Some results on the eigenfunctions of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra D_{4}

J. Fernández Núñez, W. García Fuertes

Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, E-33007 Oviedo, Spain

and A.M. Perelomov ^{∗}

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

Abstract We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra D_{4} in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated to them such as recurrence relations and generating functions.

1 Introduction

Integrable models play a prominent role in theoretical physics. The reason is not only the direct phenomenological interest of some of them, but also the fact that they often provide some deep insights into the mathematical structure of the theories in which they arise. Sometimes, they even reveal unexpected relations among different physical or mathematical theories. In classical mechanics, integrability not only shows up itself in some of the most important and time-honored problems, such as the Keplerian motion or the Lagrange or Kovalevskaya top. It appears also in a plethora of new hypothetical, highly nontrivial sytems discovered mainly during the three last decades of the past century (see [1, 2] for comprehensives reviews). Among these, the so-called Calogero-Sutherland models form a distinguised class. The first analysis of a system of this kind was performed by Calogero [3] who studied, from the quantum standpoint, the dynamics on the infinite line of a set of particles interacting pairwise by rational plus quadratic potentials, and found that the problem was exactly solvable. Soon afterwards, Sutherland [4] arrived to similar results for the quantum problem on the circle, this time with trigonometric interaction, and Moser [5] showed that the classical version of both models enjoyed integrability in the Liouville sense. The identification of the general scope of these discoveries comes with the work of Olshanetsky and Perelomov [6, 7], who realized that it was possible to associate models of this kind to all the root sytems of the simple Lie algebras, and that all these models were integrable, both in the classical and in the quantum framework [8, 9]. Nowadays, there is a widespread interest in this type of integrable systems, and many mathematical and physical applications for them have been found, see for instance [10].

The eigenfunctions of the Calogero-Sutherland Hamiltonian associated to the root system of a simple Lie algebra L are proportional to some polynomials which form a complete orthogonal system in the quantum Hilbert space. For the specials values κ_{α} = 1, where g_{α} = κ_{α}(κ_{α} − 1) are the coupling constants, they coincide with the irreducible characters of L. For L = A_{n}, these polynomials provide natural generalizations to n variables of the classical orthogonal polynomials in one indeterminate. In particular, for the case with a trigonometric potential, one obtains a generalized system of Gegenbauer polynomials. As it was shown in the papers [11, 12, 13], these generalized Gegenbauer polynomials obey a set of recurrence relations which constitute a κ-deformation of the Clebsch-Gordan series of the algebra. The finding of these recurrence relations opened the way to obtain many concrete results on the system of polynomials, as for example explicit expressions, ladder operators or generating functions [14, 15]. The recurrence relations are also the key ingredient to formulate a perturbative approach to most general among the Calogero-Sutherland models, that involving the Weierstrass ℘-function as potential [16].

∗On leave of absence from the Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia. Current E-mail address: perelomo@dftuz2.unizar.es

The aim of this paper is to extend some of the results which have been obtained for A_{n} to the polynomials related to other simple algebras. We think that it is a good idea to begin with a concrete case. We choose to work in the first place the problem associated to D_{4} because of the triality symmetry exhibited by this algebra, which will help us in simplifying of the treatment. The organization of the paper is as follows. In Sect. 2, we explain how to express the Calogero-Sutherland Hamiltonian in terms of the fundamental characters of the algebra and how to solve the Schrödinger equation. Then, in Sect. 3, we obtain the main recurrence relations among the polynomials and use them to give algorithms to calculate some subsets of them. Sect. 4 is devoted to find the generating functions for some classes of characters and monomials functions of D_{4}. More recurrrence relations and some other relevant results are included in Sect. 5, and finally, in Sect. 6, we give some brief conclusions. Also, we offer two appendices. In Appendix A, for the convenience of the reader, we collect some of the basic facts about D_{4} which we use in the main text. In Appendix B we list some polynomials, characters and monomial functions.

2 The eigenvalue problem

The Hamiltonian operator for the trigonometric Calogero-Sutherland model related to the root system of a simple Lie algebra of rank r has the form

H = \frac{1}{2}(p, p) + ∑_{α∈R^{+}} κ_{α}(κ_{α} − 1) sin^{−2}(α, q),  (1)

where q = (q_{1}, …, q_{r}), p = (p_{1}, …, p_{r}), ( , ) is the usual euclidean inner product in R^{r}, R^{+} is the set of positive roots of the algebra, and κ_{α} are constants such that κ_{α} = κ_{β} if || α ||=|| β ||. In particular, for the case of the algebra D_{4} (see Appendix A), this leads to the following Schrödinger equation:

HΨ^{κ} = E(κ) Ψ^{κ},
H = −\frac{1}{2} Δ + κ(κ − 1) (∑_{j<k}^{4} sin^{−2}(q_{j} − q_{k}) + ∑_{j<k}^{4} sin^{−2}(q_{j} + q_{k})) , Δ = ∑_{j=1}^{4} \frac{∂^{2}}{∂q_{j}^{2}} .  (2)

The q coordinates are assumed to take values in the [0, π] interval, and therefore the equation can be interpreted as describing the dynamics of a system of four particles moving on the circle. Let us notice that there is not translational invariance. We recapitulate some important facts about this model which follow from the general structure of the quantum Calogero-Sutherland models related to Lie algebras [9]. The ground state energy and the (non-normalized) wavefunction are

E_{0}(κ) = 2(ρ, ρ) κ^{2} = 28 κ^{2},
Ψ_{0}^{κ}(q) = {∏_{j<k}^{4} sin(q_{j} − q_{k}) sin(q_{j} + q_{k})}^{κ} ,  (3)

where ρ is the standard Weyl vector, ρ = \frac{1}{2} ∑_{α∈R^{+}} α, with the sum extended over all the positive roots of D_{4}. The excited states depend on a four-tuple of quantum numbers m = (m_{1}, m_{2}, m_{3}, m_{4})

H Ψ_{m}^{κ} = E_{m}(κ) Ψ_{m}^{κ},
E_{m}(κ) = 2(λ + κρ, λ + κρ),  (4)

where λ is the highest weight of the irreducible representation of D_{4} labelled by m, i.e., λ = ∑_{i=1}^{4} m_{i}λ_{i}, and λ_{i} are the fundamental weights of D_{4}. By substitution in (4) of

Ψ_{m}^{κ}(q) = Ψ_{0}^{κ}(q) Φ_{m}^{κ}(q),  (5)

we are led to the eigenvalue problem

−Δ^{κ}Φ_{m}^{κ} = ε_{m}(κ) Φ_{m}^{κ}  (6)

with

Δ^{κ} = \frac{1}{2}Δ + κ ∑_{j<k}^{4} (ctg(q_{j} − q_{k}) (\frac{∂}{∂q_{j}} − \frac{∂}{∂q_{k}} ) + ctg(q_{j} + q_{k}) (\frac{∂}{∂q_{j}} + \frac{∂}{∂q_{k}} ))  (7)

and

ε_{m}(κ) = E_{m}(κ) − E_{0}(κ) = 2(λ, λ + 2κρ).  (8)

Introducing the inverse Cartan matrix A_{jk}^{−1} = (λ_{j}, λ_{k}), we can give a more explicit expression for ε_{m}(κ):

ε_{m}(κ) = 2 ∑_{j,k=1}^{4} A_{jk}^{−1} m_{j}m_{k} + 4κ ∑_{j,k=1}^{4} A_{jk}^{−1} m_{j} = 2(m_{1}^{2} + m_{3}^{2} + m_{4}^{2}) + 4m_{2}^{2} + 2(m_{1}m_{3} + m_{1}m_{4} + m_{3}m_{4})
+ 4m_{2}(m_{1} + m_{3} + m_{4}) + 12κ(m_{1} + m_{3} + m_{4}) + 20κm_{2}.  (9)

