Integrable Highest Weight Representations of Affine Lie algebras

In this section \(\mathfrak{g}\) can be an arbitrary Kac-Moody Lie Algebra made with a symmetrizable, indecomposable Cartan matrix.

Suppose that \(V\) is a representation with a weight space decomposition as in Roots and Weights. Let \(\alpha\) be a real root, and let \(\mathfrak{g}_\alpha\) be the corresponding root space, that is, the one-dimensional weight space for \(\alpha\) in the adjoint representation of \(\mathfrak{g}\) on itself. Then \(-\alpha\) is also a root. The two one-dimensional spaces \(\mathfrak{g}_\alpha\) and \(\mathfrak{g}_{-\alpha}\) generate a Lie algebra isomorphic to \(\mathfrak{sl}_2\). The module \(V\) is called integrable if for each such \(\alpha\) the representation of \(\mathfrak{sl}_2\) obtained this way integrates to a representation of the Lie group \(\operatorname{SL}_2\). Since this group contains an element that stabilizes \(\mathfrak{h}\) and induces the corresponding simple reflection on the weight lattice, integrability implies that the weight multiplicities are invariant under the action of the Weyl group.

If the Kac-Moody Lie algebra \(\mathfrak{g}\) is finite-dimensional then the integrable highest weight representations are just the irreducible finite-dimensional ones. For a general Kac-Moody Lie algebra the integrable highest weight representations are the analogs of the finite-dimensional ones, that is, with WeylCharacterRing elements. Their theory has many aspects in common with the finite-dimensional representations of finite-dimensional simple Lie algebras, such as the parametrization by dominant weights, and generalizations of the Weyl denominator and character formulas, due to Macdonald and Kac respectively.

If \(\Lambda\) is a dominant weight, then the irreducible highest weight module \(L(\Lambda)\) defined in Roots and Weights is integrable. Moreover every highest weight integrable representation arises this way, so these representations are in bijection with the cone \(P^+\) of dominant weights.

The affine case

Now we assume that \(\mathfrak{g}\) is affine. The integrable highest weight representations (and their crystals) are extremely interesting. Integrable highest weight representations of \(\mathfrak{g}\) arise in a variety of contexts, from string theory to the modular representation theory of the symmetric groups, and the theory of modular forms. The representation \(L(\Lambda_0)\) is particularly ubiquitous and is called the basic representation.

Therefore in [KMPS] (published in 1990) tabulated data for many of these representations. They wrote

We present a vast quantity of numerical data in tabular form, this being the only source for such information. The computations are tedious and not particularly straightforward when it is necessary to carry them out individually. We hope the appearance of this book will spur interest in a field that has become, in barely 20 years, deeply rewarding and full of promise for the future. It would indeed be gratifying if these tables were to appear to the scientists of 2040 as obsolete as the dust-gathering compilations of transcendental functions appear for us today because of their availability on every pocket calculator.

As we will explain, Sage can reproduce the contents of these tables. Moreover the tables in [KMPS] are limited to the untwisted types, but Sage also implements the twisted types.

Although Sage can reproduce the tables in the second volume of [KMPS], the work remains very useful. The first volume is a down-to-earth and very helpful exposition of the theory of integrable representations of affine Lie algebras with explicit examples and explanations of the connections with mathematical physics and vertex operators.

The support of an integrable highest weight representation

Let \(\Lambda \in P^+\) and let \(V = L(\Lambda)\) be the integrable representation with highest weight \(\Lambda\). If \(\mu\) is another weight, let \(\operatorname{mult}(\mu)\) denote the multiplicity of the weight \(\mu\) in \(L(\Lambda)\). Define the support of the representation \(\operatorname{supp}(V)\) to be the set of \(\mu\) such that \(\operatorname{mult}(\mu) > 0\).

