Generating functions for characters and weight multiplicities of irreducible 𝓈𝓁(4)-modules

José Ferńandez Núñez; Wifredo García Fuertes; Askold M. Perelomov

doi:10.1080/14029251.2018.1503436

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Volume 25, Issue 4, July 2018, Pages 618 - 632

Generating functions for characters and weight multiplicities of irreducible 𝓈𝓁(4)-modules

Authors

José Ferńandez Núñez

Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, 33007-Oviedo, Spain,nonius@uniovi.es

Wifredo García Fuertes

Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, 33007-Oviedo, Spain,wifredo@uniovi.es

Askold M. Perelomov

Institute for Theoretical and Experimental Physics, Moscow, Russia,aperelomo@gmail.com

Received 22 January 2018, Accepted 14 May 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1503436 How to use a DOI?
Keywords: Calogero-Sutherland models; Generating functions; Lie algebra representation theory
Abstract: Generating functions for the characters of the irreducible representations of simple Lie algebras are rational functions where both the numerator and denominator can be expressed as polynomials in the characters corresponding to the fundamental weights. They encode much information on the representation theory of the algebra, but their explicit expressions are in general very complicated. In fact, it seems that rank three is the highest rank tractable. In this paper, we use a method based on the quantum Calogero-Sutherland model to compute the full generating function for the characters of irreducible modules over the complex Lie algebra 𝔰𝔩 (4), and exploit this result to obtain also generating functions giving the multiplicities of some low order weights in all representations. We have applied the same method to compute the generating function for the characters of the modules the other rank three simple Lie algebras, but in these cases the full expressions are very long and appear only in the arXiv version of the paper (arXiv:1705.03711 [math-ph]). Nevertheless, when the generating functions are limited to some particular subsets of characters, the results are quite simple and we present them here.
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Generating functions for characters are very useful tools for the study of representations of Lie algebras. They were introduced and shown to be rational functions in [21], and since then different procedures for computing these objects and extracting practical information from them have been developed, see for instance [17] and references therein. We have recently shown how the theory of the quantum integrable Calogero-Sutherland models can be used to obtain the generating functions [5] in an efficient manner, and we have obtained, proceeding on this basis, several results regarding characters and weight multiplicities of the representations of the rank two simple Lie algebras [6,7]. Compared to other schemes, the approach based on the Calogero-Sutherland model is advantageous in that cumbersome combinatorial recipes involving the Weyl group are bypassed. This makes the method a rather convenient one to be applied to any simple Lie algebra.

Nevertheless, an unavoidable fact which is independent of the method employed is that, as the rank increases, the final result for the full generating function of characters becomes quickly exceedingly complicated. Thus, in general, for higher rank algebras only the generating functions for some restricted sets of characters, typically with only one or two non-vanishing Dynkin indices, are tractable without falling into an excessive clumsiness. Apart from rank two algebras, the only exception seems to be the case of the complex Lie algebra 𝔰𝔩 (4), in which the Weyl orbits of the fundamental weights, the Weyl formula for dimensions, and the Calogero-Sutherland Hamiltonian, are still quite simple, these circumstances suggesting that the whole generating function for characters is not too unwieldy to be computed. As far as we know, this generating function has not been explicitly written down in previous works. It seems thus worth to use the method of [5] to obtain it and to explore some other results which can be deduced from it.

Results of this type are interesting in themselves, but also due to the fact that 𝔰𝔩(4)≃𝔬(6) is a Lie algebra with applications to important physical systems. Let us mention, among several others, the grand unified Pati-Salam model with gauge group SU(2)_L × SU (2)_R × SU (4) [23], which has been much investigated owing to the fact that it fits well with the string or M-theory framework and it is also suitable to explain phenomenological issues like neutrino oscillations or baryogenesis [22]; the effective SU (4) hadronic models which, in spite of the large breaking of SU (4) flavor symmetry, can be used to study the phenomenology of charmed particles, see for instance [10]; the SU (4)-Kondo effect, due to the interplay between spin and orbital electronic degrees of freedom, which has recently aroused remarkable interest [11] because of its role in condensed matter settings such as quantum dots, carbon nanotubes or nanowires; or the AdS₅/CFT₄ string-gauge equivalence [15], in which SU (4) is the R-symmetry in the supersymmetric quantum field theory side of the duality and SO(6) is the symmetry of the effective IIB gauged supergravity on the string side.

2. The generating function for characters

This section is devoted to the computation of the generating function for the characters of the irreducible representations of 𝔰𝔩 (4) by means of the approach developed in [5]. This approach needs a background on Calogero-Sutherland models [1,26], which is succinctly explained in Subsection 2.1. The computation of the generating function is done in Subsection 2.2.