The main problem is to solve equation (6). As it has been shown for the case of the algebra A_{n} [11, 12, 13], the best way to do that is to use a set of independent variables which are invariant under the Weyl symmetry of the Hamiltonian, namely, the characters of the four fundamental representations of the algebra D_{4}. Unfortunately, the expression of these characters in terms of the q-variables (which play the role of coordinates on the maximal torus of D_{4}) is not very simple. Denoting the character of the irreducible representation of maximal weight λ_{j} as z_{j}, we find

z_{1} = ∑_{j=1}^{4} x_{j} + ∑_{j=1}^{4} x_{j}^{−1}

z_{2} = ∑_{i<j}^{4} x_{i}x_{j} + ∑_{i<j}^{4}(x_{i}x_{j})^{−1} + ∑_{i,j}^{4} x_{i}^{−1}x_{j}

z_{3} = x̄ ∑_{i=1}^{4} \frac{1}{x_{i}} + \frac{1}{x̄} ∑_{i=1}^{4} x_{i},

z_{4} = x̄ + \frac{1}{x̄} + \frac{1}{x̄} ∑_{i<j}^{4} x_{i}x_{j}

where x_{j} = e^{2iq_{j}}, and x̄ = \sqrt{x_{1}x_{2}x_{3}x_{4}}. These expressions make the direct change of variables from q_{i} to z_{k} quite cumbersome. We refrain from trying that approach, and choose an indirect route which has the added advantage of being also applicable to other algebras in which the expressions for the characters are even more involved. We can infer from (7) the structure of Δ^{κ} when written in the z-variables:

Δ^{κ} = ∑_{j,k=1}^{4} a_{jk}(z_{i}) ∂_{z_{j}} ∂_{z_{k}} + ∑_{j=1}^{4} [b_{j}^{(0)}(z_{i}) + κb_{j}^{(1)}(z_{i})] ∂_{z_{j}}.  (10)

On the other hand, as it is well known [17], the Φ_{m}^{κ} are polynomials which, with some precise partial ordering for the monomials to be described later, start as follows:

Φ_{m}^{κ}(z_{i}) = P_{m}^{κ}(z_{i}) = z_{1}^{m_{1}}z_{2}^{m_{2}}z_{3}^{m_{3}}z_{4}^{m_{4}} + · · · .  (11)

Therefore, making use of (9), we conclude that

a_{jk}(z_{i}) = 2 A_{jk}^{−1} z_{j}z_{k} + lower order terms,
b_{j}^{(r)}(z_{i}) = c_{j}^{(r)}z_{j} + d_{j}^{(r)}, r = 0, 1.  (12)

Now, to obtain the full expressions for these coefficients, we rely on the fact that, for κ = 1, the P_{m}^{κ} polynomial gives the character of the irreducible representation of D_{4} with maximal weight ∑_{i=1}^{4} m_{i}λ_{i}, while for κ = 0 the same polynomial is the corresponding symmetric monomial function [9]. Both, characters and monomial functions, can be computed by using the information available in the literature (see, for instance, the “Reference Chapter” of [18]). In fact, the following short list of polynomials

P_{2,0,0,0}^{(1)}(z) = z_{1}^{2} − z_{2} − 1,
P_{1,1,0,0}^{(1)}(z) = z_{1}z_{2} − z_{3}z_{4},
P_{1,0,1,0}^{(1)}(z) = z_{1}z_{3} − z_{4},
P_{0,2,0,0}^{(1)}(z) = z_{2}^{2} − z_{1}z_{3}z_{4} + z_{2},
P_{2,0,0,0}^{(0)}(z) = z_{1}^{2} − 2z_{2}

is all we need to obtain Δ^{κ}. By substituting these polynomials in (6) and using (9), (10), (12) and the triality symmetry (that here implies that the final expression for Δ^{κ} should be invariant under permutations of the indices 1,3,4), we get enough simple linear algebraic equations to fix all the coefficients. We give here only the final result:

\frac{1}{2}Δ^{κ} = (z_{1}^{2} − 2z_{2} − 8) ∂_{z_{1}}^{2} + [2 z_{2}^{2} − 4 (z_{1}^{2} + z_{3}^{2} + z_{4}^{2}) − 2 z_{1} z_{3} z_{4} + 8 z_{2}] ∂_{z_{2}}^{2} + (z_{3}^{2} − 2z_{2} − 8) ∂_{z_{3}}^{2}
+ (z_{4}^{2} − 2z_{2} − 8) ∂_{z_{4}}^{2} + (2z_{1}z_{2} − 6z_{3}z_{4} − 8z_{1}) ∂_{z_{1}} ∂_{z_{2}} + (z_{1} z_{3} − 8 z_{4}) ∂_{z_{1}} ∂_{z_{3}}
+ (z_{1}z_{4} − 8z_{3}) ∂_{z_{1}} ∂_{z_{4}} + (2z_{2}z_{3} − 6z_{1}z_{4} − 8z_{3}) ∂_{z_{2}} ∂_{z_{3}} + (2z_{2}z_{4} − 6z_{1}z_{3} − 8z_{4}) ∂_{z_{2}} ∂_{z_{4}}
+ (z_{3}z_{4} − 8z_{1}) ∂_{z_{3}} ∂_{z_{4}} + (6κ + 1) z_{1} ∂_{z_{1}} + [2(5κ + 1) z_{2} + 8(κ − 1)] ∂_{z_{2}} + (6κ + 1) z_{3} ∂_{z_{3}}
+ (6κ + 1) z_{4} ∂_{z_{4}}.  (13)

Once the explicit expression for the operator Δ^{κ} in the z variables is given, the Schrödinger equation can be solved iteratively. By direct application of Δ^{κ} to z^{m} ≡ z_{1}^{m_{1}} z_{2}^{m_{2}} z_{2}^{m_{3}} z_{4}^{m_{4}}, we find

Δ^{κ}z^{m} = ε_{m}(κ) z^{m} − ∑_{i=1}^{4} a_{m}^{i} z^{m−α_{i}} − ∑_{j∈I} b_{m}^{j} z^{m−(α_{2}+α_{j})} − ∑_{ij∈T} c_{m}^{ij} z^{m−(α_{2}+α_{i}+α_{j})}
− 2 a_{m}^{2} ∑_{ij∈T} z^{m−(2α_{2}+α_{i}+α_{j})} − d_{m} z^{m−(α_{1}+2α_{2}+α_{3}+α_{4})}
− 4 ∑_{j∈I} a_{m}^{j} z^{m−(α_{1}+2α_{2}+α_{3}+α_{4}+α_{j})},  (14)

where the sets of indices are I = {1, 3, 4} and T = {13, 14, 34}, and

a_{m}^{i} = 4 m_{i}(m_{i} − 1),  b_{m}^{j} = 12m_{2}m_{j},
c_{m}^{ij} = 16 m_{i} m_{j},  d_{m}(κ) = 16 m_{2} (2 − m_{2} − κ + ∑_{j∈I} m_{j}) .

All monomials in Δ^{κ}z^{m} take the form z^{m−μ} with μ as a positive root. Thus, the polynomial P_{m}^{κ} has the form

P_{m}^{κ}(z) = ∑_{μ∈Q^{+}(m)} c_{μ}z^{m−μ},  (15)

where we choose the normalization c_{0} = 1 and, if Q^{+} is the cone of positive roots,

Q^{+}(m) = {μ ∈ Q^{+} | z^{m−μ} is well defined if z_{1} z_{2} z_{3} z_{4} = 0} .  (16)

The above-mentioned partial ordering of monomials is given simply by the height of μ, i. e. z^{m−μ_{1}} > z^{m−μ_{2}} if ht(μ_{1}) < ht(μ_{2}). From (14), the coefficients c_{μ} obey the iterative formula

c_{μ} = \frac{N_{μ}}{ε_{m−μ}(κ) − ε_{m}(κ)}  (17)

with

N_{μ} = ∑_{i=1}^{4} a_{m−(μ−α_{i})}^{i} c_{μ−α_{i}} + ∑_{j∈I} b_{m−(μ−α_{2}−α_{j})}^{j} c_{μ−(α_{2}+α_{j})} + ∑_{ij∈T} c_{m−(μ−α_{2}−α_{i}−α_{j})}^{ij} c_{μ−(α_{2}+α_{i}+α_{j})}
+ 2 ∑_{ij∈T} a_{m−(μ−2α_{2}−α_{i}−α_{j})}^{2} c_{μ−(2α_{2}+α_{i}+α_{j})} + d_{m−(μ−α_{1}−2α_{2}−α_{3}−α_{4})} c_{μ−(α_{1}+2α_{2}+α_{3}+α_{4})}
+ 4 ∑_{j∈I} a_{m−(μ−α_{1}−2α_{2}−α_{3}−α_{4}−α_{j})}^{j} c_{μ−(α_{1}+2α_{2}+α_{3}+α_{4}+α_{j})}.