If \(\operatorname{mult}(\mu) > 0\) then \(\lambda-\mu\) is a linear combination of the simple roots with nonnegative integer coefficients. Moreover \(\operatorname{supp}(V)\) is contained in the paraboloid

\[(\Lambda+\rho | \Lambda+\rho) - (\mu+\rho | \mu+\rho) \geq 0\]

where \((\, | \,)\) is the invariant inner product on the weight lattice and \(\rho\) is the Weyl vector (Untwisted Affine Kac-Moody Lie Algebras). Moreover if \(\mu \in \operatorname{supp}(V)\) then \(\Lambda - \mu\) is an element of the root lattice \(Q\) ([Kac], Propositions 11.3 and 11.4).

We organize the weight multiplicities into sequences called string functions or strings as follows. By [Kac], Proposition 11.3 or Corollary 11.9, for fixed \(\mu\) the function \(\operatorname{mult}(\mu - k\delta)\) of \(k\) is an increasing sequence. We adjust \(\mu\) by a multiple of \(\delta\) to the beginning of the positive part of the sequence. Thus we define \(\mu\) to be maximal if \(\operatorname{mult}(\mu) \neq 0\) but \(\operatorname{mult}(\mu + \delta) = 0\).

Since \(\delta\) is fixed under the action of the affine Weyl group, and since the weight multiplicities are Weyl group invariant, the function \(k \mapsto \operatorname{mult}(\mu - k \delta)\) is unchanged if \(\mu\) is replaced by \(w(\mu)\) for some Weyl group element \(w\). Now every Weyl orbit contains a dominant weight. Therefore in enumerating the string we may assume that the weight \(\mu\) is dominant. There are only a finite number of dominant maximal weights. Thus there are only a finite number of such strings to be computed.

Modular Forms

Remarkably, [KacPeterson] showed that each string is the set of Fourier coefficients of a weakly holomorphic modular form; see also [Kac] Chapters 12 and 13. Here weakly holomorphic modular means that the form is allowed to have poles at cusps.

To this end we define the modular characteristic

\[m_\Lambda = \frac{|\Lambda+\rho|^2}{2(k+h^\vee)} - \frac{|\rho|^2}{2h^\vee}.\]

Here \(k = (\Lambda | \delta)\) is the level of the representation and \(h^\vee\) is the dual Coxeter number (Labels and Coxeter Number). If \(\mu\) is a weight, define

\[m_{\Lambda,\mu} = m_\Lambda - \frac{|\mu|^2}{2k}.\]

Let \(\Lambda\) be a weight, which we may assume maximal. Then Kac and Peterson defined the string function

\[c_\mu^\Lambda = q^{m_{\Lambda,\mu}} \sum_{n\in\ZZ} \operatorname{mult}(\mu - n\delta) q^n.\]

Although these do arise as partition functions in string theory, the term “string” here does not refer to physical strings.

The string function \(c_\mu^\Lambda\) is a weakly holomorphic modular form, possibly of half-integral weight. See [Kac], Corollary 13.10, or [KacPeterson]. It can have poles at infinity, but multiplying \(c_\mu^\Lambda\) by \(\eta(\tau)^{\dim\,\mathfrak{g}^\circ}\) gives a holomorphic modular form (for some level). Here \(\eta\) is the Dedekind eta function:

\[\eta(\tau) = q^{1/24} \prod_{k=1}^\infty(1-q^k), \qquad q = e^{2\pi i \tau}.\]

The weight of this modular form \(\eta(\tau)^{\dim\,\mathfrak{g}^\circ} c^\Lambda_\lambda\) is the number of positive roots of \(\mathfrak{g}^\circ\).

Sage methods for integrable representations

In this section we will show how to use Sage to compute with integrable highest weight representations of affine Lie algebras. For further documentation, see the reference manual IntegrableRepresentation.