2.1. Review of the theory of quantum Calogero-Sutherland models

Calogero-Sutherland models are a class of mechanical systems which enjoy the existence of a complete set of mutually commuting integrals of motion and are, therefore, integrable systems in the sense of Liouville. The first analysis of a system of this kind was performed by Calogero [1] 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 [26] arrived to similar results for the quantum problem on the circle, this time with trigonometric interaction; and later Moser [16] proved, in terms of Lax pairs, that the classical counterparts of these models also enjoyed integrability. The identification of the general scope of these discoveries came with the work of Olshanetsky and Perelomov [18], who realized that it is possible to associate models of this kind to all the root systems of the simple Lie algebras, and that all these models are integrable, both in the classical and the quantum framework, for interactions of the type rational (or inverse-square), q⁻²; rational + quadratic, q⁻² + ω²q²; trigonometric, sin⁻²q; hyperbolic, sinh⁻²q; and the most general, given by the Weierstrass elliptic function 𝒫(q). Nowadays, there is a widespread interest in this kind of integrable systems, and many mathematical and physical applications for them have been found, see for instance [2]

Here, we limit ourselves to review the most salient features of the trigonometric model which are useful for our present purposes and refer the reader to [19] for a more detailed treatment. Let 𝒜 be a complex simple Lie algebra of rank r. As is well known [20], the roots α₁, α₂, …, α_r and fundamental weights λ₁, λ₂,…,λ_r of 𝒜 can be conveniently represented by elements of a vector space V whose dimension is r or r + 1 depending on the algebra. The Hamiltonian of Calogero-Sutherland model associated to 𝒜 has the form

H=12p2+U(q)

where the coordinates q and momenta p are elements of V. The potential term is

U(q)=∑α∈ℛ+κα(κα-1)sin-2〈α,q〉,

where ℛ⁺ is the set of positive roots of 𝒜 and ⟨⋅, ⋅⟩ is the Euclidean scalar product on V. The constants κ_α must be chosen in such a way that the couplings gα2=κα(κα-1) are equal for roots of equal length. It turns out that the energy eigenfunctions of this Hamiltonian depend on r quantum numbers m = (m₁, m₂,…,m_r) and are of the form Ψmκ=Ψ0κ⋅Φmκ , where

Ψ0κ(q)=∏α∈ℛ+sinκα〈α,q〉

is the wave function of the ground state and the ΦmK are solutions of the related Schrödinger equation

ΔκΦmκ(q)=ε(m;κ)Φmκ(q), (2.1)

where Δ^κ is the linear differential operator

Δκ=-12∑j=1l∂qj2-∑α∈ℛ+〈α,α〉καcot〈α,q〉〈α,∂q〉

and the eigenvalues are

ε(m;κ)=2〈λ+2ρ(κ),λ〉 (2.2)

for 2p(κ)=∑α∈ℛ+καα and λ the highest weight λ = m₁ λ₁ + m₂ λ₂ + ⋯ + m_rλ_r defined by m.

The most relevant fact for us is that if we tune all coupling constants κ_α to one, the eigenfunctions of this Schrödinger operator Δ^κ are precisely the characters χ_m of the irreducible representations of the algebra [19]

Φm1(q)=χm(q)=∑wnwe2i〈λj,q〉, (2.3)

where the sum extends to all weights w entering in the representation and n_w is the multiplicity of the weight w. This comes about as follows. Although the potential vanishes for κ_α → 1, there is a remnant of the interaction in that, to take the limit consistently, we have to choose fermionic boundary conditions ensuring that the wave functions are zero when sin ⟨α, q⟩ = 0 for any positive root. As a consequence, the wave function Ψm1 is given by a Weyl-alternating sum of free-particle exponentials which turns out to coincide exactly with the numerator of the Weyl character formula. The ground state wave function, on the other hand, can be rewritten as the denominator of the Weyl formula, and the Φm1 are the characters of the irreducible modules of the Lie algebra thereby. (The particles are free also for κ_α = 0, but in that case bosonic boundary conditions are appropriate and the Φm0 are the monomial symmetric functions associated to the root system; the Φmκ for other values of the couplings are systems of orthogonal polynomials which interpolate between the monomial symmetric functions and the characters.)

Thus, we can obtain the characters by solving the second order differential equation

Δ1χm(q)=ε(m;1)χm(q) (2.4)

Furthermore, if we change variables and describe the dynamical system by means of the characters z_k = 𝒳_{λ_k} of the fundamental representations R_{λ_k}, k = 1,2,…,r, the differential operator Δ¹ takes the form

Δz1=∑j,k=1rajk(z)∂zj∂zk+∑j=1rbj(z)∂zj, (2.5)

with a_jk(z) and b_j(z) polynomials in the z_k with integer coefficients, and the Schrödinger equation can be solved by iterative methods [24]. This operator can be given an explicit form taking into account that:

•
bj(z)=Δz1zj=ε(0,…,1(j,…,0;1)zj, and
•
Δz1(zjzk)=2ajk(z)+bj(z)zk+bk(z)zj,

while z_jz_k is the character of the tensor product R_{λ_j} ⊗ R_{λ_k}

Hence, knowing all the quadratic Clebsch-Gordan series of the algebra we will be able to determine the a_jk(z) coefficients. For more explicit details, see for instance [4] and references therein. Once we know the definite expression for the operator Δz1 it is possible to compute the characters χ_m as polynomials in the z-variables by solving the Schrödinger equation (2.1) in a recursive way.

