Orthogonal Polynomials of Compact Simple Lie Groups

Abstract

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 ℝ^𝑛 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,…,𝑒_𝑛}, the orthonormal basis of ℝ^𝑛. 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⁺⊂ℝ^𝑛. The fundamental region 𝐹(𝐺)⊂ℝ^𝑛 is the convex hull of the vertices {0,(𝜔₁/𝑞₁),…,(𝜔_𝑛/𝑞_𝑛)}, where 𝑞_𝑗, 𝑗=1,𝑛 are comarks of the highest root.

Definition 2.1. The 𝐶-function 𝐶_𝜆(𝑥) is defined as 𝐶_𝜆(𝑥)∶=_{𝜇∈𝑊𝜆(𝐺)}𝑒^{2𝜋𝑖⟨𝜇,𝑥⟩},𝑥∈ℝ^𝑛,𝜆∈𝑃⁺. (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 𝑆_𝜆(𝑥)∶=_{𝜇∈𝑊𝜆(𝐺)}(−1)^𝑝(𝜇)𝑒^{2𝜋𝑖⟨𝜇,𝑥⟩},𝑥∈ℝ^𝑛,𝜆∈𝑃⁺⁺, (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∈ℝ^𝑛 is given in 𝛼̌-basis, hence the orbit functions have the following forms: 𝐶_𝜆(𝑥)=_{𝜇∈𝑊𝜆}𝑒^{2𝜋𝑖∑𝑛𝑗=1𝜇𝑗𝑥𝑗}=_{𝜇∈𝑊𝜆}^𝑛_𝑗=1𝑒^{2𝜋𝑖𝜇𝑗𝑥𝑗},𝑆_𝜆(𝑥)=_{𝜇∈𝑊𝜆}(−1)^𝑝(𝜇)𝑒^{2𝜋𝑖∑𝑛𝑗=1𝜇𝑗𝑥𝑗}=_{𝜇∈𝑊𝜆}(−1)^{𝑛𝑝(𝜇)}_𝑗=1𝑒^{2𝜋𝑖𝜇𝑗𝑥𝑗}. (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: 𝜒_𝜆(𝑥)=𝑆_𝜆+𝜌(𝑥)𝑆_𝜌(𝑥)=_𝜇𝑚^𝜆_𝜇𝐶_𝜇(𝑥),𝑥∈ℝ^𝑛,𝜆,𝜇∈𝑃⁺,𝜌=^𝑛_𝑘=1𝜔_𝑘. (2.4) The positive integer 𝑚^𝜆_𝜇 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 𝔗∶ℝ^𝑛⟶ℂ^𝑛;𝑥₁,…,𝑥_𝑛⟼𝑋₁,…,𝑋_𝑛, (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 𝐹 with the weight function det⁻¹(𝐷(𝑋)/𝐷(𝑥)), where 𝐹 is the image of the fundamental region 𝐹 under the transformation 𝔗.

(i) The first type of transformation is rather straightforward 𝑋_𝑗∶=𝑒^{2𝜋𝑖𝑥𝑗},𝑥_𝑗∈ℝ,𝑗=1,2,…,𝑛. (3.2)

Polynomial summands are products ∏^𝑛_𝑗=1𝑋^𝜇𝑗_𝑗, 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 𝐴₁,𝐵_𝑛(𝑛≥3),𝐶_𝑛(𝑛≥2),𝐷_2𝑛(𝑛≥2),𝐸₇,𝐸₈,𝐹₄,𝐺₂ (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: 𝑒^{2𝜋𝑖⟨𝜇,𝑥⟩}+𝑒^{−2𝜋𝑖⟨𝜇,𝑥⟩}=2cos2𝜋𝜇,𝑥∈𝐶_𝜆(𝑥),𝑒^{2𝜋𝑖⟨𝜇,𝑥⟩}−𝑒^{−2𝜋𝑖⟨𝜇,𝑥⟩}=2𝑖sin2𝜋𝜇,𝑥∈𝑆_𝜆(𝑥), (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 𝛼̌-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: 𝑋_𝑗∶=𝑋_𝑗(𝑥)∶=𝐶_𝜔𝑗(𝑥),𝑗=1,2,…,𝑛,𝑥∈ℝ^𝑛 (3.5) completed by one more variable, the lowest 𝑆 function, 𝐒∶=𝑆_𝜌(𝑥),𝑥∈ℝ^𝑛,𝜌=(1,1,…,1)=^𝑛_𝑚=1𝜔_𝑚. (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 𝑋_𝑗𝐶_{(𝑎1,𝑎2,…,𝑎𝑛)} 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 𝑋_𝑘∶=𝜒_𝜔𝑘(𝑥),𝑘=1,2,…,𝑛. (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, 𝐶_𝜔𝑘(𝑥)=𝜒_𝜔𝑘(𝑥) for all 𝑘=1,…,𝑛 and for all 𝑥∈ℝ^𝑛. Already for the rank two cases other than 𝐴₂ there is a difference. Indeed, (4.1) reads as follows: 𝐶₂:𝑋₁=𝜒_𝜔1(𝑥)=𝐶_𝜔1(𝑥),𝑋₂=𝜒_𝜔2(𝑥)=𝐶_𝜔2(𝑥)𝐺+2,₂:𝑋₁=𝜒_𝜔1(𝑥)=𝐶_𝜔1(𝑥)+𝐶_𝜔2(𝑥)+2,𝑋₂=𝜒_𝜔2(𝑥)=𝐶_𝜔2(𝑥)+1. (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].

