Recursive algebraic construction of two infinite families of polynomials in n 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.

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 rank n, 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 a C- 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 A_n 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 from n fundamental representations. Thus the C polynomials have n variables that are the C-orbit functions, one for each fundamental weight ω_j. 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 Appendix A 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].

Let ℝⁿ 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 - and -bases. In addition we use {e₁, …, e_n}, 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⁺⁺.

Hereafter W = W(G) is the Weyl group of size |W_λ|, and W_λ 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), …, (ω_n/q_n)}, where q_j, are comarks of the highest root.

Definition 2.1. The C-function C_λ(x) is defined as

(2.1)

Occasionally it is useful to scale up C_λ of nongeneric λ by the stabilizer of λ in W.

Definition 2.2. The S-function S_λ(x) is defined as

(2.2)

where p(μ) is the number of elementary reflections necessary to obtain μ from λ.

In this paper, we always suppose that λ, μ ∈ P are given in ω-basis and x ∈ ℝⁿ is given in

-basis, hence the orbit functions have the following forms:

(2.3)

There is a fundamental relation between the C- and S-orbit functions for simple Lie group G of any type and rank, called the Weyl character formula:

(2.4)

The positive integer

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 C_λ(x)- and S_λ(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)

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 domain with the weight function det⁻¹(D(X)/D(x)), where is the image of the fundamental region F under the transformation 𝔗.

(i) The first type of transformation is rather straightforward

(3.2)

Polynomial summands are products , where μ_j ∈ ℤ are components of the orbit points relative to a suitable basis. Under this transformation orbit function, C_λ(x) and S_λ(x), given by (2.3), become Laurent polynomials in n variables X_j, 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 in C polynomials, and 1 or −1 in S polynomials.

(ii) The W-orbits of the Lie groups

(3.3)

have an additional property ±μ ∈ W_λ(L) for all μ ∈ W_λ(L), then the pair of corresponding terms of the function of W_λ(L) can be combined so that C_λ(x) and S_λ(x) become linear combinations of cosines and sines:

(3.4)

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 that λ are given in ω-basis and x is given in -basis, then, using common trigonometric identities, cosines and sines can be expressed through the cosines and sines of 2πkx_i, k ∈ ℕ, i = 1, …, n. What immediately represents our orbit functions as polynomials of Chebyshev polynomials of the first T_k(x_i) and second U_k(x_i) 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 choose C functions of the n fundamental weights as the n new variables:

(3.5)

completed by one more variable, the lowest S function,

(3.6)

The recursive construction of C-polynomials begins by multiplying the variables X_j 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 with “sufficiently large” a₁, a₂, …, a_n (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 irreducible C-function and any character χ_λ of a simple Lie group G can be represented as a polynomial of C-functions of the fundamental weights ω₁, …, ω_n, that is, a polynomial in the variables X₁, X₂, …, X_n.

The recursive construction of S-polynomials starts by multiplying the variables S and X_j 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 S_λ/S should be used instead of S_λ.

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 in S- 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 of C and S functions introduced in this section are partial cases of the Macdonald symmetric polynomials.

All C- 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 Appendices A–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 Appendices A–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 the A_n cases. The crucial substitution (3.5) can be replaced by

(4.1)

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 A_n the two coincide,

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)

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].

Acknowledgments

This work supported in part by the Natural Sciences and Engineering Research Council of Canada, MITACS, and by the MIND Research Institute. M. Nesterenko and A. Tereszkiewicz are grateful for the hospitality extended to her at the Centre de Recherches Mathématiques, Université de Montréal, where a part of the work was carried out.

Appendices

A. Orbit Functions of A₁, Their Polynomial Forms, and Chebyshev Polynomials

A number of multivariate generalizations of classical Chebyshev polynomials are available in the literature [34–39]; the aim of this section is to show in all details how Chebyshev polynomials appear as particular case of the multivariate polynomials proposed in this paper. First we recall that well-known classical Chebyshev polynomials can be obtained independently using only the properties of C- and S-orbit functions of the Lie group A₁, see [40] for details. The C-polynomials generated by our approach are naturally normalized in a different way than the classical polynomials (they coincide with the form of Dickson polynomials).