2.2. Computation of the generating function for 𝔰𝔩 (4)

For the particular case of 𝔰𝔩 (4), the model can be interpreted as describing the quantum dynamics of a system of four particles moving on a circle. The one-dimensional coordinates are assembled into the four-tuple q = (q₁, q₂, q₃, q₄) with the constraint that the center of mass is fixed at the origin, i.e., q₁ + q₂ + q₃ + q₄ = 0. The particles interact through a pairwise potential of trigonometric form whose strength is governed by a single coupling constant κ. The eigenfunctions Φmκ and the eigenvalues ε(m; κ) are indexed by the 3-tuples of non-negative integers m = (m₁, m₂, m₃) –the quantum numbers,– and m₁λ₁ + m₂λ₂ + m₃λ₃ are the highest weights of the irreducible representations of the algebra, with λ₁, λ₂, λ₃ being the fundamental weights. The set of independent Weyl-invariant variables z₁, z₂, z₃, namely the characters of the three fundamental representations R_{λ_j} of 𝔰𝔩 (4), are related to the q-variables by

z1=χ1,0,0=x1+x2x1+x3x2+1x3,z2=χ0,1,0=x2+x3x1+x2x1x3+x1x3x2+x1x3+1x2z3=χ0,0,1=x3+x2x3+x1x2+1x1,, (2.6)

where xj=e2i〈λj,q〉 , the fundamental weights λ_j of 𝔰𝔩 (4) given as four-tuples in the standard way. Higher order characters are polynomials in the z-variables with integer coefficients. The change of variables from the coordinates on the circle to the fundamental characters, see [9], leads to the following form Δz1 of the Hamiltonian for coupling κ = 1 in terms of the z-variables:

Δz1=12[(3z12-8z2)∂z12+(4z22-8z1z3-16)∂z22+(3z32-8z2)∂z32+(4z1z2-24z3)∂z1∂z2+(2z1z3-32)∂z1∂z3+(4z2z3-24z1)∂z2∂z3+15z1∂z1+20z2∂z2+15z3∂z3]; (2.7)

the eigenvalues (2.2), on the other hand, are

ε(m;1)=12(3m12+4m22+3m32+4m1m2+2m1m3+4m2m3+12m1+16m2+12m3). (2.8)

In this setup, as explained in [5], the generating function for characters

G(t1,t2,t3;z1,z2,z3)=∑m1=0∞∑m2=0∞∑m3=0∞t1m1t2m2t3m3χm1,m2,m3(z1,z2,z3) (2.9)

is a rational function

G(t1,t2,t3;z1,z2,z3)=N(t1,t2,t3;z1,z2,z3)D(t1,t2,t3;z1,z2,z3) (2.10)

that can be obtained by applying (2.4) to the generating function, which yields the differential equation

(Δt-Δz1)G(tk;zk)=0,(2.11)

where the differential operator Δ_t is what ε(m; 1) turns into after making the substitution mi → t_i∂_{t_i} in (2.8). The computation goes through the following four steps:

Step (i): The denominator of the generating function is
D(t1,t2,t3;z1,z2,z3)=D1×D2×D3, (2.12)
where Di=Πj(1-tix1nj1x2nj2x3nj3) with the product extended to all the weights ∑k=13njkλk entering in the Weyl orbit of the fundamental representation R_{λ_i}. Looking at (2.6) and bearing in mind that the fundamental representations of 𝔰𝔩 (4) contain one single Weyl orbit, the result follows. After changing variables back from the x_j to the z_k, it can be written as
D1=1-t1z1+t12z2-t13z3+t14,D2=1+t26-(t2+t25)z2+(t22+t24)(z1z3-1)+t23(2z2-z12-z32),D3=1-t3z3+t32z2-t33z1+t34. (2.13)
Step (ii): The Weyl formula for dimensions gives for the representation R_λ, where λ=∑imiλi ,
dimRλ=112(m1+1)(m2+1)(m3+1)(m1+m2+2)(m2+m3+2)(m1+m2+m3+3). (2.14)

To obtain the generating function for dimensions E(t₁, t₂, t₃) it suffices to perform the change mi→ti∂ti in this formula and to apply the resulting differential operator to ∏i=13(1-ti)-1 . One finds in this way that
E(t1,t2,t3)=P(t1,t2,t3)(1-t1)4(1-t2)6(1-t3)4, (2.15)
where
P(t1,t2,t3)=1-4t1t2-4t2t3-t1t3-t22+t12t2+t2t32+4t1t22+4t22t3+6t1t2t3-t12t23-t23t32-4t12t22t3-4t1t22t32-6t1t23t3+t1t24t3+t12t22t32+4t1t23t32+4t12t23t3-t12t24t32.