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: 𝐶_𝑚(𝑥)=𝑒^{2𝜋𝑖𝑚𝑥}+𝑒^{−2𝜋𝑖𝑚𝑥}=2cos(2𝜋𝑚𝑥),𝑥∈ℝ,𝑚∈ℤ^>0,𝑆_𝑚(𝑥)=𝑒^{2𝜋𝑖𝑚𝑥}−𝑒^{−2𝜋𝑖𝑚𝑥}=2𝑖sin(2𝜋𝑚𝑥),𝑥∈ℝ,𝑚∈ℤ^>0. (A.1)

We introduce new variables 𝑋 and 𝑆 as follows: 𝑋∶=𝐶₁(𝑥)=𝑒^{2𝜋𝑖𝑥}+𝑒^{−2𝜋𝑖𝑥}=2cos(2𝜋𝑥),𝑆∶=𝑆₁(𝑥)=𝑒^{2𝜋𝑖𝑥}−𝑒^{−2𝜋𝑖𝑥}=2𝑖sin(2𝜋𝑥). (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 𝑋𝐶_𝑚=𝐶_𝑚+1+𝐶_𝑚−1,𝑆𝐶_𝑚=𝑆_𝑚+1−𝑆_𝑚−1,𝑋𝑆_𝑚=𝑆_𝑚+1+𝑆_𝑚−1,𝑆𝑆_𝑚=𝐶_𝑚+1−𝐶_𝑚−1,𝑚⩾1. (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.