Along with the explicit expressions for the roots given in Appendix A, it is suitable for the implementation on a symbolic computer program. A list of polynomials obtained through the use of this formula is offered in Appendix B.

3 The structure of the recurrence relations

As it is well known, all the systems of orthogonal polynomials in one indeterminate z, such that P_{m}(z) = z^{m} + ⋯ satisfy a recursive formula z P_{m}(z) = a_{m} P_{m+1}(z) + b_{m} P_{m}(z) + c_{m} P_{m−1}(z). In particular, the orthogonal polynomials associated to the trigonometric Calogero-Sutherland model for the case of two particles and Lie algebra A_{1} are the classical Gegenbauer polynomials, whose recursive formula is known to be

z P_{m}^{κ}(z) = P_{m+1}^{κ}(z) + \frac{m(m − 1 + 2κ)}{(m − 1 + κ)(m + κ)} P_{m−1}^{κ}(z).

This formula is reminiscent of the Clebsch-Gordan series for A_{1}. In fact, for κ = 1 it reduces exactly to this Clebsch-Gordan series: the polynomials are the characters of A_{1} and the coefficents are equal to one. Immediately the question arises about the existence of analogous recurrence relations, i.e., with the structure of κ-deformations of the corresponding Clebsch-Gordan series, for the polynomials related to Calogero-Sutherland models associated to other simple Lie algebras. As it was shown in [11], the answer turns out to be in the afirmative for all root systems, but to obtain the expressions for the deformed coefficients it is necessary to proceed through a case-by-case analysis. Once the coefficients are known, many applications are possible. The aim of this section is to fix the structure of the basic recurrence relations for the case of D_{4} and to give a simple illustration of their use.

We want to study the formulas for z_{i}P_{m}^{κ}(z), i = 1, 2, 3, 4. Therefore, as P_{m}^{(1)}(z) = z_{i} for m_{j} = (δ_{ji}), and the recursive formulas are deformations of the Clebsch-Gordan series, we need to know the weights of the irreducible representations whose integral dominant weights are λ_{1}, λ_{2}, λ_{3} and λ_{4}. For the case of λ_{1}, λ_{3} and λ_{4}, these representations have dimension eight. On the other hand, if we act on the highest weight with the Weyl group in the way explained in the Appendix A, we obtain eight different weights. Thus, these representations include only one orbit of the Weyl group and we are done. For the case of λ_{2}, the representation has dimension 28 and the orbit of the Weyl group containing λ_{2} has only 24 elements. But λ_{2} = α_{12}^{+}, the highest root, and thus this representation is the adjoint one and includes a second orbit: the Cartan subalgebra, with four elements of weight zero. Let us summarize.

• Weights in z_{1}: ±λ_{1}, ±(λ_{1} − λ_{2}), ±(λ_{2} − λ_{3} − λ_{4}), ±(λ_{3} − λ_{4}).

• Weights in z_{2}: ±λ_{2}, ±(λ_{2} − 2λ_{j}), ±(2λ_{2} − λ_{1} − λ_{3} − λ_{4}), ±(λ_{2} + λ_{i} − λ_{j} − λ_{k}), ±(λ_{i} + λ_{j} − λ_{k}), ±(λ_{2} − λ_{1} − λ_{3} − λ_{4}), 0, with i, j, k ∈ I.

• Weights in z_{3}: ±λ_{3}, ±(λ_{3} − λ_{2}), ±(λ_{2} − λ_{1} − λ_{4}), ±(λ_{1} − λ_{4}).

• Weights in z_{4}: ±λ_{4}, ±(λ_{4} − λ_{2}), ±(λ_{2} − λ_{1} − λ_{3}), ±(λ_{1} − λ_{3}).

With these weights, the structure of the recurrence relations results to be as follows:

z_{1} P_{m_{1},m_{2},m_{3},m_{4}}^{κ}(z) = P_{m_{1}+1,m_{2},m_{3},m_{4}}^{κ}(z) + a_{m}^{1}(κ) P_{m_{1}−1,m_{2},m_{3},m_{4}}^{κ}(z) + b_{m}^{1}(κ) P_{m_{1}+1,m_{2}−1,m_{3},m_{4}}^{κ}(z)
+ c_{m}^{1}(κ) P_{m_{1}−1,m_{2}+1,m_{3},m_{4}}^{κ}(z) + d_{m}^{1}(κ) P_{m_{1},m_{2}+1,m_{3}−1,m_{4}−1}^{κ}(z)
+ e_{m}^{1}(κ) P_{m_{1},m_{2}−1,m_{3}+1,m_{4}+1}^{κ}(z) + f_{m}^{1}(κ) P_{m_{1},m_{2},m_{3}+1,m_{4}−1}^{κ}(z)
+ g_{m}^{1}(κ) P_{m_{1},m_{2},m_{3}−1,m_{4}+1}^{κ}(z)
z_{2} P_{m_{1},m_{2},m_{3},m_{4}}^{κ}(z) = P_{m_{1},m_{2}+1,m_{3},m_{4}}^{κ}(z) + A_{m}(κ) P_{m_{1},m_{2}−1,m_{3},m_{4}}^{κ}(z) + B_{m}(κ)^{1±} P_{m_{1}±2,m_{2}∓1,m_{3},m_{4}}^{κ}(z)
+ B_{m}(κ)^{3±} P_{m_{1},m_{2}∓1,m_{3}±2,m_{4}}^{κ}(z) + B_{m}(κ)^{4±} P_{m_{1},m_{2}∓1,m_{3},m_{4}±2}^{κ}(z)
+ C_{m}(κ)^{±} P_{m_{1}∓1,m_{2}±2,m_{3}∓1,m_{4}∓1}^{κ}(z) + D_{m}(κ)^{1±} P_{m_{1}±1,m_{2}±1,m_{3}∓1,m_{4}∓1}^{κ}(z)
+ D_{m}(κ)^{3±} P_{m_{1}∓1,m_{2}±1,m_{3}±1,m_{4}∓1}^{κ}(z) + D_{m}(κ)^{4±} P_{m_{1}∓1,m_{2}±1,m_{3}∓1,m_{4}±1}^{κ}(z)
+ E_{m}(κ)^{1±} P_{m_{1}∓1,m_{2},m_{3}±1,m_{4}±1}^{κ}(z) + E_{m}(κ)^{3±} P_{m_{1}±1,m_{2},m_{3}∓1,m_{4}±1}^{κ}(z)
+ E_{m}(κ)^{4±} P_{m_{1}±1,m_{2},m_{3}±1,m_{4}∓1}^{κ}(z) + F_{m}(κ)^{±} P_{m_{1}±1,m_{2}∓1,m_{3}±1,m_{4}±1}^{κ}(z)
+ G_{m}(κ) P_{m_{1},m_{2},m_{3},m_{4}}^{κ}(z),
z_{3} P_{m_{1},m_{2},m_{3},m_{4}}^{κ}(z) = P_{m_{1},m_{2},m_{3}+1,m_{4}}^{κ}(z) + a_{m}^{3}(κ) P_{m_{1},m_{2},m_{3}−1,m_{4}}^{κ}(z) + b_{m}^{3}(κ) P_{m_{1},m_{2}−1,m_{3}+1,m_{4}}^{κ}(z)
+ c_{m}^{3}(κ) P_{m_{1},m_{2}+1,m_{3}−1,m_{4}}^{κ}(z) + d_{m}^{3}(κ) P_{m_{1}−1,m_{2}+1,m_{3},m_{4}−1}^{κ}(z)
+ e_{m}^{3}(κ) P_{m_{1}+1,m_{2}−1,m_{3},m_{4}+1}^{κ}(z) + f_{m}^{3}(κ) P_{m_{1}+1,m_{2},m_{3},m_{4}−1}^{κ}(z)
+ g_{m}^{3}(κ) P_{m_{1}−1,m_{2},m_{3},m_{4}+1}^{κ}(z),
z_{4} P_{m_{1},m_{2},m_{3},m_{4}}^{κ}(z) = P_{m_{1},m_{2},m_{3},m_{4}+1}^{κ}(z) + a_{m}^{4}(κ) P_{m_{1},m_{2},m_{3},m_{4}−1}^{κ}(z) + b_{m}^{4}(κ) P_{m_{1},m_{2}−1,m_{3},m_{4}+1}^{κ}(z)
+ c_{m}^{4}(κ) P_{m_{1},m_{2}+1,m_{3},m_{4}−1}^{κ}(z) + d_{m}^{4}(κ) P_{m_{1}−1,m_{2}+1,m_{3}−1,m_{4}}^{κ}(z)
+ e_{m}^{4}(κ) P_{m_{1}+1,m_{2}−1,m_{3}+1,m_{4}}^{κ}(z) + f_{m}^{4}(κ) P_{m_{1}−1,m_{2},m_{3}+1,m_{4}}^{κ}(z)
+ g_{m}^{4}(κ) P_{m_{1}+1,m_{2},m_{3}−1,m_{4}}^{κ}(z),