A further simplification is possible, but we have written E(t₁, t₂, t₃) in such a way that the denominator comes from the substitution in (2.12) of the fundamental characters z₁, z₂ and z₃ by their dimensions.
Step (iii): We will compute the numerator N(t₁, t₂, t₃; z₁, z₂, z₃) of the generating function of characters (2.10) by tentatively assuming that it contains only the terms appearing in the numerator P(t₁, t₂, t₃) of (2.15), now with coefficients depending of the z-variables. Under such hypothesis, we can expand N / D as a series in the t-variables and compare with the right-hand side of (2.9), so that we will be able to fix the coefficients in N(t₁, t₂, t₃; z₁, z₂, z₃) provided that the expressions of the characters of some 𝔰𝔩 (4)-modules with small values of m_i (in fact, m_i ≤ 4 for the case at hand) are known. To obtain these is not a difficult task: as we have said, they are polynomials in the z-variables and, given the simple structure of the Hamiltonian Δz1 2.7, they can be computed by recursively solving the eigenvalue equation (2.1)^a. Thus we obtain
N(t1,t2,t3;z1,z2,z3)=1-z3t1t2-z1t2t3-t1t3-t22+t12t2+t2t32+z1t1t22+z3t22t3+z2t1t2t3-t12t23-t23t32-z1t12t22t3-z3t1t22t32-z2t1t23t3+t1t24t3+t12t22t32+z1t1t23t32+z3t12t23t3-t12t24t32. (2.16)
Step (iv): We need to be sure that the conjecture to limit the number of unknown coefficients in step (iii) is correct. For this purpose, we have to verify that G = N/D (2.10) does indeed satisfy the differential equation (2.11). This is a matter of directly plugging (2.10) into (2.11) and doing the derivatives. In this way, one can check that (2.11) is fulfilled. Thus (2.10) with (2.12), (2.13), and (2.16) is the correct generating function for characters of the irreducible modules of 𝔰𝔩 (4).

3. Generating functions for weight multiplicities

Once we have the generating function for characters, it is possible to use it to obtain some other results. Let us consider, in particular, generating functions of the form

An1,n2,n3(t1,t2,t3)=∑m1=0∞∑m2=0∞∑m3=0∞μm1,m2,m3(n1,n2,n3)t1m1t2m2t3m3, (3.1)

where μ_m_{₁, m₂, m₃}(n₁, n₂, n₃) is the multiplicity of the weight n₁λ₁ + n₂λ₂ + n₃λ₃ in the representation R_m_{₁λ₁, m₂λ₁, m₃λ₁} of 𝔰𝔩 (4). The way in which these generating functions can be computed is described in [6,7], see also [3]: after expressing G(t₁, t₂, t₃; z₁, z₂, z₃) in the x-variables by means of (2.6), they are given by the triple integral

An1,n2,n3(t1,t2,t3)=1(2πi)3∮dx3∮dx1∮dx2G(t1,t2,t3;x1,x2,x3)x11+n1x21+n2x31+n3, (3.2)

where the integration contours are along the unit circles on the complex x₁, x₂ and x₃-planes. We will give here explicit expressions of A_n_{₁, n₂, n₃(}t₁, t₂, t₃) for n₁ + n₂ + n₃ ≤ 2. In all these cases, the integrations (3.2), which are readily performed by means of the residue theorem, go along the same pattern. First, the integral in x₂ acquires contributions from poles arising at x₂ = t₁, x₂ = t₁x₃, x₂ = t₃x₁ and x₂ = t₂x₁x₃; then, for the integral in x₁ there are poles at x₁ = t₃, x1=t12x3 , x₁ = t₂x₃ and x1=t22x3-1 ; finally, the poles contributing to the last integral are located at x₃ = t₁, x₃ = t₂t₃, x3=t33 , x1=t23/2 and x1=t2-3/2 , except for the case (n₁, n₂, n₃) = (2,0,0), where an additional pole at x₃ = 0 occurs. After the residues are evaluated, we find the final results for (3.1) in the form

An1,n2,n3(t1,t2,t3)=Nn1,n2,n3(t1,t2,t3)D0(t1,t2,t3), (3.3)

where the denominator is in all cases

D0(t1,t2,t3)=(1-t14)(1-t34)(1-t1t3)2(1-t12t2)(1-t2t32)(1-t22)2

and the numerators are given in the Appendix.