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 𝑆_𝑚(𝑋)=𝜒_𝑚(𝑥)=𝑆_𝑚+1(𝑥)𝑆(𝑥)=𝐶_𝑚(𝑥)+𝐶_𝑚−2(𝑥)𝐶+⋯+²(𝑥)+𝐶₀𝐶if𝑚even,₃(𝑥)+𝐶₁(𝑥)if𝑚odd. (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 𝐶²₂=𝐶₄+2 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): 𝑋₁∶=𝐶_(1,0)𝑥₁,𝑥₂,𝑋₂∶=𝐶_(0,1)𝑥₁,𝑥₂,𝑆∶=𝑆_(1,1)𝑥₁,𝑥₂. (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, 𝑋²₁, 𝑋₁𝑋₂, and 𝑋²₂, 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 #(𝑎,𝑏)=(𝑎+2𝑏)mod3. (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)mod3. Consequently, there are three classes of polynomials corresponding to #=0,1,2. During multiplication, the congruence numbers add up mod3. 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, 𝑋₁𝐶_(𝑎,𝑏)=𝐶_{(𝑎+1,𝑏)}+𝐶_{(𝑎−1,𝑏+1)}+𝐶_{(𝑎,𝑏−1)}𝑋,𝑎,𝑏⩾2,₂𝐶_(𝑎,𝑏)=𝐶_{(𝑎,𝑏+1)}+𝐶_{(𝑎+1,𝑏−1)}+𝐶_{(𝑎−1,𝑏)},𝑎,𝑏⩾2. (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: 𝑋²₁=𝐶_(2,0)+2𝑋₂,𝑋²₂=𝐶_(0,2)+2𝑋₁,𝑋₁𝑋₂=𝐶_(1,1)𝑋+3,₁𝐶_(1,1)=𝐶_(2,1)+2𝐶_(0,2)+2𝑋₁,𝑋₂𝐶_(1,1)=𝐶_(1,2)+2𝐶_(2,0)+2𝑋₂, (B.4) for 𝑎⩾2:𝑋₁𝐶_(𝑎,1)=𝐶_(𝑎+1,1)+𝐶_(𝑎−1,2)+2𝐶_(𝑎,0),𝑋₂𝐶_(𝑎,1)=𝐶_(𝑎,2)+2𝐶_(𝑎+1,0)+𝐶_(𝑎−1,1),𝑋₁𝐶_(𝑎,0)=𝐶_(𝑎+1,0)+𝐶_(𝑎−1,1),𝑋₂𝐶_(𝑎,0)=𝐶_(𝑎,1)+𝐶_(𝑎−1,0), (B.5) for 𝑏⩾2:𝑋₁𝐶_(1,𝑏)=𝐶_(2,𝑏)+2𝐶_(0,𝑏+1)+𝐶_(1,𝑏−1),𝑋₂𝐶_(1,𝑏)=𝐶_(1,𝑏+1)+𝐶_(2,𝑏−1)+2𝐶_(0,𝑏),𝑋₁𝐶_(0,𝑏)=𝐶_(1,𝑏)+𝐶_(0,𝑏−1),𝑋₂𝐶_(0,𝑏)=𝐶_(0,𝑏+1)+𝐶_(1,𝑏−1). (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 𝑆_(𝑎,𝑏)𝑋₁,𝑋₂=𝜒_(𝑎,𝑏)=𝑆𝑥,𝑦^{(𝑎+1,𝑏+1)}𝑥,𝑦𝑆_(1,1)𝑥,𝑦=𝐶_(𝑎,𝑏)𝑥,𝑦+_𝜆𝑚_𝜆𝐶_𝜆𝑥,𝑦. (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 𝜒_(0,0)=𝐶_(0,0)𝜒=1,_(1,1)=𝐶_(1,1)+2𝐶_(0,0),𝜒_(3,0)=𝐶_(3,0)+𝐶_(1,1)+𝐶_(0,0),𝜒_(0,3)=𝐶_(0,3)+𝐶_(1,1)+𝐶_(0,0),𝜒_(2,2)=𝐶_(2,2)+𝐶_(0,3)+𝐶_(3,0)+2𝐶_(1,1)+3𝐶_(0,0),𝜒_(1,4)=𝐶_(1,4)+𝐶_(2,2)+2𝐶_(0,3)+𝐶_(3,0)+2𝐶_(1,1)+2𝐶_(0,0),𝜒_(3,3)=𝐶_(3,3)+𝐶_(4,1)+𝐶_(1,4)+2𝐶_(2,2)+2𝐶_(0,3)+2𝐶_(3,0)+3𝐶_(1,1)+4𝐶_(0,0),𝜒_(6,0)=𝐶_(6,0)+𝐶_(4,1)+𝐶_(2,2)+𝐶_(0,3)+𝐶_(3,0)+𝐶_(1,1)+𝐶_(0,0). (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: 𝜒_(1,0)=𝐶_(1,0),𝜒_(0,2)=𝐶_(0,2)+𝐶_(1,0),𝜒_(2,1)=𝐶_(2,1)+𝐶_(0,2)+2𝐶_(1,0),𝜒_(1,3)=𝐶_(1,3)+𝐶_(2,1)+2𝐶_(0,2)+2𝐶_(1,0),𝜒_(4,0)=𝐶_(4,0)+𝐶_(2,1)+𝐶_(0,2)+𝐶_(1,0),𝜒_(0,5)=𝐶_(0,5)+𝐶_(1,3)+𝐶_(2,1)+𝐶_(0,2)+𝐶_(1,0). (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. 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 #(𝑎,𝑏)=𝑎mod2. (C.1) The dimension 𝑑_(𝑎,𝑏) of an irreducible representation of 𝐶₂ with the highest weight (𝑎,𝑏) is given by 𝑑_(𝑎,𝑏)=16(𝑎+1)(𝑏+1)(2𝑎+𝑏+3)(𝑎+𝑏+2). (C.2)