In the following example, we work with the integrable representation with highest weight \(2 \Lambda_0\) for \(\widehat{\mathfrak{sl}}_2\), that is, \(A_1^{(1)}\). First we create a dominant weight in the extended weight lattice, then create the IntegrableRepresentation class. We compute the strings. There are two, since there are two dominant maximal weights. One of them is the highest weight \(2\Lambda_0\), and the other is \(2\Lambda_1 - \delta\):

sage: L = RootSystem("A1~").weight_lattice(extended=True)
sage: Lambda = L.fundamental_weights()
sage: delta = L.null_root()
sage: W = L.weyl_group(prefix="s")
sage: s0, s1 = W.simple_reflections()
sage: V = IntegrableRepresentation(2*Lambda[0])
sage: V.strings()
{2*Lambda[0]: [1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236],
 2*Lambda[1] - delta: [1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196, 287]}
sage: mw1, mw2 = V.dominant_maximal_weights(); mw1, mw2
(2*Lambda[0], 2*Lambda[1] - delta)

We see there are two dominant maximal weights, \(2 \Lambda_0\) and \(2 \Lambda_1 - \delta\). We obtain every maximal weight from these by applying Weyl group elements. These lie inside the paraboloid described in The support of an integrable highest weight representation. Here are a few more maximal weights:

sage: pairs = [(s0*s1*s0, mw1), (s0*s1, mw2), (s0, mw1), (W.one(), mw2),
....:          (W.one(), mw1), (s1, mw2), (s1*s0, mw1), (s1*s0*s1, mw2)]
sage: [w.action(mw) for (w, mw) in pairs]
[-6*Lambda[0] + 8*Lambda[1] - 8*delta,
 -4*Lambda[0] + 6*Lambda[1] - 5*delta,
 -2*Lambda[0] + 4*Lambda[1] - 2*delta,
 2*Lambda[1] - delta,
 2*Lambda[0],
 4*Lambda[0] - 2*Lambda[1] - delta,
 6*Lambda[0] - 4*Lambda[1] - 2*delta,
 8*Lambda[0] - 6*Lambda[1] - 5*delta]

We confirm that the string function for one in the Weyl orbit is the same as that for mw2, calculated above:

sage: s1.action(mw2)
4*Lambda[0] - 2*Lambda[1] - delta
sage: [V.mult(s0.action(mw2)-k*delta) for k in [0..10]]
[1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196]

String functions of integrable representations often appear in the Online Encyclopedia of Integer Sequences:

sage: [oeis(x) for x in V.strings().values()]    # optional - internet
[0: A233758: Bisection of A006950 (the even part).,
 0: A233759: Bisection of A006950 (the odd part).]

Reading what the OEIS tells us about the sequence OEIS sequence A006950, we learn that the two strings are the odd and even parts of the series

\[\prod_{k=1}^\infty \frac{1+q^{2k-1}}{1-q^{2k}} = \prod_{k=1}^\infty \frac{1-q^{2k}}{(1-q^k)(1-q^{4k})} = q^{1/8} \frac{\eta(2\tau)}{\eta(\tau)\eta(4\tau)}\]

This is not a modular form because of the factor \(q^{1/8}\) in front of the ratio of eta functions.

Let us confirm what the Online Encyclopedia tells us by computing the above product:

sage: PS.<q> = PowerSeriesRing(QQ)
sage: prod([(1+q^(2*k-1))/(1-q^(2*k)) for k in [1..20]])
1 + q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 5*q^6 + 7*q^7 + 10*q^8
 + 13*q^9 + 16*q^10 + 21*q^11 + 28*q^12 + 35*q^13 + 43*q^14
 + 55*q^15 + 70*q^16 + 86*q^17 + 105*q^18 + 130*q^19 + O(q^20)

We see the values of the two strings interspersed in this product, with the \(2 \Lambda_0\) string values in the even positions and the \(2 \Lambda_1 - \delta\) values in the odd positions.

