x)=Cω1(x)+Cω2(x)+2,X2=χω2(x)=Cω2(x)+1.
Since products of characters decompose into their sum, the recursive construction can proceed, but the polynomials will be different.For simplicity of formulation, we insisted throughout this paper that the underlying Lie group be simple. The extension to compact semisimple Lie groups and their Lie algebras is straightforward. Thus, orbit functions are products of orbit functions of simple constituents, and different types of orbit functions can be mixed.Polynomials formed from other orbit functions (E-, E--, E+-, S+-, S--, C+-, C--functions) by the same substitution of variables should be equally interesting once n>1. These functions have been studied in [20, 25, 26, 33].



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