where B_{m}(κ)^{1±} P_{m_{1}±2,m_{2}∓1,m_{3},m_{4}}^{κ}(z) means B_{m}(κ)^{1+} P_{m_{1}+2,m_{2}−1,m_{3},m_{4}}^{κ}(z)+B_{m}(κ)^{1−} P_{m_{1}−2,m_{2}+1,m_{3},m_{4}}^{κ}(z), etc, and it is understood that all polynomials involving negative quantum numbers are zero. The recurrence relations reflect triality in the fact that not all the coefficients appearing in these formulas are independent. There are coincidences upon permutations of the quantum numbers, for instance

a_{m_{1},m_{2},m_{3},m_{4}}^{1} = a_{m_{3},m_{2},m_{1},m_{4}}^{3} = a_{m_{4},m_{2},m_{3},m_{1}}^{4},  (18)

and similarly for b_{m}^{j}, c_{m}^{j}, d_{m}^{j}, e_{m}^{j}, f_{m}^{j}, g_{m}^{j}. In the same fashion, we have also

B_{m_{1},m_{2},m_{3},m_{4}}^{1±} = B_{m_{3},m_{2},m_{1},m_{4}}^{3±} = B_{m_{4},m_{2},m_{3},m_{1}}^{4±}  (19)

and similarly for D_{m}^{j±}, E_{m}^{j±}.

As an example, let us consider a simple case in which only one of the quantum numbers is nonvanishing, namely,

z_{1} P_{m,0,0,0}^{κ}(z) = P_{m+1,0,0,0}^{κ}(z) + a_{m}(κ) P_{m−1,0,0,0}^{κ}(z) + c_{m}(κ) P_{m−1,1,0,0}^{κ}(z),  (20)

where we write a_{m}(κ) = a_{m,0,0,0}^{1}(κ) and c_{m}(κ) = c_{m,0,0,0}^{1}(κ). Using formulae

P_{m,0,0,0}^{κ}(z) = z_{1}^{m} − \frac{m(m − 1) [4κ^{2} + 4(m − 2)κ + (m − 1)(m − 2)]}{(m − 1 + κ)(m − 1 + 3κ)(m − 2 + κ)} z_{1}^{m−2} − \frac{m(m − 1)}{m − 1 + κ}z_{1}^{m−2} z_{2} + ⋯
P_{m,1,0,0}^{κ}(z) = z_{1}^{m} z_{2} + \frac{4 κ(κ − 1)(m − 2 + 2κ)}{(m + 1 + 5κ)(m + 2κ)(m − 1 + κ)} z_{1}^{m} + ⋯,

we obtain the coeffficients in (20)

a_{m}(κ) = \frac{m(m + 2κ)(m − 1 + 4κ)(m − 1 + 6κ)}{(m − 1 + κ)(m − 1 + 3κ)(m + 3κ)(m + 5κ)},
c_{m}(κ) = \frac{m(m − 1 + 2κ)}{(m + κ)(m − 1 + κ)}.

As a byproduct of triality, we can also write other two recurrence relations with the same coefficients:

z_{3}P_{0,0,m,0}^{κ}(z) = P_{0,0,m+1,0}^{κ}(z) + a_{m}(κ) P_{0,0,m−1,0}^{κ}(z) + c_{m}(κ) P_{0,1,m−1,0}^{κ}(z)
z_{4} P_{0,0,0,m}^{κ}(z) = P_{0,0,0,m+1}^{κ}(z) + a_{m}(κ) P_{0,0,0,m−1}^{κ}(z) + c_{m}(κ) P_{0,1,0,m−1}^{κ}(z).  (21)

The first of these recurrence relations can be used to devise an algorithm for the calculation of the polynomials of the form P_{m,0,0,0}^{κ}(z) and P_{m,1,0,0}^{κ}(z). By multiplying (20) by the differential operator Δ^{κ} − ε_{m−1,1,0,0}(κ), the term involving P_{m−1,1,0,0}^{κ} cancels. Using the explicit expressions (9), (13), we find

P_{m+1,0,0,0}^{κ} = \frac{1}{4(m + κ)} [Δ^{κ}, z_{1}] P_{m,0,0,0}^{κ}(z) − \frac{1 + 4κ}{2(m + κ)} z_{1}P_{m,0,0,0}^{κ}(z)
+ \frac{m(m + 2κ)(m − 1 + 4κ)(m − 1 + 6κ)}{(m − 1 + κ)(m − 1 + 3κ)(m + κ)(m + 3κ)} P_{m−1,0,0,0}^{κ}(z),

where, from (13),

[Δ^{κ}, z_{1}] = 4 (z_{1}^{2} − 2z_{2} − 8) ∂_{z_{1}} + 2 (z_{1}z_{3} − 8z_{4}) ∂_{z_{3}} + 2 (z_{1}z_{4} − 8z_{3}) ∂_{z_{4}}
+ 4 (z_{1}z_{2} − 3z_{3}z_{4} − 4z_{1}) ∂_{z_{2}} + 2(6κ + 1) z_{1}.

where, from (13),

[Δ^{κ}, z_{1}] = 4 (z_{1}^{2} − 2z_{2} − 8) ∂_{z_{1}} + 2 (z_{1}z_{3} − 8z_{4}) ∂_{z_{3}} + 2 (z_{1}z_{4} − 8z_{3}) ∂_{z_{4}}
+ 4 (z_{2}z_{2} − 3z_{3}z_{4} − 4z_{1}) ∂_{z_{2}} + 2(6κ + 1) z_{1}.

Once the polynomials P_{m,0,0,0}^{κ}(z) are known, the recurrence relation (20) provides a formula for each P_{m,1,0,0}^{κ}(z):

c_{m+1}(κ) P_{m,1,0,0}^{κ}(z) = z_{1} P_{m+1,0,0,0}^{κ}(z) − P_{m+2,0,0,0}^{κ}(z) − a_{m+1}(κ) P_{m,0,0,0}^{κ}(z).  (22)

4 Some generating functions

We present in this section the generating functions for some characters and symmetric monomial functions. Let us consider first the case of the monomial functions with only one non-vanishing quantum number in the form P_{m,0,0,0}^{(0)}(z). The generating function for this subset is

F_{0}(t, z) = ∑_{m=0}^{∞} t^{m} P_{m,0,0,0}^{(0)}(z).  (23)

In terms of the x variables, the general expression for these monomial functions is

P_{m,0,0,0}^{(0)}(x) = ∑_{j=1}^{4} (x_{j}^{m} + x_{j}^{−m}) ,  (24)

and, in particular, we define P_{0,0,0,0}^{(0)}(z) = 8. In these variables, the computation of F_{0}(t, x) only requires to sum the geometric series:

F_{0}(t, x) = ∑_{j=1}^{4} (\frac{1}{1 − tx_{j}} + \frac{1}{1 − \frac{t}{x_{j}}} ) .  (25)

The change to the original z variables can be done by the inspection of the coefficients of the powers of t in both the numerator and denominator of this rational expression, with the result

F_{0}(t, z) = \frac{N_{0}(t, z)}{D(t, z)},  (26)

where

N_{0}(t, z) = 8 − 7z_{1} t + 6z_{2} t^{2} − 5(z_{3}z_{4} − z_{1}) t^{3} + 4(z_{3}^{2} + z_{4}^{2} − 2z_{2} − 2) t^{4} − 3(z_{3}z_{4} − z_{1}) t^{5}
+ 2z_{2} t^{6} − z_{1} t^{7},
D(t, z) = 1 − z_{1} t + z_{2} t^{2} − (z_{3}z_{4} − z_{1}) t^{3} + (z_{3}^{2} + z_{4}^{2} − 2z_{2} − 2) t^{4} − (z_{3}z_{4} − z_{1}) t^{5}
+ z_{2} t^{6} − z_{1} t^{7} + t^{8}.  (27)

