What is known about string functions for $\widehat{\mathfrak{sl}_2}$?

Question

Let $\Lambda$ be a dominant integral weight for $\widehat{\mathfrak{sl}_2}$. The string function associated to a maximal weight $\lambda$ of $L(\Lambda)$ is the series $$ a^{\Lambda}_\lambda = \sum_{k \geq 0} \operatorname{dim}(L(\Lambda)_{\lambda - k \delta}) q^k. $$ Computing the character for $L(\Lambda)$ is equivalent to computing all it's string functions for representatives $\lambda \in \max(\Lambda)/W$.

In Kac's book Infinite Dimensional Lie Algebras the string function $a^{\Lambda_0}_{\Lambda_0}$ is computed as $$ a^{\Lambda_0}_{\Lambda_0} = \dfrac{1}{(q;q)_\infty} .$$ In Wakimoto's book Lectures on Infinite-dimensional Lie algebras the string functions for $\Lambda = 2\Lambda_0$ is computed (Example 2.1.2 II).

1. Since both Kac's book and Wakimoto's book are quite old, I would like to know what is known about other string functions for $\widehat{\mathfrak{sl_2}}$ beyond these two special cases. Is there a known formula for level $\ell$ string functions for $\widehat{\mathfrak{sl_2}}$?

2. Can one describe the set $\max(\Lambda)/W$?

I apologize in advance if this is well-known, I am a novice at Kac-Moody algebras.

Answer 1

A standard reference is the book Structure of the Standard Modules for the Affine Lie Algebra $A_1^{(1)}$ by J. Lepowski and M. Primc, 1985. I do not have access to it a the moment, and I do not remember if there are explicit expressions for the string functions $a^\Lambda_\lambda$.

I first learned about string functions when studying Quasi-particles models for the representations of Lie algebras and geometry of flag manifold by B. Feigin and A. V. Stoyanovsky, 1993, which, for one thing, relates principle subspaces and modules and it contains explicit expressions.

Based on the latter, we wrote a more physics style interpretation, arXiv:cond-mat/0409369. The generalisation to the $\widehat{\mathfrak sl}_n$ case was published in Comm. Math. Phys., arXiv:math/0504364.

Edit to clarify and answer part 2. of the question.

The paper by Feigin and Stoyanovsky (FS) calculates the character of module $V_{l,k}$, formula (2.6.2') on page 17 (that is for $\Lambda = l \omega_1$ at level $k$). This character is not written as a sum over string functions. This we do in our paper arXiv:cond-mat/0409369, starting from (46) (i.e., (2.6.2') in FS), resulting in (49) after some changes in summation variables; (49) is an explicit sum over string functions.

To answer part 2. of the question, it is easiest to use the connection between the string functions $a^\Lambda_\lambda$ and the $Z_k$ parafermion CFTs, as explained in Modular invariant partition functions for parafermionic field theories by D. Gepner and Z. Qiu. They denote the string functions as $c^l_m$. One has $0 \leq l \leq k$, and the string functions satisfy $c^l_m = c^l_{m+2k} = c^{k-l}_{m+k} = c^l_{-m}$. Given these relations, one finds the minimal set of string functions necessary to obtain the characters for all $\Lambda$ at a given level.