In multiplying the polynomials, congruence numbers add up mod2. 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 𝑋₁∶=𝐶_(1,0)𝑥,𝑦,𝑋₂∶=𝐶_(0,1)𝑥,𝑦,𝑆∶=𝑆_(1,1)𝑥,𝑦,𝑆_(𝑎,𝑏)𝑋₁,𝑋₂=𝜒_(𝑎,𝑏)𝑥,𝑦 (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 𝑋₁𝐶_(𝑎,𝑏)=𝐶_{(𝑎+1,𝑏)}+𝐶_{(𝑎−1,𝑏+1)}+𝐶_{(𝑎+1,𝑏−1)}+𝐶_{(𝑎−1,𝑏)}𝑋,𝑎,𝑏⩾2,₂𝐶_(𝑎,𝑏)=𝐶_{(𝑎,𝑏+1)}+𝐶_{(𝑎+2,𝑏−1)}+𝐶_{(𝑎−2,𝑏+1)}+𝐶_{(𝑎,𝑏−1)},𝑎⩾3,𝑏⩾2. (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: 𝑋₁𝐶_(𝑎,1)=𝐶_(𝑎+1,1)+𝐶_(𝑎−1,2)+2𝐶_(𝑎+1,0)+𝐶_(𝑎−1,1)𝑋,𝑎⩾2,₁𝐶_(𝑎,0)=𝐶_(𝑎+1,0)+𝐶_(𝑎−1,1)+𝐶_(𝑎−1,0)𝑋,𝑎⩾2,₁𝐶_(1,𝑏)=𝐶_(2,𝑏)+2𝐶_(0,𝑏+1)+𝐶_(2,𝑏−1)+2𝐶_(0,𝑏)𝑋,𝑏⩾2,₁𝐶_(0,𝑏)=𝐶_(1,𝑏)+𝐶_(1,𝑏−1)𝑋,𝑏⩾2,₁𝐶_(1,1)=𝐶_(2,1)+2𝐶_(0,2)+2𝐶_(2,0)+2𝑋₂,𝑋₁𝑋₂=𝐶_(1,1)+2𝑋₁;𝑋²₁=𝐶_(2,0)+2𝑋₂𝑋+4,₂𝐶_(𝑎,1)=𝐶_(𝑎,2)+2𝐶_(𝑎+2,0)+𝐶_(𝑎−2,2)+2𝐶_(𝑎,0)𝑋,𝑎⩾3,₂𝐶_(𝑎,0)=𝐶_(𝑎,1)+𝐶_(𝑎−2,1)𝑋,𝑎⩾3,₂𝐶_(2,𝑏)=𝐶_(2,𝑏+1)+𝐶_(4,𝑏−1)+2𝐶_(0,𝑏+1)+𝐶_(2,𝑏−1)𝑋,𝑏⩾2,₂𝐶_(1,𝑏)=𝐶_(1,𝑏+1)+𝐶_(3,𝑏−1)+𝐶_(1,𝑏−1)+𝐶_(1,𝑏)𝑋,𝑏⩾2,₂𝐶_(0,𝑏)=𝐶_(0,𝑏+1)+𝐶_(2,𝑏−1)+𝐶_(0,𝑏−1)𝑋,𝑏⩾2,₂𝐶_(2,1)=𝐶_(2,2)+2𝐶_(4,0)+𝐶_(0,4)+2𝐶_(0,2)+2𝐶_(2,0),𝑋₂𝐶_(1,1)=𝐶_(1,2)+2𝐶_(3,0)+𝐶_(1,1)+2𝑋₁,𝑋₂𝐶_(2,0)=𝐶_(2,1)+2𝑋₂;𝑋²₂=𝐶_(0,2)+2𝐶_(2,0)+4. (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.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: 𝜒#=0:_(0,0)=𝐶_(0,0)𝜒=1,_(0,1)=1+𝐶_(0,1)=1+𝑋₂,𝜒_(2,0)=2+𝑋₂+𝐶_(2,0),𝜒_(0,2)=2+𝑋₂+𝐶_(2,0)+𝐶_(0,2),𝜒_(2,1)=3+3𝑋₂+2𝐶_(2,0)+𝐶_(0,2)+𝐶_(2,1),𝜒_(0,3)=2+2𝑋₂+𝐶_(2,0)+𝐶_(0,2)+𝐶_(2,1)+𝐶_(0,3),𝜒_(4,0)=3+2𝑋₂+2𝐶_(2,0)+𝐶_(0,2)+𝐶_(2,1)+𝐶_(4,0),𝜒_(2,2)=5+4𝑋₂+4𝐶_(2,0)+3𝐶_(0,2)+2𝐶_(2,1)+𝐶_(0,3)+𝐶_(4,0)+𝐶_(2,2),𝜒#=1:_(1,0)=𝐶_(1,0)=𝑋₁,𝜒_(1,1)=2𝑋₁+𝐶_(1,1),𝜒_(3,0)=2𝑋₁+𝐶_(1,1)+𝐶_(3,0),𝜒_(1,2)=3𝑋₁+2𝐶_(1,1)+𝐶_(3,0)+𝐶_(1,2),𝜒_(3,1)=4𝑋₁+3𝐶_(1,1)+2𝐶_(3,0)+𝐶_(1,2)+𝐶_(3,1),𝜒_(1,3)=4𝑋₁+3𝐶_(1,1)+2𝐶_(3,0)+2𝐶_(1,2)+𝐶_(3,1)+𝐶_(1,3). (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.