There is an alternative approach. As the monomial functions are eigenfunctions of Δ^{(0)} with eigenvalues ε_{m,0,0,0}(0) = 2m^{2}, we have

\frac{1}{2} Δ^{(0)} F_{0}(t, z) = ∑_{m=0}^{∞} m^{2}t^{m} P_{m,0,0,0}^{(0)}(z),

and, therefore, we can write a differential equation for F_{0}(t, z):

[\frac{1}{2}Δ^{(0)} − (t ∂_{t})^{2}] F_{0}(t, z) = 0,  F_{0}(0, z) = 8.  (28)

One can verify by substitution that (26) satisfies this equation. When F_{0}(t, z) is known, we can easily obtain the generating function

G_{0}(t, z) = ∑_{m=0}^{∞} t^{m} P_{m,1,0,0}^{(0)}(z)  (29)

by only recalling (20), which for κ = 0 is simply

z_{1}P_{m,0,0,0}^{(0)}(z) = P_{m+1,0,0,0}^{(0)}(z) + P_{m−1,0,0,0}^{(0)}(z) + P_{m−1,1,0,0}^{(0)}(z).  (30)

This gives

G_{0}(t, z) = \frac{M_{0}(t, z)}{D(t, z)}  (31)

with

M_{0}(t, z) = z_{2} − 4 + (6 z_{1} − 3 z_{3}z_{4}) t
+ (−8 − 2 z_{1}^{2} − 10 z_{2} − z_{2}^{2} + 4 z_{3}^{2} + 2 z_{1}z_{3}z_{4} + 4 z_{4}^{2}) t^{2}
+ (10 z_{1} + 5 z_{1}z_{2} − 3 z_{1}z_{3}^{2} − 4 z_{3}z_{4} + z_{2}z_{3}z_{4} − 3 z_{1}z_{4}^{2}) t^{3}
+ (8 z_{2} − 4 z_{1}^{2} + 2 z_{2}^{2} − z_{2}z_{3}^{2} + 4 z_{1}z_{3}z_{4} − z_{2}z_{4}^{2}) t^{4}
+ (−6 z_{1} − 6 z_{1}z_{2} − z_{3}z_{4} + z_{2}z_{3}z_{4}) t^{5} + (8 + 6z_{1}^{2} + 2 z_{2} − z_{2}^{2}) t^{6}
+ (−10 z_{1} + z_{1}z_{2}) t^{7} + (4 − z_{2}) t^{8}.

The computation of the generating functions for the characters P_{m,0,0,0}^{(1)} and P_{m,1,0,0}^{(1)} goes through similar arguments. In this case, the eigenvalues are ε_{m,0,0,0}(1) = 2m^{2} + 12m. Hence,

F_{1}(t, z) = ∑_{m=0}^{∞} t^{m}P_{m,0,0,0}^{(1)}(z) ,  P_{0,0,0,0}^{(1)}(z) ≡ 1  (32)

is the solution of the equation

[\frac{1}{2}Δ^{(1)} − (t ∂_{t})^{2} − 6t ∂_{t}] F_{1}(t, z) = 0,  F_{1}(0, z) = 1.  (33)

The Weyl character formula implies that the denominator of F_{1}(t, z) should be the same D(t, z) found before. Thus, we try an Ansatz

F_{1}(t, z) = \frac{N_{1}(t, z)}{D(t, z)}  (34)

and obtain the simple answer

N_{1}(t, z) = 1 − t^{2}.  (35)

Applying the recurrence relation (20)) we obtain the generating function G_{1}(t, z) for the characters P_{m,1,0,0}^{(1)}:

G_{1}(t, z) = \frac{1}{D(t, z)} \{z_{2} − z_{3}z_{4}t + (z_{3}^{2} + z_{4}^{2} − 2z_{2} − 1) t^{2} − (z_{3}z_{4} − z_{1}) t^{3} + z_{2} t^{4} − z_{1} t^{5} + t^{6}\} .  (36)

5 More recurrence relations and other results

In this Section, we give the remaining recurrence relations involving the product of a fundamental character times a polynomial with only one non-vanishing quantum number. We also comment the existence of some peculiar values for κ for which the polynomials associated to some special excited states are proportional to integer powers of the fundamental state wavefunction.

To obtain the mentioned recurrence relations, it is necessary to compute the coefficients of a limited number of terms of the polynomials involved. Once the form of these terms is known, we can obtain the coefficients in the recurrence relations solving a system of linear algebraic equations. We do not give here the full expressions for the coefficients of the required terms, because some of them are too long, and only list them:

P_{1,0,m,0}^{κ}(z) = z_{1}z_{3}^{m} + A z_{3}^{m−1}z_{4} + ⋯,
P_{0,m,0,0}^{κ}(z) = z_{2}^{m} + B z_{2}^{m−1} + C z_{2}^{m−2} + D z_{1}z_{2}^{m−2}z_{3}z_{4} + E z_{1}z_{2}^{m−3}z_{3}z_{4}
+ F (z_{1}^{2}z_{2}^{m−2} + z_{2}^{m−2}z_{3}^{2} + z_{2}^{m−2}z_{4}^{2}) + ⋯,
P_{1,m,0,0}^{κ}(z) = z_{1}z_{2}^{m} + G z_{1}z_{2}^{m−1}z_{4} + H z_{2}^{m−1}z_{3}z_{4} + ⋯,
P_{0,m,1,1}^{κ}(z) = z_{2}^{m}z_{3}z_{4} + I z_{1}z_{2}^{m} + ⋯,
P_{m,0,0,0}^{κ}(z) = z_{1}^{m} + J z_{1}^{m−2} + K z_{1}^{m−2}z_{2} + ⋯,
P_{m,1,0,0}^{κ}(z) = z_{1}^{m}z_{2} + L z_{1}^{m−2}z_{2} + N z_{1}^{m−1}z_{3}z_{4} + M z_{1}^{m} + ⋯,
P_{m,0,1,1}^{κ}(z) = z_{1}^{m}z_{3}z_{4} + N z_{1}^{m−1}z_{2} + O z_{1}^{m+1} + ⋯,
P_{1,m,1,1}^{κ}(z) = z_{1}z_{2}^{m}z_{3}z_{4} + P z_{2}^{m} + Q z_{1}z_{2}^{m−1}z_{3}z_{4} + R (z_{1}^{2}z_{2}^{m} + z_{2}^{m}z_{3}^{2} + z_{2}^{m}z_{4}^{2}) + S z_{2}^{m+1} + ⋯,
P_{2,m,0,0}^{κ}(z) = z_{1}^{2}z_{2}^{m} + T z_{2}^{m} + U z_{1}z_{2}^{m−1}z_{3}z_{4} + W z_{2}^{m+1} + · · · .

The use of the quantities denoted A to W in the previous formulas in the general structure of the recurrence relations give the following results:

• Formulae of type z_{1}P_{0,0,m,0}^{κ}(z):

z_{1} P_{0,0,m,0}^{κ}(z) = P_{1,0,m,0}^{κ}(z) + b_{m}(κ)P_{0,0,m−1,1}(z)
z_{1} P_{0,0,0,m}^{κ}(z) = P_{1,0,0,m}^{κ}(z) + b_{m}(κ)P_{0,0,1,m−1}(z)
z_{3} P_{m,0,0,0}^{κ}(z) = P_{m,0,1,0}^{κ}(z) + b_{m}(κ)P_{m−1,0,0,1}(z)
z_{3} P_{0,0,0,m}^{κ}(z) = P_{0,0,1,m}^{κ}(z) + b_{m}(κ)P_{1,0,0,m−1}(z)
z_{4} P_{m,0,0,0}^{κ}(z) = P_{m,0,0,1}^{κ}(z) + b_{m}(κ)P_{m−1,0,1,0}(z)
z_{4} P_{0,0,m,0}^{κ}(z) = P_{0,0,m,1}^{κ}(z) + b_{m}(κ)P_{1,0,m−1,0}(z)

with

b_{m}(κ) = \frac{m(m − 1 + 4κ)}{(m − 1 + κ)(m + 3κ)} .