To compute \(c^{2\Lambda_0}_\lambda\), which is guaranteed to be a modular form, we must compute the modular characteristics. We are interested in the cases where \(\lambda\) is one of the two dominant maximal weights:

sage: [V.modular_characteristic(x) for x in [2*Lambda[0], 2*Lambda[1]-delta]]
[-1/16, 7/16]

This gives us the string functions

\[\begin{split}\begin{aligned} c^{2\Lambda_0}_{2\Lambda_0} & = q^{-1/16}(1+q+3q^2+5q^3+10q^4+16q^5+\cdots),\\ c^{2\Lambda_0}_{2\Lambda_1-\delta} & = q^{7/16}(1+2q+4q^2+7q^3+13q^4+21q^5+\cdots). \end{aligned}\end{split}\]

These are both weakly holomorphic modular forms. Any linear combination of these two is also a weakly holomorphic modular form. For example we may replace \(\tau\) by \(\tau/2\) in our previous identity and get

\[c^{2\Lambda_0}_{2\Lambda_0} + c^{2\Lambda_0}_{2\Lambda_1-\delta} = \frac{\eta(\tau)}{\eta(\tau/2)\eta(2\tau)}.\]

Many more examples may be found in [KacPeterson] and [KMPS].

Let \(V\) be the integrable highest weight representation with highest weight \(\Lambda\). If \(\mu\) is in the support of \(V\) then \(\Lambda - \mu\) is of the form \(\sum_i n_i\alpha_i\) where \(\alpha_i\) are the simple roots. Sage employs an internal representation of the weights as tuples \((n_0, n_1, \ldots)\). You can convert weights to and from this notation as follows:

sage: L = RootSystem(['E',6,2]).weight_lattice(extended=True)
sage: Lambda = L.fundamental_weights()
sage: delta = L.null_root()
sage: V = IntegrableRepresentation(Lambda[0])
sage: V.strings()
{Lambda[0]: [1, 2, 7, 14, 35, 66, 140, 252, 485, 840, 1512, 2534]}
sage: V.to_weight((1,2,0,1,0))
Lambda[0] - 3*Lambda[1] + 4*Lambda[2] - 2*Lambda[3] + Lambda[4] - delta
sage: V.from_weight(Lambda[0] - 3*Lambda[1] + 4*Lambda[2] - 2*Lambda[3] + Lambda[4] - delta)
(1, 2, 0, 1, 0)
sage: V.from_weight(Lambda[0]-delta)
(1, 2, 3, 2, 1)

In reporting the strings, one may set the optional parameter depth to get more or fewer values. In certain cases even the first coefficient of the string is significant. See [JayneMisra2014] and [KimLeeOh2017].

Catalan numbers (OEIS sequence A000108):

sage: P = RootSystem(['A',12,1]).weight_lattice(extended=true)
sage: Lambda = P.fundamental_weights()
sage: IntegrableRepresentation(2*Lambda[0]).strings(depth=1) # long time
{2*Lambda[0]: [1],
 Lambda[1] + Lambda[12] - delta: [1],
 Lambda[2] + Lambda[11] - 2*delta: [2],
 Lambda[3] + Lambda[10] - 3*delta: [5],
 Lambda[4] + Lambda[9] - 4*delta: [14],
 Lambda[5] + Lambda[8] - 5*delta: [42],
 Lambda[6] + Lambda[7] - 6*delta: [132]}

Catalan triangle numbers (OEIS sequence A000245):

sage: sorted(IntegrableRepresentation(Lambda[0]+Lambda[2]).strings(depth=1).values()) # long time
[[1], [3], [9], [12], [28], [90], [297]]

Central binomial coefficients (OEIS sequence A001700, OEIS sequence A128015):

sage: P = RootSystem(['B',8,1]).weight_lattice(extended=true)
sage: Lambda = P.fundamental_weights()
sage: sorted(IntegrableRepresentation(Lambda[0]+Lambda[1]).strings(depth=1).values()) # long time
[[1], [1], [1], [3], [3], [10], [10], [35], [35], [126]]