Explicit formulas for the weight multiplicities of the representations of simple Lie algebras are in general difficult to obtain and are known only in particular cases, see for instance the recent paper [14], devoted to study this subject for the so-called fundamental string representations of the classical algebras. Regarding this point, we should mention that a possible application of the generating function An1,n2,n3(t1,t2,t3) is to obtain closed formulas for the multiplicities μ_m_{₁, m₂, m₃}(n₁, n₂, n₃) by proceeding as done for rank two algebras in [7] (for an alternative approach, applied also to these algebras, see [25]). Nevertheless, in the case of 𝔰𝔩 (4) the expressions given above are somewhat complicated and the procedure turns out to be considerably cumbersome, as are indeed other approaches: see for instance [27], or [13] for a recent computation of μ_m_{₁, m₂, m₃}(0,0,0). Thus, we have studied the case of the real weights in the previous list by means of the Kostant multiplicity formula [12], see [8] for a pedagogic exposition,

μm1,m2,m3(n1,n2,n3)=∑w∈W(-1)w𝒵[w(∑i=13(mi+1)λi))-∑i=13(ni+1)λi],

where W is the Weyl group and 𝒵[∑i=13kiαi]≡𝒵[k1,k2,k3] is the Kostant partition function for 𝔰𝔩 (4). This function gives the number of different ways in which a vector of the root lattice can be expressed as a linear combination of the positive roots with non-negative integer coefficients. The generating function for 𝒵[k_1,k_2,k₃] is

∑k1=0∞∑k2=0∞∑k3=0∞t1k1t2k2t3k3𝒵[k1,k2,k3]=1(1-t1)(1-t2)(1-t3)(1-t1t2t3)(1-t1t2)(1-t2t3)

Thus, 𝒵[k_1,k_2,k₃] is symmetric under interchange of k₁ and k₃, and its expression for k₁ ≤ k₃ can eventually found to be

6𝒵[k1,k2,k3]=(k2+1)(k2+2)(k2+3),6𝒵[k1,k2,k3]=(k1+1)(k1+2)(3k2-2k1+3),6𝒵[k1,k2,k3]=(k1+1)(k1+2)(3k3-k1+3),6𝒵[k1,k2,k3]=(k2-k3+1)(k2-k3+2)(2k2-3k1+k3+3) -(k1-k2+k3)(3k1+3k3-12k2+2k12+2k32-k22-k1k2-2k1k3-k2k3-11),

for, respectively, the cases

i) k1≥k2;i) k1≥k2, k3≥k2;iii) k3≤k2, k1≤k2-k3;iv) k3<k2, k1>k2-k3.

With this, and taking advantage of the symmetry under m₁ ↔ m₃ to state the results only for the case m₁ ≤ m₃, one finds the following formulas:

• μ_m_{₁, m₂, m₃}(0,0,0) ≠ 0 only if m₃ − m₁ = 2m₂ + 4p with p integer, and in this case

μm1,m2,m3(0,0,0)=a(m1,m2,m3)if p≥0,8μm1,m2,m3(0,0,0)=(m1+1)[b(m1,m2,m3)+8]if p<0;

• μ_m_{₁, m₂, m₃}(0,1,0) ≠ 0 only if m₃ − m₁ = 2(m₂ −1) + 4p with p integer, and in this case

μm1,m2,m3(0,1,0)=a(m1,m2,m3)-2(m1+1)δp,0if p≥0,8μm1,m2,m3(0,1,0)=(m1+1)[b(m1,m2,m3)+4]if p<0;

• μ_m_{₁, m₂, m₃}(1,0,1) ≠ 0 only if m₃ − m₁ = 2m₂ + 4p with p integer, and in this case

μm1,m2,m3(1,0,1)=a(m1,m2,m3)-(m1+1)δp,0if p≥0,8μm1,m2,m3(1,0,1)=(m1+1)b(m1,m2,m3)if p<0;

• μ_m_{₁, m₂, m₃}(0,2,0) ≠ 0 only if m₃ − m₁ = 2(m₂ −2) + 4p with p integer, and in this case

μm1,m2,m3(0,2,0)=a(m1,m2,m3)-2(m1+1)(δp,1+3δp,0)+δp,1δm2,0+δp,0δm2,2if p≥0,8μm1,m2,m3(0,2,0)=(m1+1)[b(m1,m2,m3)-8]+8δm1,m3if p<0,

where

a(m1,m2,m3)=12(m1+1)(m2+1)(m1+m2+2),b(m1,m2,m3)=4[(m2+1)(m3+1)-1]-(m1-m3)2.

The derivation of these expressions from the Kostant multiplicity formula is a laborious process: the Weyl group of 𝔰𝔩 (4) has order 24, and hence there are many different cases which must be separately considered and then assembled together. Thus, to give a detailed description of the proof of these results is pretty tedious. Nevertheless, once they are written down, the generating functions for multiplicities given above provide a practical way to check that they are correct. In each case, with the help of a program for symbolic computations, it is easy to expand the generating function as a Taylor series in t-variables up to some high order and to subtract from this expansion the corresponding series built with the μ_m_{₁, m₂, m₃}(n₁, n₂, n₃) coefficients. One then finds that the difference is zero, as it should be. This application illustrates one the benefits of working out explicit formulas like (2.10) or (3.3): despite their awkward appearance, they are considerably useful tools to check at once a number of other results concerning the representations of the algebra.