The orbit functions of A₁ are of two types:

(A.1)

We introduce new variables X and S as follows:

(A.2)

Polynomials can now be constructed recursively in the degrees of X and S by calculating the decompositions of products of appropriate orbit functions. “Generic” recursion relations are those where one of the first degree polynomials, X or S, multiplies the generic polynomial C_m or S_m, that is, m ≥ 1. Omitting the dependence on x from the symbols, we have the generic recursion relations

(A.3)

When solving recursion relations for C polynomials, we need to start from the lowest ones; several results are in Table 1. Hence we conclude that C_m = 2T_m, for m = 0,1, ….

Table 1. The irreducible C and S polynomials of A₁ of degrees up to 8.

C polynomials		S polynomials
# = 0		# = 0
C₂	X² − 2	S₀	1
C₄	X⁴ − 4X² + 2	S₂	X² − 1
C₆	X⁶ − 6X⁴ + 9X² − 2	S₄	X⁴ − 3X² + 1
C₈	X⁸ − 8X⁶ + 20X⁴ − 16X² + 2	S₆	X⁶ − 5X⁴ + 6X² − 1
# = 1		# = 1
C₁	X	S₁	X
C₃	X³ − 3X	S₃	X³ − 2X
C₅	X⁵ − 5X³ + 5X	S₅	X⁵ − 4X³ + 3X
C₇	X⁷ − 7X⁵ + 14X³ − 7X	S₇	X⁷ − 6X⁵ + 10X³ − 4X

The character χ_m(x) of an irreducible representation of A₁ of dimension m + 1 is known explicitly for all m⩾0. There are two ways to write the character: as the ratio of S-functions, and as the sum of C-functions. Explicitly, that is

(A.4)

Note that (A.4) is the Chebyshev polynomial of the second kind U_m(x).

Remark A.1. The main argument in favor of our normalization of Chebyshev polynomials is that polynomials C_m from Table 1 are Dickson polynomials (it is well known that they are equivalent to Chebyshev polynomials over the complex numbers). It is easy to prove (see e.g., [40]) that Weyl group of A_n is equivalent to S_n+1, therefore it is natural to consider multivariate C-polynomials of A_n as n-dimensional generalizations of Dickson polynomials (as permutation polynomials). Also our form of Dickson-Chebyshev polynomials makes them the lowest special case of (2.4) without additional adjustments and it appears more “natural” because, for example, the equality would not hold for T₂ and T₄.

B. Recursion Relations for A₂ Orbit Functions and Polynomials

The variables of the A₂ polynomials are the C functions of the lowest dominant weights ω₁ = (1,0) and ω₂ = (0,1):

(B.1)

We omit writing (x₁, x₂) at the symbols of orbit functions for simplicity of notations.

In addition to the obvious polynomials X₁, X₂, , X₁X₂, and , we recursively find the rest of the A₂-polynomials. The degree of the polynomial C_(a,b) equals a + b. The degree of S_(a,b) is also a + b provided ab ≠ 0, otherwise the S-polynomials are zero.

Due to the A₂ outer automorphism, polynomials C_(a,b) and C_(b,a) are related by the interchange of variables X₁↔X₂ (i.e. C_(a,b)(X₁, X₂) = C_(b,a)(X₂, X₁)).

In general, each term in an irreducible polynomial, equivalently each weight of an orbit, must belong to the same congruence class specified by the congruence number #. For A₂-weight (a, b), we have

(B.2)

Hence, irreducible orbit functions have a well-defined value of #. For A₂-orbit functions, we have #(C_(a,b)) = #(S_(a,b)) = (a + 2b)mod 3. Consequently, there are three classes of polynomials corresponding to # = 0,1, 2. During multiplication, the congruence numbers add up mod 3. A product of irreducible orbits decomposes into the sum of orbits belonging to the same congruence class. The sizes of the irreducible orbits of W(A₂) are found in [30]. The dimension d_(a,b) of the representation of A₂ with the highest weight (a, b) is given by d_(a,b) = (1/2)(a + 1)(b + 1)(a + b + 2).