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: 𝑋₁∶=𝐶_(1,0)𝑥,𝑦,𝑋₂∶=𝐶_(0,1)𝑥,𝑦,𝑆=𝑆_(1,1)𝑥,𝑦,𝑆_(𝑎,𝑏)𝑋₁,𝑋₂=𝜒_(𝑎,𝑏)𝑥,𝑦. (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. 𝑋For𝑎⩾3,𝑏⩾4:₁𝐶_(𝑎,𝑏)=𝐶_{(𝑎+1,𝑏)}+𝐶_{(𝑎−1,𝑏+3)}+𝐶_{(𝑎+2,𝑏−3)}+𝐶_{(𝑎−2,𝑏+3)}+𝐶_{(𝑎+1,𝑏−3)}+𝐶_{(𝑎−1,𝑏)}.𝑋For𝑎⩾2,𝑏⩾3:₂𝐶_(𝑎,𝑏)=𝐶_{(𝑎,𝑏+1)}+𝐶_{(𝑎+1,𝑏−1)}+𝐶_{(𝑎−1,𝑏+2)}+𝐶_{(𝑎+1,𝑏−2)}+𝐶_{(𝑎−1,𝑏+1)}+𝐶_{(𝑎,𝑏−1)}. (D.2)

Specializing the first of the generic relations to either 𝑎∈{0,1,2} or 𝑏∈{0,1,2,3}, we have 𝑋₁𝐶_(2,𝑏)=𝐶_(3,𝑏)+𝐶_(1,𝑏+3)+𝐶_(4,𝑏−3)+2𝐶_(0,𝑏+3)+𝐶_(3,𝑏−3)+𝐶_(1,𝑏),𝑋₁𝐶_(1,𝑏)=𝐶_(2,𝑏)+2𝐶_(0,𝑏+3)+𝐶_(3,𝑏−3)+𝐶_(2,𝑏−3)+𝐶_(1,𝑏)+2𝐶_(0,𝑏),𝑋₁𝐶_(0,𝑏)=𝐶_(1,𝑏)+𝐶_(2,𝑏−3)+𝐶_(1,𝑏−3),𝑋₁𝐶_(𝑎,3)=𝐶_(𝑎+1,3)+𝐶_(𝑎−1,6)+2𝐶_(𝑎+2,0)+𝐶_(𝑎−2,6)+2𝐶_(𝑎+1,0)+𝐶_(𝑎−1,3),𝑋₁𝐶_(𝑎,2)=𝐶_(𝑎+1,2)+𝐶_(𝑎−1,5)+𝐶_(𝑎+1,1)+𝐶_(𝑎−2,5)+𝐶_(𝑎,1)+𝐶_(𝑎−1,2),𝑋₁𝐶_(𝑎,1)=𝐶_(𝑎+1,1)+𝐶_(𝑎−1,4)+𝐶_(𝑎−2,4)+𝐶_(𝑎−1,1)+𝐶_(𝑎,2)+𝐶_(𝑎−1,2),𝑋₁𝐶_(𝑎,0)=𝐶_(𝑎+1,0)+𝐶_(𝑎−1,3)+𝐶_(𝑎−2,3)+𝐶_(𝑎−1,0),𝑋₁𝐶_(2,3)=𝐶_(3,3)+𝐶_(1,6)+2𝐶_(4,0)+2𝐶_(0,6)+2𝐶_(3,0)+𝐶_(1,3),𝑋₁𝐶_(2,2)=𝐶_(3,2)+𝐶_(1,5)+𝐶_(0,5)+𝐶_(1,2)+𝐶_(2,1)+𝐶_(3,1),𝑋₁𝐶_(2,1)=𝐶_(3,1)+𝐶_(1,4)+2𝐶_(0,4)+𝐶_(1,1)+𝐶_(2,2)+𝐶_(1,2),𝑋₁𝐶_(2,0)=𝐶_(3,0)+𝐶_(1,3)+2𝐶_(0,3)+𝑋₁,𝑋₁𝐶_(1,3)=𝐶_(2,3)+2𝐶_(0,6)+2𝐶_(3,0)+2𝐶_(2,0)+𝐶_(1,3)+2𝐶_(0,3),𝑋₁𝐶_(1,2)=𝐶_(2,2)+2𝐶_(0,5)+𝐶_(1,2)+2𝐶_(0,2)+𝐶_(2,1)+𝐶_(1,1),𝑋₁𝐶_(1,1)=𝐶_(2,1)+2𝐶_(0,4)+𝐶_(1,2)+2𝐶_(0,2)+𝐶_(1,1)+2𝑋₂,𝑋₁𝐶_(0,3)=𝐶_(1,3)+2𝐶_(2,0)+2𝑋₁,𝑋₁𝐶_(0,2)=𝐶_(1,2)+𝐶_(1,1)+2𝑋₂,𝑋₁𝑋₁=𝐶_(2,0)+2𝐶_(0,3)+2𝑋₁𝑋+6,₁𝑋₂=𝐶_(1,1)+2𝐶_(0,2)+2𝑋₂. (D.3)