• Formulae of type z_{1} P_{0,m,0,0}^{κ}(z):

z_{1} P_{0,m,0,0}^{κ}(z) = P_{1,m,0,0}^{κ}(z) + d_{m}(κ)P_{1,m−1,0,0}(z) + e_{m}(κ)P_{0,m−1,1,1}^{κ}(z)
z_{3} P_{0,m,0,0}^{κ}(z) = P_{0,m,1,0}^{κ}(z) + d_{m}(κ)P_{0,m−1,1,0}(z) + e_{m}(κ)P_{1,m−1,0,1}^{κ}(z)
z_{4} P_{0,m,0,0}^{κ}(z) = P_{0,m,0,1}^{κ}(z) + d_{m}(κ)P_{0,m−1,0,1}(z) + e_{m}(κ)P_{1,m−1,1,0}^{κ}(z)

with

d_{m}(κ) = \frac{2m(m + κ)(m − 1 + 3κ)(m − 1 + 4κ)(2m − 1 + 6κ)}{(m − 1 + κ)(m − 1 + 2κ)(m + 3κ)(2m − 1 + 5κ)(2m + 5κ)},
e_{m}(κ) = \frac{m(m − 1 + 3κ)}{(m − 1 + κ)(m + 2κ)} .

• Formulae of type z_{2} P_{m,0,0,0}^{κ}(z):

z_{2} P_{m,0,0,0}^{κ}(z) = P_{m,1,0,0}^{κ}(z) + f_{m}(κ) P_{m−2,1,0,0}(z) + g_{m}(κ) P_{m−1,0,1,1}^{κ}(z) + h_{m}(κ) P_{m,0,0,0}^{κ}(z)
z_{2} P_{0,0,m,0}^{κ}(z) = P_{0,1,m,0}^{κ}(z) + f_{m}(κ) P_{0,1,m−2,0}(z) + g_{m}(κ)P_{1,0,m−1,1}^{κ}(z) + h_{m}(κ) P_{0,0,m,0}^{κ}(z)
z_{2} P_{0,0,0,m}^{κ}(z) = P_{0,1,0,m}^{κ}(z) + f_{m}(κ) P_{0,1,0,m−2}(z) + g_{m}(κ) P_{1,0,1,m−1}^{κ}(z) + h_{m}(κ) P_{0,0,0,m}^{κ}(z)

with

f_{m}(κ) = \frac{m(m − 1)(m − 2 + 2κ)(m + 2κ)(m − 1 + 4κ)(m − 1 + 5κ)}{(m − 2 + κ)(m − 1 + κ)^{2}(m − 1 + 3κ)(m + 3κ)(m + 4κ)},
g_{m}(κ) = \frac{m(m − 1 + 3κ)}{(m − 1 + κ)(m + 2κ)},
h_{m}(κ) = \frac{4 [−3κ^{3} + 5κ^{2} + (6m − 1)κ + (m^{2} − 1)]}{(m − 1 + κ)(1 + 3κ)(m + 1 + 5κ)} .

• Formula for z_{2} P_{0,m,0,0}^{κ}(z):

z_{2} P_{0,m,0,0}^{κ}(z) = P_{0,m+1,0,0}^{κ}(z) + k_{m}(κ) P_{0,m−1,0,0}(z) + p_{m}(κ) P_{1,m−1,1,1}^{κ}(z) + q_{m}(κ) P_{1,m−2,1,1}^{κ}(z)
+ r_{m}(κ) [P_{2,m−1,0,0}^{κ}(z) + P_{0,m−1,2,0}^{κ}(z) + P_{0,m−1,0,2}^{κ}(z)] + s_{m}(κ) P_{0,m,0,0}^{κ}(z)

with

k_{m}(κ) = \frac{4m(m + κ)^{2}(m + 2κ)(m − 1 + 3κ)(m − 1 + 4κ)^{2}(2m − 1 + 4κ)(m − 1 + 5κ)(2m − 1 + 6κ)}{(m − 1 + κ)(m − 1 + 2κ)^{2}(m + 3κ)^{2}(m + 4κ)(2m − 2 + 5κ)(2m − 1 + 5κ)^{2}(2m + 5κ)},
p_{m}(κ) = \frac{m(m − 1 + 2κ)}{(m − 1 + κ)(m + κ)},
q_{m}(κ) = \frac{2m(m − 1)(m + κ)^{2}(m − 2 + 2κ)(m − 1 + 3κ)^{3}(2m − 1 + 6κ)}{(m − 2 + κ)(m − 1 + κ)^{2}(m − 1 + 2κ)^{2}(m + 2κ)^{2}(2m − 1 + 5κ)(2m + 5κ)},
r_{m}(κ) = \frac{m(m + κ)(m − 1 + 3κ)(m − 1 + 4κ)}{(m − 1 + κ)(m − 1 + 2κ)(m + 2κ)(m + 3κ)},
s_{m}(κ) = \frac{−4t_{m}(κ)}{(κ + 1)(m − 1 + κ)(m + 1 + 4κ)(2m − 1 + 5κ)(2m + 1 + 5κ)},
t_{m}(κ) = (−1 + 5m^{2} − 4m^{4}) + (2 + 25m − 7m^{2} − 40m^{3} + 2m^{4})κ + (20 − 35m − 123m^{2} + 20m^{3})κ^{2},
+ (−22 − 115m + 63m^{2})κ^{3} + (−19 + 65m)κ^{4} + 20κ^{5}.

Finally, we mention that for κ = −\frac{1}{2}(n − 1), n ∈ N, the polynomials associated to the dominant weight which is n times the Weyl vector ρ are proportional to a power of the ground state wavefunction, namely

P_{nρ}^{− \frac{1}{2} (n−1)} = (−1)^{n} 2^{12n} \{∏_{j<k}^{4} sin(q_{j} − q_{k}) sin(q_{j} + q_{k})\}^{n} .

This formula can be verified quite easily by direct application of Δ^{− \frac{1}{2} (n−1)} in the form (7) to the righthand side: one finds that the Schrödinger equation (6) with the appropriate eigenvalue is satisfied. The most convenient way to fix the proportionality constant is by performing an analytic continuation to complex q_{i} and considering the region x_{i} ∈ R and x_{1} ≫ x_{2} ≫ x_{3} ≫ x_{4} ≫ 0. Then, the polynomials are dominated by the leading order term, P_{nρ}^{− \frac{1}{2} (n−1)} ≃ z_{1}^{n}z_{2}^{n}z_{3}^{n}z_{4}^{n}, and, on the other hand, using the formulas for the fundamental characters displayed in Section 2. one finds z_{1}z_{2}z_{3}z_{4} ≃ x_{1}^{3}x_{2}^{2}x_{3} and ∏_{j<k}^{4} sin(q_{j} − q_{k}) sin(q_{j} + q_{k}) ≃ −2^{−12}x_{1}^{3}x_{2}^{2}x_{3}. This gives the proportionality constant written above.

6 Conclusions

In this paper, we have shown how to solve the Schrödinger equation for the trigonometric Calogero-Sutherland model related to the Lie algebra D_{4} and we have explored some properties of the energy eigenfunctions. The main point is that the use of a Weyl-invariant set of variables, the characters of the fundamental representations, leads to a formulation of the Schrödinger equation by means of a second order differential operator which is simple enough to make feasible a recursive method for the treatment of the spectral problem. The eigenfunctions provide a complete system of orthogonal polynomials in four variables, and these polynomials obey recurrence relations which are extensions of the Clebsch-Gordan series of the algebra. The structure of some of these recurrence relations has been fixed and, for particular cases, the coefficients involved have been computed. Also, some generating functions for the polynomials with parameter κ = 1 and κ = 0 have been obtained. These generating functions can give some hints about the form of the generating function for general κ, see [20].

Acknowledgements

A.M.P. would like to express his gratitude to the Max Planck Institut für Mathematik for hospitality. The work of J.F.N. and W.G.F. has been partially supported by the University of Oviedo, Vicerrectorado de Investigación, grant MB-03-514-1.

Appendix A: Summary of results on the Lie algebra D_{4}

In this appendix, we review some standard facts about the root and weight systems of the Lie algebra D_{4} that the reader could find useful to follow the main text. More extensive and sound treatments of these topics can be found in many excellent textbooks, see for instance [18], [19].

The most convenient explicit representation of D_{4} is

D_{4} = \{( \begin{matrix}m & b \\ c & −m^{t} \end{matrix} ) | m, b, c real 4 × 4 matrices and b^{t} = −b, c^{t} = −c\} .