B.1. Recursion Relations for C-Function Polynomials of A₂

There are two 4-term generic recursion relations for C functions. They are obtained as the decomposition of the products of X and Y, each being a sum of three exponential functions, with a generic C-function which is the sum of |W(A₂)| = 6 exponential terms,

(B.3)

Before generic recursion relations can be used, the special recursion relations for particular values a, b ∈ {0,1} need to be solved recursively starting from the lowest ones:

(B.4)

for a⩾2:

(B.5)

for b⩾2:

(B.6)

Using the symmetry of orbit functions with respect to the permutation of the components of dominant weights, we obtain analogous polynomials C_(a,0) and C_(a,1) for all a ∈ ℕ. Then the 4-term special recursion relations are solved yielding C_(2,b) and C_(a,2) for all a, b ∈ ℕ. After that, the generic recursion relations should be used.

B.2. The Character of A₂

In the A₂ case the general formula (2.4) is specialized

(B.7)

The summation extends over the dominant weights that have positive multiplicities m_λ in the case of χ_(a,b). The coefficients (dominant weight multiplicities) are tabulated in [27] for the 50 first χ_(a,b) in each congruence class of A₂. The first few characters for the congruence class # = 0 are

(B.8)

The equalities must satisfy two relatively simple conditions: (i) the dominant weights on both sides must have the same congruence number (B.2), and (ii) the number of exponential terms in a character χ_(a,b) is known to be the dimension of the irreducible representation (a, b). Therefore, the sizes of the orbit functions on the right side have to add up to the dimension. For # = 1:

(B.9)

For # = 2, it suffices to interchange the component of all dominant weights in the equalities for # = 1. Thus no independent calculation is needed, see Table 2 for the solution.

Table 2. The irreducible C- and S-polynomials of A₂ of degree up to 4. From any polynomial C_(a,b) or S_(a,b) we obtain C_(b,a) or S_(b,a), respectively, by interchanging X₁ and X₂.

C polynomials		S polynomials
# = 0		# = 0
C_(1,1)	X₁X₂ − 3	S_(1,1)	X₁X₂ − 1
C_(3,0)		S_(0,3)
C_(2,2)		S_(2,2)
# = 1		# = 1
C_(1,0)	X₁	S_(1,0)	X₁
C_(0,2)		S_(0,2)
C_(2,1)		S_(2,1)
C_(1,3)		S_(1,3)
C_(4,0)		S_(4,0)

C. Recursion Relations for C₂ Orbit Functions

There are two congruence classes of C₂ orbit functions/polynomials. For C₂ weight (a, b) (dominant or not), we have

(C.1)

The dimension d_(a,b) of an irreducible representation of C₂ with the highest weight (a, b) is given by

(C.2)

In multiplying the polynomials, congruence numbers add up mod 2. Character in the case of C₂ is given by (2.4), where the C and S functions are those of C₂, as are the coefficients m_λ (also tabulated in [27]).

We denote the variables of the C₂-polynomials by

(C.3)

often omitting (x, y) from the symbols. The variable S cannot be built out of X₁ and X₂. Although the variables are denoted by the same symbols as in the case of A₂ (and also G₂ below), they are very different. Thus X₁ and X₂ contain 4 exponential terms and S contains 8 terms. The congruence number # of X₁ and S is 1, while that of X₂ is 0.

C.1. Recursion Relations for C Functions of C₂

The two generic recursion relations for C-functions of C₂ are

(C.4)

The special recursion relations for C-functions involving low values of a and b have to be solved first starting from the lowest ones:

(C.5)

The 3- and 4-term recursion relations are solved independently, giving us C_(0,b), C_(a,0), C_(1,b), and C_(a,1) for all a and b, for example, see Table 3.

Table 3. The irreducible C polynomials of C₂ of degree up to 4.