Specializing the second of the generic relations to either 𝑎∈{0,1} or 𝑏∈{0,1,2}, we have 𝑋₂𝐶_(1,𝑏)=𝐶_(1,𝑏+1)+𝐶_(2,𝑏−1)+2𝐶_(0,𝑏+2)+𝐶_(2,𝑏−2)+2𝐶_(0,𝑏+1)+𝐶_(1,𝑏−1),𝑋₂𝐶_(0,𝑏)=𝐶_(0,𝑏+1)+𝐶_(1,𝑏−1)+𝐶_(1,𝑏−2)+𝐶_(0,𝑏−1),𝑋₂𝐶_(𝑎,1)=𝐶_(𝑎,2)+2𝐶_(𝑎+1,0)+𝐶_(𝑎−1,3)+𝐶_(𝑎−1,2)+2𝐶_(𝑎,0)+𝐶_(𝑎,1),𝑋₂𝐶_(𝑎,0)=𝐶_(𝑎,1)+𝐶_(𝑎−1,2)+𝐶_(𝑎−1,1),𝑋₂𝐶_(1,2)=𝐶_(1,3)+𝐶_(2,1)+2𝐶_(0,4)+2𝐶_(0,3)+𝐶_(1,1)+2𝐶_(2,0),𝑋₂𝐶_(1,1)=𝐶_(1,2)+2𝐶_(2,0)+2𝐶_(0,3)+2𝐶_(0,2)+𝐶_(1,1)+2𝑋₁,𝑋₂𝐶_(0,2)=𝐶_(0,3)+𝐶_(1,1)+2𝑋₁+𝑋₂,𝑋₂𝑋₂=𝐶_(0,2)+2𝑋₁+2𝑋₂+6. (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.

D.2. Recursion Relations for 𝑆 Functions of 𝐺₂

Generic recursion relations for 𝑆 polynomials differ very little from those for 𝐶 polynomials. 𝑋For𝑎⩾3,𝑏⩾4:₁𝑆_(𝑎,𝑏)=𝑆_{(𝑎+1,𝑏)}+𝑆_{(𝑎−1,𝑏+3)}+𝑆_{(𝑎+2,𝑏−3)}+𝑆_{(𝑎−2,𝑏+3)}+𝑆_{(𝑎+1,𝑏−3)}+𝑆_{(𝑎−1,𝑏)}.𝑋For𝑎⩾2,𝑏⩾3:₂𝑆_(𝑎,𝑏)=𝑆_{(𝑎,𝑏+1)}+𝑆_{(𝑎+1,𝑏−1)}+𝑆_{(𝑎−1,𝑏+2)}+𝑆_{(𝑎+1,𝑏−2)}+𝑆_{(𝑎−1,𝑏+1)}+𝑆_{(𝑎,𝑏−1)}. (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.

