Generating functions for characters and weight multiplicities of irreducible 𝓈𝓁(4)-modules
- 10.1080/14029251.2018.1503436How to use a DOI?
- Calogero-Sutherland models; Generating functions; Lie algebra representation theory
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 (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.
- © 2018 The Authors. Published by Atlantis and Taylor & Francis
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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 , and since then different procedures for computing these objects and extracting practical information from them have been developed, see for instance  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  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 (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  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 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) , 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 ; 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 ; the SU (4)-Kondo effect, due to the interplay between spin and orbital electronic degrees of freedom, which has recently aroused remarkable interest  because of its role in condensed matter settings such as quantum dots, carbon nanotubes or nanowires; or the AdS5/CFT4 string-gauge equivalence , 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 (4) by means of the approach developed in . 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  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  arrived to similar results for the quantum problem on the circle, this time with trigonometric interaction; and later Moser  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 , 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 
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  for a more detailed treatment. Let 𝒜 be a complex simple Lie algebra of rank r. As is well known , the roots α1, α2, …, αr and fundamental weights λ1, λ2,…,λr 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
The most relevant fact for us is that if we tune all coupling constants κα to one, the eigenfunctions of this Schrödinger operator Δκ are precisely the characters χm of the irreducible representations of the algebra 
Thus, we can obtain the characters by solving the second order differential equation
Furthermore, if we change variables and describe the dynamical system by means of the characters zk = 𝒳λk of the fundamental representations Rλk, k = 1,2,…,r, the differential operator Δ1 takes the form
Hence, knowing all the quadratic Clebsch-Gordan series of the algebra we will be able to determine the ajk(z) coefficients. For more explicit details, see for instance  and references therein. Once we know the definite expression for the operator it is possible to compute the characters χm 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 (4)
For the particular case of (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 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 Rλj of (4), are related to the q-variables by
In this setup, as explained in , the generating function for characters
Step (i): The denominator of the generating function is(2.12)where with the product extended to all the weights entering in the Weyl orbit of the fundamental representation Rλi. Looking at (2.6) and bearing in mind that the fundamental representations of (4) contain one single Weyl orbit, the result follows. After changing variables back from the xj to the zk, it can be written as(2.13)
Step (ii): The Weyl formula for dimensions gives for the representation Rλ, where ,(2.14)
To obtain the generating function for dimensions E(t1, t2, t3) it suffices to perform the change in this formula and to apply the resulting differential operator to . One finds in this way that(2.15)where
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 (4)-modules with small values of mi (in fact, mi ≤ 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 2.7, they can be computed by recursively solving the eigenvalue equation (2.1)a. Thus we obtain(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 (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
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 , 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 is to obtain closed formulas for the multiplicities μm1, m2, m3(n1, n2, n3) by proceeding as done for rank two algebras in  (for an alternative approach, applied also to these algebras, see ). Nevertheless, in the case of (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 , or  for a recent computation of μm1, 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 , see  for a pedagogic exposition,
Thus, 𝒵[k1,k2,k3] is symmetric under interchange of k1 and k3, and its expression for k1 ≤ k3 can eventually found to be
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:
• μm1, m2, m3(0,0,0) ≠ 0 only if m3 − m1 = 2m2 + 4p with p integer, and in this case
The derivation of these expressions from the Kostant multiplicity formula is a laborious process: the Weyl group of (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 μm1, 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 (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 χm1, 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,
Step (i): Assuming that the generating function GR is rational, the denominator is now(4.1)where the weights entering in D13 are those in the Weyl orbit Rλ1 + λ3. These can be read from the corresponding monomial symmetric function
Step (ii): For real representations, the dimensions (2.14) are
Given this formula, we can proceed as in Section 2 to shape the generating function for dimensions. It turns out to be
Step (iii): We next compute the numerator NR(t1, t2; z1, z2, z3) of GR by provisionally assuming that the only non-vanishing coefficients correspond to the monomials appearing in the numerator of ER(t1, t2). After using the eigenvalue equation (2.1) to figure out the real characters needed, we get
Step (iv): There only remains to find out if(4.2)solves the differential equation
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  has by now been used to obtain a variety of results concerning characters and weight multiplicities in the case of rank two algebras in  and  and to study the case of the rank three algebra 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, and (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 Rm1λ1 and Rm3λ3 of (7) are, respectively,
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
We give here the form of the numerators Nn1,n2,n3 (t1,t2,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.
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.
Cite this article
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 UR - https://doi.org/10.1080/14029251.2018.1503436 DO - 10.1080/14029251.2018.1503436 ID - FerńandezNúñez2021 ER -