C polynomials
# = 0
C_(0,1)	X₂
C_(2,0)
C_(2,1)
C_(4,0)
C_(0,2)
C_(0,3)
C_(2,2)
C_(0,4)
# = 1
C_(1,0)	X₁
C_(1,1)	X₁X₂ − 2X₁
C_(3,0)
C_(3,1)
C_(1,2)
C_(1,3)

C.2. Recursion Relations for S Functions of C₂

The generic relations for S functions are readily obtained from those of C functions by replacing C by S, and by making appropriate sign changes.

All C functions of C₂ are real valued. Here are a few examples of C₂ characters:

(C.6)

Using these characters and Table 3, we can calculate all irreducible S polynomials of degree up to four with respect to the variables X₁ and X₂, see Table 4. Note that χ_(0,4) yields the polynomial of order five.

Table 4. The irreducible S polynomials of C₂ of degree up to 4.

S polynomials
# = 0
S_(0,1)	1 + X₂
S_(2,0)
S_(0,2)
S_(2,1)
S_(0,3)
S_(4,0)
S_(2,2)
# = 1
S_(1,0)	X₁
S_(1,1)	X₁X₂
S_(3,0)
S_(1,2)
S_(3,1)
S_(1,3)

D. Recursion Relations for G₂ Orbit Functions

All G₂ weights fall into the same congruence class # = 0. Thus there are no congruence classes to distinguish in G₂. The variables are the orbit functions of the two fundamental weights:

(D.1)

D.1. Recursion Relations for C Functions of G₂

There are two generic recursion relations for C polynomials of G₂, each containing one product term and six C polynomials.

(D.2)

Specializing the first of the generic relations to either a ∈ {0,1, 2} or b ∈ {0,1, 2,3}, we have

(D.3)

Specializing the second of the generic relations to either a ∈ {0,1} or b ∈ {0,1, 2}, we have

(D.4)

Remark D.1. It can be seen from Table 5 that order of C_(a,b) polynomial sometimes exceeds a + b.

Table 5. The irreducible C_(a,b) polynomials of G₂.

C polynomials
C_(1,0)	X₁
C_(0,1)	X₂
C_(0,2)
C_(1,1)
C_(0,3)
C_(2,0)
C_(1,2)
C_(2,1)
C_(1,3)
C_(3,0)
C_(2,2)
C_(3,1)
C_(2,3)
C_(3,2)

D.2. Recursion Relations for S Functions of G₂

Generic recursion relations for S polynomials differ very little from those for C polynomials.

(D.5)

The S polynomials need not be calculated independently. They can be read off the tables [27] as the characters of G₂ representations, see Table 6.

Table 6. The irreducible S_(a,b) polynomials of G₂ with a + b ⩽ 5.

S polynomials
S_(0,1)	X₂ + 1
S_(1,0)	X₁ + X₂ + 2
S_(0,2)
S_(1,1)	X₁X₂ + 2X₁ + 2X₂ + 4
S_(2,0)
S_(1,2)
S_(2,1)
S_(2,2)

Here are all G₂-characters χ_(a,b) with a + b ⩽ 3:

(D.6)

E. Recursion Relations for Lie Algebras of Rank 3

E.1. Recursion Relations for C-Functions of A₃

There are 4 congruence classes of A₃ defined by

(E.1)

The variables of the A₃ polynomials are chosen to be

(E.2)

For C functions the generic recursion relations are the following ones, where we assume a, b, c⩾2:

(E.3)

Note that the first and the third relations are easily obtained from each other by interchanging the first and third component of all dominant weights. Thus X₁↔X₃ and (a, b, c)↔ (c, b, a).

The special recursion relations are obtained from the same products, where some of the components a, b, c of the generic dominant weight take special values 1 and 0. The explicit form of these relations is available in [30] and here we skip them in order to save the space, instead of this we adduce all their solutions of form of Table 7.

Table 7. The irreducible C polynomials and S polynomials of A₃.