This gives dim D_{4} = 28. One can choose the following linear basis:

M_{jk} = E_{j,k} − E_{4+j,4+k},  j, k = 1, 2, 3, 4
B_{jk} = E_{j,4+k} − E_{k,4+j},  j, k = 1, 2, 3, 4, j < k
C_{jk} = E_{4+j,k} − E_{4+k,j},  j, k = 1, 2, 3, 4, j < k

with (E_{i,j})_{kl} = δ_{ik}δ_{jl}. The Cartan subalgebra is

H = {h = ∑_{i=1}^{4} c_{i}M_{ii} | c_{i} ∈ R}

and this confirms that the rank of D_{4} is four. The matrix commutators

[h, M_{jk}] = (c_{j} − c_{k})M_{jk},
[h, B_{jk}] = (c_{j} + c_{k})B_{jk},
[h, C_{jk}] = −(c_{j} + c_{k})C_{jk}

allow us to classify the 24 roots in two groups

α_{jk}(h) = c_{j} − c_{k},  j ≠ k,
α_{jk}^{±}(h) = ±(c_{j} + c_{k}),  j < k.

One can extract the following basis of simple roots

α_{1} ≡ ( 1, −1, 0, 0 ) = α_{12}, α_{2} ≡ ( 0, 1, −1, 0 ) = α_{23},
α_{3} ≡ ( 0, 0, 1, −1 ) = α_{34}, α_{4} ≡ ( 0, 0, 1, 1 ) = α_{34}^{+},

where we have given the decomposition of these roots in the basis of H^{∗} dual to diag(M_{ii}), i=1,2,3,4. The euclidean relations among the simple roots are

(α_{i}, α_{i}) = 2,  i = 1, 2, 3, 4,
(α_{2}, α_{i}) = −1,  i = 1, 3, 4,
(α_{i}, α_{j}) = 0,  i = 1, 3, 4.

Thus, the Cartan matrix reads

A = ( \begin{matrix}2 & −1 & 0 & 0 \\ −1 & 2 & −1 & −1 \\ 0 & −1 & 2 & 0 \\ 0 & −1 & 0 & 2 \end{matrix} ) .

The positive roots are α_{ij}, α_{ij}^{+}, i < j, and they can be classified by heights as indicated in the table. The

Height Positive roots
1  α_{1}, α_{2}, α_{3}, α_{4}
2  α_{13} = α_{1} + α_{2}, α_{24} = α_{2} + α_{3}, α_{24}^{+} = α_{2} + α_{4}
3  α_{14} = α_{1} + α_{2} + α_{3}, α_{14}^{+} = α_{1} + α_{2} + α_{4}, α_{23}^{+} = α_{2} + α_{3} + α_{4}
4  α_{13}^{+} = α_{1} + α_{2} + α_{3} + α_{4}
5  α_{12}^{+} = α_{1} + 2α_{2} + α_{3} + α_{4}

Table 1: Heights of positive roots.

Weyl group is easy to describe. The Weyl reflection on the hyperplane in H^{∗} orthogonal to the root α is s_{α}(v) = v − 2\frac{(α,v)}{(α,α)}α. Applying this formula to α_{ij}, α_{ij}^{±}, one readily finds that the most general Weyl reflection consists in a permutation of the components of v in the e_{i} basis plus an even number of changes of the signs of these components. This gives |W | = 192 for the order of the Weyl group. The fundamental weights λ_{k} can be obtained from the equation α_{i} = ∑_{j=1}^{4} A_{ji}λ_{j}. They are

λ_{1} = \frac{1}{2}(2α_{1} + 2α_{2} + α_{3} + α_{4}) = \frac{1}{2} ( 2, 0, 0, 0 ),
λ_{2} = \frac{1}{2}(2α_{1} + 4α_{2} + 2α_{3} + 2α_{4}) = \frac{1}{2} ( 2, 2, 0, 0 ),
λ_{3} = \frac{1}{2}(α_{1} + 2α_{2} + 2α_{3} + α_{4}) = \frac{1}{2} ( 1, 1, 1, −1 ),
λ_{4} = \frac{1}{2}(α_{1} + 2α_{2} + α_{3} + 2α_{4}) = \frac{1}{2} ( 1, 1, 1, 1 ),

and the geometry of the weight system is summarized by the relations

∥ λ_{1} ∥=∥ λ_{3} ∥=∥ λ_{4} ∥= 1,  ∥ λ_{2} ∥= \sqrt{2},
(λ_{i}, λ_{2}) = 1, i = 1, 3, 4,  (λ_{i}, λ_{j}) = \frac{1}{2}, i, j = 1, 3, 4.

The Weyl vector is

ρ = \frac{1}{2} ∑_{α∈R^{+}} = ∑_{j=1}^{4} λ_{j} = α = 3α_{1} + 5α_{2} + 3α_{3} + 3α_{4} = ( 3, 2, 1, 0 ),

and the Weyl formula for dimensions applied to the irreducible representation associated to the integral dominant weight m = m_{1}λ_{1} + m_{2}λ_{2} + m_{3}λ_{3} + m_{4}λ_{4} gives

dim r(m) = ∏_{α∈R^{+}} \frac{(α, m + ρ)}{(α, ρ)} = \frac{P}{1440}

with

P = ∏_{i=1}^{4}(m_{i} + 1) ∏_{j}(m_{2} + m_{j} + 2) ∏_{j<k}(m_{2} + m_{j} + m_{k} + 3) (m_{1} + m_{2} + m_{3} + m_{4}) (m_{1} + 2m_{2} + m_{3} + m_{4})

where the indices j, k take the values 1, 3, 4. In particular, for the fundamental representations, one finds:

dim r(λ_{1}) = 8,  dim r(λ_{2}) = 28,
dim r(λ_{3}) = 8,  dim r(λ_{4}) = 8.

Appendix B: Some polynomials, characters and monomial functions

We list here all the polynomials, characters and monomial functions with total degree lower or equal to three up to triality.

Polynomials

P_{1,0,0,0}^{κ}(z) = z_{1}
P_{0,1,0,0}^{κ}(z) = z_{2} + \frac{4(κ − 1)}{5κ + 1},
P_{2,0,0,0}^{κ}(z) = z_{1}^{2} − \frac{2}{1 + κ} z_{2} − \frac{8κ}{(1 + κ)(1 + 3κ)}
P_{0,2,0,0}^{κ}(z) = z_{2}^{2} − \frac{2}{1 + κ} z_{1}z_{3}z_{4} − \frac{2(−1 + κ)}{(1 + κ)(1 + 2κ)} (z_{1}^{2} + z_{3}^{2} + z_{4}^{2}) + \frac{4(−3 + 5κ + 6κ^{2} + 4κ^{3})}{(1 + κ)(1 + 2κ)(3 + 5κ)} z_{2} +
+ \frac{16(−1 + κ)(3 + 10κ + 3κ^{2} + 2κ^{3})}{(1 + κ)(1 + 2κ)(2 + 5κ)(3 + 5κ)}
P_{1,1,0,0}^{κ}(z) = z_{1}z_{2} − \frac{3}{1 + 2κ} z_{3}z_{4} + \frac{4(−1 + κ)(−1 + 2κ)}{(1 + 2κ)(2 + 5κ)}z_{1}
P_{1,0,1,0}^{κ}(z) = z_{1}z_{3} − \frac{4}{1 + 3κ}z_{4}
P_{3,0,0,0}^{κ}(z) = z_{1}^{3} − \frac{6}{2 + κ}z_{1}z_{2} + \frac{6}{(1 + κ)(2 + κ)}z_{3}z_{4} − \frac{12(1 + 2κ + 2κ^{2})}{(1 + κ)(2 + κ)(2 + 3κ)} z_{1}
P_{0,3,0,0}^{κ}(z) = z_{2}^{3} − \frac{6}{2 + κ} z_{1}z_{2}z_{3}z_{4} + \frac{6}{(1 + κ)(2 + κ)} (z_{1}^{2}z_{3}^{2} + z_{1}^{2}z_{4}^{2} + z_{3}^{2}z_{4}^{2}) − \frac{3(2 + κ + κ^{2})}{(1 + κ)^{2}(2 + κ)} (z_{1}^{2}z_{2} + z_{2}z_{3}^{2} + z_{2}z_{4}^{2})
+ \frac{6(10 + 17κ + 21κ^{2} + 10κ^{3} + 2κ^{4})}{5(1 + κ)^{3}(2 + κ)} z_{2}^{2} − \frac{3(30 + 53κ + 4κ^{2} − 15κ^{3} + 8κ^{4})}{5(1 + κ)^{4}(2 + κ)} z_{1}z_{3}z_{4}