4. Generating function for the characters of real representations

The generating function obtained in Section 2 collects together the characters of all irreducible representations of 𝔰𝔩 (4). It can be of interest to have also generating functions for particular subsets of characters. The simplest examples are the generating functions for characters with only one or two non-vanishing Dynkin indices, which follow directly from (2.10) when the appropriate t-variables are taken to vanish. A more interesting distinction is between the characters of complex and real representations, the latter being those with highest weight symmetric under interchange of z₁ and z₃, i.e., of the form χ_m_{₁, m₂, m₃}. The general four-step procedure used in Section 2 can be also applied to construct the generating function for characters of this type,

GR(t1,t2;z1,z2,z3)=∑m1=0∞∑m2=0∞t1m1t2m2χm1,m2,m1(z1,z2,z3),

as follows:

Step (i): Assuming that the generating function G_R is rational, the denominator is now
DR(t1,t2;z1,z2,z3)=D13×D2, (4.1)
where the weights entering in D₁₃ are those in the Weyl orbit R_{λ₁ + λ₃}. These can be read from the corresponding monomial symmetric function
Mλ1+λ3=x1x3+x1x2x3+x12x2+x2x3x1+x22x1x3+x32x2+c.c.,
and lead to the expression
D13=1+t112-(t1+t111)(z1z3-4)+(t12+t110)(z12z2-2z22-4z1z3+z2z32+10)-(t13+t19)(z14-7z12z2+8z22+13z1z3+z1z22z3-7z2z32+z34-20)+(t14+t18)d48+(t15+t17)d57+t16d6
where
d48=z24-3z14+18z12z2-16z22-24z1z3+z13z2z3-8z1z22z3-z12z32+18z2z32+z1z2z33-3z34+31,d57=4z24-6z14+29z12z2-24z22-z12z23-34z1z3+5z13z2z3-19z1z22z3-2z12z32+29z2z32-z23z32-z13z33+5z1z2z33-6z34+40,d6=z12z22z32-7z14+34z12z2-28z22-2z12z23+6z24-40z1z3+6z13z2z3-24z1z22z3+34z2z32-2z23z32-2z13z33+6z1z2z33-7z34+44.
Step (ii): For real representations, the dimensions (2.14) are
dimRm1λ1+m2λ2+m1λ3=112(m1+1)2(m2+1)(m1+m2+2)2(2m1+m2+3)

Given this formula, we can proceed as in Section 2 to shape the generating function for dimensions. It turns out to be
ER(t1,t2)=(1-t1)6(1-t2)PR(t1,t2)(1-t1)12(1-t2)6
where
PR(t1,t2)=1+9t1+9t12+t13+t2-17t1t2-39t12t2-5t13t2+5t1t22+39t12t22+17t13t22-t14t22-t1t23-9t12t23-9t13t23-t14t23.
Step (iii): We next compute the numerator N_R(t₁, t₂; z₁, z₂, z₃) of G_R by provisionally assuming that the only non-vanishing coefficients correspond to the monomials appearing in the numerator of E_R(t₁, t₂). After using the eigenvalue equation (2.1) to figure out the real characters needed, we get
NR=1+t19-t22+t1t24-t110t22+t110t24+3(t1+t18+t12t24+t19t24)+n1(t1t2+t19t23)+n2(t1t22+t19t22)-z2(t19t2+t1t23)-n3(t12+t17+t13t24+t18t24)+n4(t12t2+t18t23)+n5(t12t22+t18t22)-2z2(t18t2+t12t23)+n6(t13+t16+t14t24+t17t24)+n7(t13t2+t17t23)+n8(t13t22+t17t22)+n9(t17t2+t13t23)+n10(t14+t15+t15t24+t16t24)+n11(t14t2+t16t23)+n12(t14t22+t16t22)+n13(t16t2+t14t23)+n14(t15t2+t15t23)+n15t15t22,
where the coefficients are
n1=z2-z12-z32,n2=2z1z3-4,n3=z22-6,n4=2z1z2z3-3z12+2z2-3z32,n5=z22-z12z2+6z1z3-z2z32-9,n6=z12z2-3z22-2z1z3+z2z32+10,n7=8z1z2z3-5z12+2z2-z23-z13z3-5z32-z1z33,n8=z14-5z12z2+4z22+14z1z3-5z2z32+z34-16,n9=z23-z12-2z2-z32,n10=2z12z2-4z22-2z1z3-z12z32+2z2z32+12,n11=10z1z2z3-6z12+2z2-2z23-z13z3-6z32-z1z33,n12=2z1z22z3+2z14-10z12z2+7z22+22z1z3-z12z32-10z2z32+2z34-22,n13=4z1z2z3-3z12-2z2-z12z22+2z23-3z32-z22z32,n14=8z1z2z3-5z12-z12z22-z13z3-5z32+z12z2z32-z22z32-z1z33,n15=4z1z22z3+3z14-12z12z2+8z22+24z1z3-z13z2z3-12z2z32-z1z2z33+3z34-24.
Step (iv): There only remains to find out if
GR(t1,t2;z1,z2,z3)=NR(t1,t2;z1,z2,z3)DR(t1,t2;z1,z2,z3) (4.2)
solves the differential equation
(ΔtR-Δz1)GR(t1,t2;z1,z2,z3)=0,
where the explicit form of ΔtR is derived from (2.8) in the usual way:
ΔtR=4t12∂t12+2t22∂t22+4t1t2∂t1∂t2+16t1∂t1+10t2∂t2