C polynomials		S polynomials
# = 0		# = 0
C_(1,0,1)	−4 + X₁X₃	S_(1,0,1)	−1 + X₁X₃
C_(0,2,0)		S_(0,2,0)
C_(0,1,2)		S_(0,1,2)
C_(2,1,0)		S_(2,1,0)
# = 1		# = 1
C_(1,0,0)	X₁	S_(1,0,0)	X₁
C_(0,1,1)	−3X₁ + X₂X₃	S_(0,1,1)	−X₁ + X₂X₃
C_(0,0,3)		S_(0,0,3)
C_(2,0,1)		S_(2,0,1)
C_(1,2,0)		S_(1,2,0)
# = 2		# = 2
C_(0,1,0)	X₂	S_(0,1,0)	X₂
C_(0,0,2)		S_(0,0,2)
C_(2,0,0)		S_(2,0,0)
C_(1,1,1)		S_(1,1,1)
# = 3		# = 3
C_(0,0,1)	X₃	S_(0,0,1)	X₃
C_(1,1,0)	−3X₃ + X₁X₂	S_(1,1,0)	X₁X₂ − X₃
C_(3,0,0)		S_(3,0,0)
C_(1,0,2)		S_(1,0,2)
C_(0,2,1)		S_(0,2,1)

E.2. S Polynomials of A₃

Generic recursion relations are decompositions of the following products, where we assume that a, b, c > 1:

(E.4)

To calculate S polynomials explicitly (see Table 7) we use the A₃ characters. The lowest ones from the congruence classes # = 0, # = 1, # = 2 and # = 3 are listed below:

# = 0:
χ_(0,0,0) = C_(0,0,0) = 1,
χ_(1,0,1) = 3 + C_(1,0,1),
χ_(0,2,0) = 2 + C_(1,0,1) + C_(0,2,0),
χ_(0,1,2) = 3 + 2C_(1,0,1) + C_(0,2,0) + C_(0,1,2),
χ_(2,1,0) = 3 + 2C_(1,0,1) + C_(0,2,0) + C_(2,1,0),
# = 1:
χ_(1,0,0) = C_(1,0,0) = X₁,
χ_(0,1,1) = 2X₁ + C_(0,1,1),
χ_(2,0,1) = 3X₁ + C_(0,1,1) + C_(2,0,1),
χ_(0,0,3) = X₁ + C_(0,1,1) + C_(0,0,3),
χ_(1,2,0) = 3X₁ + 2C_(0,1,1) + C_(2,0,1) + C_(1,2,0),
# = 2:
χ_(0,1,0) = C_(0,1,0) = X₂,
χ_(0,0,2) = X₂ + C_(0,0,2),
χ_(2,0,0) = X₂ + C_(2,0,0),
χ_(1,1,1) = 4X₂ + 2C_(0,0,2) + 2C_(2,0,0) + C_(1,1,1),
# = 3:
χ_(0,0,1) = C_(0,0,1) = X₃,
χ_(1,1,0) = 2X₃ + C_(1,1,0),
χ_(1,0,2) = 3X₃ + C_(1,1,0) + C_(1,0,2),
χ_(3,0,0) = X₃ + C_(1,1,0) + C_(3,0,0),
χ_(0,2,1) = 3X₃ + 2C_(1,1,0) + C_(1,0,2) + C_(0,2,1).

E.3. Recursion Relations for C and S Polynomials of B₃ and C₃

The two cases differ in many important respects in spite of the isomorphism of their Weyl groups.

We write the generic relations for the C polynomials of the Lie algebras B₃ and C₃ (resp., of the simple Lie group O(7) and Sp(6)). The generic relations for the S-polynomials are obtained by replacing the C symbol by S.

The variables are denoted by the same symbols X₁, X₂, X₃ for all algebras of rank 3, namely, , j = 1,2, 3. There are two congruence classes of (a₁, a₂, a₃) for either of the two algebras.

We have

(E.5)

For B₃, we have the generic recursion relations

(E.6)

For C₃, we have the generic recursion relations

(E.7)

Additional recursion relations for both cases are available in explicit form in [32] and here we present only their solutions in form of Tables 8 and 9.

Table 8. C polynomials of B₃ split into two congruence classes # = 0 and # = 1.