− \frac{12κ(8 + 10κ + κ^{2} + κ^{3})}{5(1 + κ)^{4}(2 + κ)}(z_{1}^{2} + z_{3}^{2} + z_{4}^{2}) + \frac{12(30 + 119κ + 159κ^{2} + 124κ^{3} + 80κ^{4} + 24κ^{5} + 4κ^{6})}{5(1 + κ)^{4}(2 + κ)(4 + 5κ)}z_{2}
+ \frac{16(−30 + 103κ + 440κ^{2} + 359κ^{3} + 98κ^{4} + 86κ^{5} + 20κ^{6} + 4κ^{7})}{5(1 + κ)^{4}(2 + κ)(3 + 5κ)(4 + 5κ)}
P_{2,1,0,0}^{κ}(z) = z_{1}^{2}z_{2} − \frac{2}{1 + κ}z_{2}^{2} − \frac{1 + 3κ}{(1 + κ)^{2}} z_{1}z_{3}z_{4} + \frac{4(−1 + κ)κ^{2}}{(1 + κ)^{2}(3 + 5κ)} z_{1}^{2} + \frac{4}{(1 + κ)^{2}} (z_{3}^{2} + z_{4}^{2})
− \frac{4(9 + 27κ + 28κ^{2} + 16κ^{3})}{(1 + κ)^{2}(2 + 3κ)(3 + 5κ)}z_{2} − \frac{16(3 + 5κ + 2κ^{3})}{(1 + κ)^{2}(2 + 3κ)(3 + 5κ)}
P_{1,2,0,0}^{κ}(z) = z_{1}z_{2}^{2} − \frac{2}{1 + κ}z_{1}^{2}z_{3}z_{4} − \frac{1 + 3κ}{(1 + κ)^{2}} z_{2}z_{3}z_{4} − \frac{2(−1 + κ)}{(1 + κ)(1 + 2κ)}z_{1}^{3} + \frac{5 − κ}{(1 + κ)^{2}} (z_{1}z_{3}^{2} + z_{1}z_{4}^{2})
+ \frac{4(−1 + κ)(9 + 19κ + 10κ^{2} + 4κ^{3})}{(1 + κ)^{2}(1 + 2κ)(4 + 5κ)}z_{1}z_{2} − \frac{4(−1 + κ)(−5 + 2κ)(1 + 3κ)}{(1 + κ)^{2}(1 + 2κ)(4 + 5κ)}z_{3}z_{4}
+ \frac{8(−9 − 57κ − 72κ^{2} + 28κ^{3} − 2κ^{4} + 4κ^{5})}{(1 + κ)^{2}(1 + 2κ)(3 + 5κ)(4 + 5κ)} z_{1}
P_{1,1,1,0}^{κ}(z) = z_{1}z_{2}z_{3} − \frac{3}{1 + 2κ}(z_{1}^{2}z_{4} + z_{3}^{2}z_{4}) − \frac{8(−1 + κ)}{(1 + 2κ)(2 + 3κ)}z_{2}z_{4} + \frac{4(12 + 23κ − 11κ^{2} + 6κ^{3})}{(1 + 2κ)(2 + 3κ)(3 + 5κ)}z_{1}z_{3}
− \frac{8(3 − 22κ + 4κ^{2})}{(1 + 2κ)(2 + 3κ)(3 + 5κ)}z_{4}
P_{1,0,1,1}^{κ}(z) = z_{1}z_{3}z_{4} − \frac{4}{1 + 3κ}(z_{1}^{2} + z_{3}^{2} + z_{4}^{2}) + \frac{12}{(1 + 2κ)(1 + 3κ)}z_{2} + \frac{16(1 + 5κ)}{(1 + 2κ)(1 + 3κ)^{2}}

Characters

P_{1,0,0,0}^{(1)}(z) = z_{1}
P_{0,1,0,0}^{(1)}(z) = z_{2}
P_{2,0,0,0}^{(1)}(z) = z_{1}^{2} − z_{2} − 1
P_{0,2,0,0}^{(1)}(z) = z_{2}^{2} + z_{2} − z_{1}z_{3}z_{4}
P_{1,1,0,0}^{(1)}(z) = z_{1}z_{2} − z_{3}z_{4}
P_{1,0,1,0}^{(1)}(z) = z_{1}z_{3} − z_{4}
P_{3,0,0,0}^{(1)}(z) = z_{1}^{3} − 2z_{1}z_{2} + z_{3}z_{4} − 2z_{1}
P_{0,3,0,0}^{(1)}(z) = z_{2}^{3} + 3z_{2}^{2} + 3z_{2} − 2z_{1}z_{2}z_{3}z_{4} + z_{1}^{2}z_{3}^{2} + z_{1}^{2}z_{4}^{2} + z_{3}^{2}z_{4}^{2}
− (z_{1}^{2} + z_{3}^{2} + z_{4}^{2})z_{2} − z_{1}z_{3}z_{4} − z_{1}^{2} − z_{3}^{2} − z_{4}^{2} + 1
P_{2,1,0,0}^{(1)}(z) = z_{1}^{2}z_{2} − z_{2}^{2} − z_{1}z_{3}z_{4} + z_{3}^{2} + z_{4}^{2} − 2z_{2} − 1
P_{1,2,0,0}^{(1)}(z) = z_{1}z_{2}^{2} − z_{1}^{2}z_{3}z_{4} − z_{2}z_{3}z_{4} + z_{1}(z_{3}^{2} + z_{4}^{2}) − z_{1}
P_{1,1,1,0}^{(1)}(z) = z_{1}z_{2}z_{3} + z_{1}z_{3} − (z_{1}^{2} + z_{3}^{2})z_{4} + z_{4}
P_{1,0,1,1}^{(1)}(z) = z_{1}z_{3}z_{4} − z_{1}^{2} − z_{3}^{2} − z_{4}^{2} + z_{2} + 2

Monomial functions

P_{1,0,0,0}^{(0)}(z) = z_{1}

P_{0,1,0,0}^{(0)}(z) = z_{2} − 4
P_{2,0,0,0}^{(0)}(z) = z_{1}^{2} − 2z_{2}
P_{0,2,0,0}^{(0)}(z) = z_{2}^{2} − 2 z_{1}z_{3}z_{4} + 2 z_{1}^{2} + 2 z_{3}^{2} + 2 z_{4}^{2} − 4 z_{2} − 8
P_{1,1,0,0}^{(0)}(z) = z_{1}z_{2} − 3z_{3}z_{4} + 2z_{1}
P_{1,0,1,0}^{(0)}(z) = z_{1}z_{3} − 4z_{4}
P_{3,0,0,0}^{(0)}(z) = z_{1}^{3} − 3z_{1}z_{2} + 3z_{3}z_{4} − 3z_{1}
P_{0,3,0,0}^{(0)}(z) = z_{2}^{3} + 6z_{2}^{2} + 9z_{2} − 3z_{1}z_{2}z_{3}z_{4} + 3z_{1}^{2}z_{3}^{2} + 3z_{1}^{2}z_{4}^{2} + 3z_{3}^{2}z_{4}^{2}
− 3(z_{1}^{2} + z_{3}^{2} + z_{4}^{2})z_{2} − 9z_{1}z_{3}z_{4} − 4
P_{2,1,0,0}^{(0)}(z) = z_{1}^{2}z_{2} − 2z_{2}^{2} − z_{1}z_{3}z_{4} + 4z_{3}^{2} + 4z_{4}^{2} − 6z_{2} − 8
P_{1,2,0,0}^{(0)}(z) = z_{1}z_{2}^{2} − 2z_{1}^{2}z_{3}z_{4} − z_{2}z_{3}z_{4} + 2z_{1}^{3} + 5z_{1}(z_{3}^{2} + z_{4}^{2}) − 9z_{1}z_{2} − 5z_{3}z_{4} − 6z_{1}
P_{1,1,1,0}^{(0)}(z) = z_{1}z_{2}z_{3} + 8z_{1}z_{3} − 3(z_{1}^{2} + z_{3}^{2})z_{4} + 4z_{2}z_{4} − 4z_{4}
P_{1,0,1,1}^{(0)}(z) = z_{1}z_{3}z_{4} − 4z_{1}^{2} − 4z_{3}^{2} − 4z_{4}^{2} + 12z_{2} + 16

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