Here are all 𝐺₂-characters 𝜒_(𝑎,𝑏) with 𝑎+𝑏⩽3: 𝜒_(1,0)=1+𝐶_(1,0)=1+𝑋₁,𝜒_(0,1)=2+𝐶_(1,0)+𝐶_(0,1)=2+𝑋₁+𝑋₂,𝜒_(2,0)=3+2𝑋₁+𝑋₂+𝐶_(2,0),𝜒_(1,1)=4+4𝑋₁+2𝑋₂+2𝐶_(2,0)+𝐶_(1,1),𝜒_(3,0)=5+4𝑋₁+3𝑋₂+2𝐶_(2,0)+𝐶_(3,0),𝜒_(0,2)=5+3𝑋₁+3𝑋₂+2𝐶_(2,0)+𝐶_(1,1)+𝐶_(3,0)+𝐶_(0,2),𝜒_(2,1)=9+8𝑋₁+6𝑋₂+5𝐶_(2,0)+3𝐶_(1,1)+2𝐶_(3,0)+𝐶_(0,2)+𝐶_(2,1),𝜒_(1,2)=10+10𝑋₁+7𝑋₂+7𝐶_(2,0)+5𝐶_(1,1)+3𝐶_(3,0)+3𝐶_(3,0)+2𝐶_(0,2)+2𝐶_(2,1)+𝐶_(4,0)+𝐶_(1,2),𝜒_(0,3)=9+7𝑋₁+7𝑋₂+5𝐶_(2,0)+4𝐶_(1,1)+4𝐶_(3,0)+3𝐶_(0,2)+2𝐶_(1,1)+𝐶_(4,0)+𝐶_(1,2),+𝐶_(3,1)+𝐶_(0,3). (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 #(𝑎,𝑏,𝑐)=𝑎+2𝑏+3𝑐mod4. (E.1) The variables of the 𝐴₃ polynomials are chosen to be 𝑋₁∶=𝐶_(1,0,0)𝑥₁,𝑥₂,𝑥₃,𝑋₂∶=𝐶_(0,1,0)𝑥₁,𝑥₂,𝑥₃,𝑋₃∶=𝐶_(0,0,1)𝑥₁,𝑥₂,𝑥₃and𝑆_{(𝑎,𝑏,𝑐)}𝑋₁,𝑋₂,𝑋₃=𝜒_{(𝑎,𝑏,𝑐)}𝑥,𝑦,𝑧. (E.2)

For 𝐶 functions the generic recursion relations are the following ones, where we assume 𝑎,𝑏,𝑐⩾2: 𝑋₁𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎+1,𝑏,𝑐)}+𝐶_{(𝑎−1,𝑏+1,𝑐)}+𝐶_{(𝑎,𝑏−1,𝑐+1)}+𝐶_{(𝑎,𝑏,𝑐−1)},𝑋₂𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏+1,𝑐)}+𝐶_{(𝑎+1,𝑏−1,𝑐+1)}+𝐶_{(𝑎−1,𝑏,𝑐+1)}+𝐶_{(𝑎+1,𝑏,𝑐−1)}+𝐶_{(𝑎−1,𝑏+1,𝑐−1)}+𝐶_{(𝑎,𝑏−1,𝑐)},𝑋₃𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏,𝑐+1)}+𝐶_{(𝑎,𝑏+1,𝑐−1)}+𝐶_{(𝑎+1,𝑏−1,𝑐)}+𝐶_{(𝑎−1,𝑏,𝑐)}. (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↔𝑋₃and(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.

E.2. 𝑆 Polynomials of 𝐴₃

Generic recursion relations are decompositions of the following products, where we assume that 𝑎,𝑏,𝑐>1: 𝑋₁𝑆_{(𝑎,𝑏,𝑐)}=𝑆_{(𝑎+1,𝑏,𝑐)}+𝑆_{(𝑎−1,𝑏+1,𝑐)}+𝑆_{(𝑎,𝑏−1,𝑐+1)}+𝑆_{(𝑎,𝑏,𝑐−1)},𝑋₂𝑆_{(𝑎,𝑏,𝑐)}=𝑆_{(𝑎,𝑏+1,𝑐)}+𝑆_{(𝑎+1,𝑏−1,𝑐+1)}+𝑆_{(𝑎−1,𝑏,𝑐+1)}+𝑆_{(𝑎+1,𝑏,𝑐−1)}+𝑆_{(𝑎−1,𝑏+1,𝑐−1)}+𝑆_{(𝑎,𝑏−1,𝑐)},𝑋₃𝑆_{(𝑎,𝑏,𝑐)}=𝑆_{(𝑎,𝑏,𝑐+1)}+𝑆_{(𝑎,𝑏+1,𝑐−1)}+𝑆_{(𝑎+1,𝑏−1,𝑐)}+𝑆_{(𝑎−1,𝑏,𝑐)}. (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, 𝑋_𝑗∶=𝐶_𝜔𝑗, 𝑗=1,2,3. There are two congruence classes of (𝑎₁,𝑎₂,𝑎₃) for either of the two algebras.