The result of this checking is positive and we can thus conclude that (4.2) is the generating function we were seeking for.

5. Conclusions and outlook

The technique for computing generating functions for characters of irreducible modules over simple Lie algebras introduced in [5] has by now been used to obtain a variety of results concerning characters and weight multiplicities in the case of rank two algebras in [6] and [7] and to study the case of the rank three algebra 𝔰𝔩≃0(6) in the present paper. These works have made obvious the versatility and usefulness of the method, which enabled us to present a number of results with potential applicability in mathematics and mathematical physics. It seems, however, that if we insist in computing the generating functions in full generality, the algebra considered in this paper is the highest rank one in which the formulas obtained through this approach are kept under a reasonable size

Thus, for instance, we have computed the generating functions of irreducible characters also for the remaining algebras of rank three, 𝔬(7) and 𝔰𝔭 (6), but the results are exceedingly complicated, with respectively 311 and 315 terms in the numerator, and with coefficients that in many cases are long expressions in z-variables. Therefore, for these algebras it is better to limit the treatment to some particular sets of characters, and we present here only a few of the simplest results. In the standard notation in which α₃ is the root of unequal length, the generating function for the characters of the representations R_m₁λ₁ and R_m₃λ₃ of 𝔬 (7) are, respectively,

1+t11+t16-(t1+t15)(z1-1)+(t12+t14)(z2-z1+1)-t13(z32-2z2-2) (5.1)

and

1+t321+t38-(t3+t37)z3+(t32+t36)(z1+z2)-(t33+t35)z1z3+t34(z12+z32-2z2-1) (5.2)

The denominator of the generating function for the representations R_m_{₁λ₁+m₃λ₃} is the product of the denominators of (5.1) and (5.2) and the numerator is

1+t1−t 32−t 13t 32+t 12t 34+t 13t 34+t1(1+t1)t 32z1−t1t3(1+t1t 32)z3.

The analogous of (5.1) and (5.2) for 𝔰𝔓 (6) are, respectively, given by

11+t16-(t1+t15)z1+(t12+t14)(z2+1)-t13(z1+z3) (5.3)

and

1-t34+t3z1-t33z11+t38+(t3+t37)(z1-z3)+(t32+t36)(z22-2z1z3)+(t33+t35)(z12-2z2-1)(z1-z3)+t34C, (5.4)

with C=z12+z14-4z12z2+2z22+2z1z3+z32-2 . In the case of representations R_m_{₁λ₁+m₃λ₃} the denominator is the product of the denominators of (5.3) and (5.4) and the numerator is

1-t13t3-t34+t13t35+t3(1+t12-t13t3-t32+t1t33+t13t33)z1 -t1t3(1+t1t3)(1+t32)z2-t1t32(1+t1t3)z3+t1t32(t1+t3)z12.

Thus, the approach based in Calogero-Sutherland model can used to deal with other higher rank classical Lie algebras, or to the exceptional ones, but for these applications it is convenient to select characters of some special types, like those above, in order to keep the results under a manageable size

Appendix

We give here the form of the numerators N_{n_1,n2,n3} (t_1,t_2,t₃) of the generating functions for weight multiplicities for the cases n₁ + n₂ + n₃ ≤ 2. The cases not explicitly written arise through the change t₁ ↔ t₃ on the appropriate numerator.