C polynomials
# = 0
C_(1,0,0)	X₁
C_(0,1,0)	X₂
C_(2,0,0)
C_(0,0,2)
C_(1,1,0)
C_(1,0,2)
C_(3,0,0)
C_(0,2,0)
C_(0,1,2)
C_(2,1,0)
# = 1
C_(0,0,1)	X₃
C_(1,0,1)	−3X₃ + X₁X₃
C_(0,1,1)	3X₃ − 2X₁X₃ + X₂X₃
C_(0,0,3)
C_(2,0,1)
C_(1,1,1)

Table 9. C polynomials of C₃ split into two congruence classes # = 0 and # = 1.

C polynomials
# = 0
C_(0,1,0)	X₂
C_(2,0,0)
C_(1,0,1)	−2X₂ + X₁X₃
C_(0,2,0)
C_(2,1,0)
C_(0,0,2)
C_(1,1,1)
C_(0,3,0)
C_(0,1,2)
# = 1
C_(1,0,0)	X₁
C_(0,0,1)	X₃
C_(1,1,0)	−4X₁ − 3X₃ + X₁X₂
C_(3,0,0)
C_(0,1,1)	4X₁ + 6X₃ − 2X₁X₂ + X₂X₃
C_(2,0,1)
C_(1,2,0)
C_(1,0,2)
C_(0,2,1)
C_(0,0,3)

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Citing Literature

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Orthogonal Polynomials of Compact Simple Lie Groups

Abstract

1. Introduction

2. Preliminaries and Conventions

3. Multivariate Orthogonal Polynomials Corresponding to Orbit Functions

4. Conclusion

Acknowledgments

Appendices

A. Orbit Functions of A₁, Their Polynomial Forms, and Chebyshev Polynomials

B. Recursion Relations for A₂ Orbit Functions and Polynomials

B.1. Recursion Relations for C-Function Polynomials of A₂

B.2. The Character of A₂

C. Recursion Relations for C₂ Orbit Functions

C.1. Recursion Relations for C Functions of C₂

C.2. Recursion Relations for S Functions of C₂

D. Recursion Relations for G₂ Orbit Functions

D.1. Recursion Relations for C Functions of G₂

D.2. Recursion Relations for S Functions of G₂

E. Recursion Relations for Lie Algebras of Rank 3

E.1. Recursion Relations for C-Functions of A₃

E.2. S Polynomials of A₃

E.3. Recursion Relations for C and S Polynomials of B₃ and C₃

References

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Orthogonal Polynomials of Compact Simple Lie Groups

Abstract

1. Introduction

2. Preliminaries and Conventions

3. Multivariate Orthogonal Polynomials Corresponding to Orbit Functions

4. Conclusion

Acknowledgments

Appendices

A. Orbit Functions of A1, Their Polynomial Forms, and Chebyshev Polynomials

B. Recursion Relations for A2 Orbit Functions and Polynomials

B.1. Recursion Relations for C-Function Polynomials of A2

B.2. The Character of A2

C. Recursion Relations for C2 Orbit Functions

C.1. Recursion Relations for C Functions of C2

C.2. Recursion Relations for S Functions of C2

D. Recursion Relations for G2 Orbit Functions

D.1. Recursion Relations for C Functions of G2

D.2. Recursion Relations for S Functions of G2

E. Recursion Relations for Lie Algebras of Rank 3

E.1. Recursion Relations for C-Functions of A3

E.2. S Polynomials of A3

E.3. Recursion Relations for C and S Polynomials of B3 and C3

References

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Information

A. Orbit Functions of A₁, Their Polynomial Forms, and Chebyshev Polynomials

B. Recursion Relations for A₂ Orbit Functions and Polynomials

B.1. Recursion Relations for C-Function Polynomials of A₂

B.2. The Character of A₂

C. Recursion Relations for C₂ Orbit Functions

C.1. Recursion Relations for C Functions of C₂

C.2. Recursion Relations for S Functions of C₂

D. Recursion Relations for G₂ Orbit Functions

D.1. Recursion Relations for C Functions of G₂

D.2. Recursion Relations for S Functions of G₂

E.1. Recursion Relations for C-Functions of A₃

E.2. S Polynomials of A₃

E.3. Recursion Relations for C and S Polynomials of B₃ and C₃