We have #𝐵₃=𝑎₃mod2,#𝐶₃=𝑎₁+𝑎₃mod2. (E.5)

For 𝐵₃, we have the generic recursion relations 𝑋₁𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎+1,𝑏,𝑐)}+𝐶_{(𝑎−1,𝑏+1,𝑐)}+𝐶_{(𝑎,𝑏−1,𝑐+2)}+𝐶_{(𝑎,𝑏+1,𝑐−2)}+𝐶_{(𝑎+1,𝑏−1,𝑐)}+𝐶_{(𝑎−1,𝑏,𝑐)}𝑋,for𝑎,𝑏⩾2,𝑐⩾3,₂𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏+1,𝑐)}+𝐶_{(𝑎+1,𝑏−1,𝑐+2)}+𝐶_{(𝑎−1,𝑏,𝑐+2)}+𝐶_{(𝑎+1,𝑏+1,𝑐−2)}+𝐶_{(𝑎−1,𝑏+2,𝑐−2)}+𝐶_{(𝑎+2,𝑏−1,𝑐)}+𝐶_{(𝑎+1,𝑏−2,𝑐+2)}+𝐶_{(𝑎−2,𝑏+1,𝑐)}+𝐶_{(𝑎−1,𝑏−1,𝑐+2)}+𝐶_{(𝑎+1,𝑏,𝑐−2)}+𝐶_{(𝑎−1,𝑏+1,𝑐−2)}+𝐶_{(𝑎,𝑏−1,𝑐)}𝑋,𝑎⩾2,𝑏,𝑐⩾3,₃𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏,𝑐+1)}+𝐶_{(𝑎,𝑏+1,𝑐−1)}+𝐶_{(𝑎+1,𝑏−1,𝑐+1)}+𝐶_{(𝑎−1,𝑏,𝑐+1)}−𝐶_{(𝑎+1,𝑏,𝑐−1)}+𝐶_{(𝑎−1,𝑏+1,𝑐−1)}+𝐶_{(𝑎,𝑏−1,𝑐+1)}+𝐶_{(𝑎,𝑏,𝑐−1)},𝑎,𝑏,𝑐⩾2. (E.6)

For 𝐶₃, we have the generic recursion relations 𝑋₁𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎+1,𝑏,𝑐)}+𝐶_{(𝑎−1,𝑏+1,𝑐)}+𝐶_{(𝑎,𝑏−1,𝑐+1)}+𝐶_{(𝑎,𝑏+1,𝑐−1)}+𝐶_{(𝑎+1,𝑏−1,𝑐)}+𝐶_{(𝑎−1,𝑏,𝑐)}𝑋,𝑎,𝑏,𝑐⩾2,₂𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏+1,𝑐)}+𝐶_{(𝑎+1,𝑏−1,𝑐)}+𝐶_{(𝑎−1,𝑏,𝑐+1)}+𝐶_{(𝑎+1,𝑏+1,𝑐−1)}+𝐶_{(𝑎−1,𝑏+2,𝑐−1)}+𝐶_{(𝑎+2,𝑏−1,𝑐)}+𝐶_{(𝑎+1,𝑏−2,𝑐+1)}+𝐶_{(𝑎−2,𝑏+1,𝑐)}+𝐶_{(𝑎+1,𝑏,𝑐−1)}+𝐶_{(𝑎−1,𝑏−1,𝑐+1)}+𝐶_{(𝑎−1,𝑏+1,𝑐−1)}+𝐶_{(𝑎,𝑏−1,𝑐)}𝑋,𝑎,𝑏⩾3,𝑐⩾2,₃𝐶_{(𝑎,𝑏,𝑐)}=𝐶_{(𝑎,𝑏,𝑐+1)}+𝐶_{(𝑎,𝑏+2,𝑐−1)}+𝐶_{(𝑎+2,𝑏−2,𝑐+1)}+𝐶_{(𝑎−2,𝑏,𝑐+1)}+𝐶_{(𝑎+2,𝑏,𝑐−1)}+𝐶_{(𝑎−2,𝑏+2,𝑐−1)}+𝐶_{(𝑎,𝑏−2,𝑐+1)}+𝐶_{(𝑎,𝑏,𝑐−1)},𝑎,𝑏⩾3,𝑐⩾2. (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.

Table 9 

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

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.