N0,0,0=1+2t12t2+t14t22+t1t3+t13t2t3+t1t22t3-2t15t22t3-t13t23t3+t12t32+2t2t32-2t14t2t32-4t12t22t32+t16t22t32-2t14t23t32+t13t33+t1t2t33-t15t2t33-2t13t22t33-t1t23t33+t15t23t33+t13t24t33-2t12t2t34+t22t34-4t14t22t34-2t12t23t34+2t16t23t34+t14t24t34-t13t2t35-2t1t22t35+t15t22t35+t13t23t35+t15t24t35+t12t22t36+2t14t23t36+t16t24t36,

N1,0,0=t1+2t13t2+t1t22+t12t3+2t2t3-t14t2t3-t12t22t3-t14t23t3+t13t32+t1t2t32-t15t2t32-3t13t22t32-t1t23t32-t15t23t32+t33+t22t33-3t14t22t33-2t12t23t33+2t16t23t33+t14t24t33-2t13t2t34-2t1t22t34-t15t22t34+t15t24t34-t14t2t35-2t12t22t35+t16t22t35+t14t23t35+t16t24t35+t13t22t36+2t15t23t36+t13t24t36,

N0,1,0=t12+t2+t14t2+t12t22+t13t3+2t1t2t3-t15t2t3-t13t22t3-t15t23t3+t32+t22t32-4t14t22t32-3t12t23t32+t16t23t32+t1t33-t1t22t33-t15t22t33+t15t24t33+t2t34-3t14t2t34-4t12t22t34+t16t22t34+t16t24t34-t1t2t35-t13t22t35-t1t23t35+2t15t23t35+t13t24t35+t14t22t36+t12t23t36+t16t23t36+t14t24t36,

N2,0,0=t12+2t14t2+t12t22+t23-t14t23+t13t3+2t1t2t3-t15t2t3-t13t22t3-t15t23t3+t14t32+t12t2t32-t16t2t32+3t22t32-6t14t22t32-4t12t23t32+2t16t23t32-t24t32+t14t24t32+t1t33-t1t22t33-t15t22t33+t15t24t33+2t2t34-4t14t2t34-4t12t22t34+t16t22t34-t23t34+t14t23t34+t16t24t34-t1t2t35-t13t22t35-t1t23t35+2t15t23t35+t13t24t35+t36-t14t36-t12t2t36+t16t2t36-2t22t36+3t14t22t36+2t12t23t36+t24t36,

N1,1,0=t13+t1t2+t15t2+t13t22+t1t23-t15t23+t14t3+2t12t2t3-t16t2t3+2t22t3-3t14t22t3-2t12t23t3+t16t23t3+t1t32-3t15t22t32-2t13t23t32-t1t24t32+t15t24t32+t12t33+2t2t33-2t14t2t33-3t12t22t33+t16t22t33+t16t24t33-2t15t2t34-3t13t22t34-2t1t23t34+2t15t23t34+t13t24t34+t35-t14t35-2t12t2t35+t16t2t35-t22t35+t16t23t35+t14t24t35-t1t22t36+2t15t22t36+2t13t23t36+t1t24t36,

N1,0,1=t14+2t12t2+t22+t1t3+t13t2t3+t1t22t3-2t15t22t3-t13t23t3+t12t32+2t2t32-2t14t2t32-3t12t22t32-2t14t23t32-t12t24t32+t16t24t32+t13t33+t1t2t33-t15t2t33-2t13t22t33-t1t23t33+t15t23t33+t13t24t33+t34-t14t34-2t12t2t34-3t14t22t34-2t12t23t34+2t16t23t34+t14t24t34-t13t2t35-2t1t22t35+t15t22t35+t13t23t35+t15t24t35+t16t22t36+2t14t23t36+t12t24t36,

N0,2,0=t14+t12t2+t16t2+t22+t12t23-t16t23+t15t3+2t13t2t3-t17t2t3+2t1t22t3-3t15t22t3-2t13t23t3+t17t23t3+t12t32+t2t32-t14t2t32-t12t22t32-2t16t22t32+t23t32-3t14t23t32-3t12t24t32+3t16t24t32+t13t33+2t1t2t33-2t15t2t33-3t13t22t33+t17t22t33-2t1t23t33+2t15t23t33+2t13t24t33-t13t33+t34-t14t34-t12t2t34-t16t2t34-3t14t22t34-3t12t23t34+3t16t23t34+t14t24t34+t1t35-t15t35-2t13t2t35+t17t2t35-3t1t22t35+2t15t22t35+2t13t23t35-t17t23t35+t15t24t35+t2t36-t14t2t36-2t12t22t36+3t16t22t36-t23t36+3t14t23t36+3t12t24t36-2t16t24t36-t1t2t37+t15t2t37+t13t22t37-t17t22t37+t1t23t37-t15t23t37-t13t24t37+t17t24t37,

Footnotes

a

To compute the characters, or to check other results of the paper, see the Mathematica notebooks attached as ancillary files to the preprint arXiv: 1705.03711.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 4
Pages: 618 - 632
Publication Date: 2021/01/06
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TY  - JOUR
AU  - José Ferńandez Núñez
AU  - Wifredo García Fuertes
AU  - Askold M. Perelomov
PY  - 2021
DA  - 2021/01/06
TI  - Generating functions for characters and weight multiplicities of irreducible 𝓈𝓁(4)-modules
JO  - Journal of Nonlinear Mathematical Physics
SP  - 618
EP  - 632
VL  - 25
IS  - 4
SN  - 1776-0852
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