String Functions for Affine Lie Algebras Integrable Modules

The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a module $L_{\mathfrak g\downarrow\mathfrak h}^\mu$ reduced to a Cartan subalgebra $\mathfrak h$ are transformed in order to place the recursion shifts $\gamma\in\Gamma _{\mathfrak a\subset\mathfrak h}$ into the fundamental Weyl chamber. The new ensembles $F\Psi$ (the “folded fans”) of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were formulated. Being considered simultaneously for the set of string functions (corresponding to the same congruence class $\Xi_{v}$ of modules) the system of recursion relations constitute an equation $\mathbf M_{(u)}^{\Xi _v}\mathbf{m}_{(u)}^{\mu}={\boldsymbol\delta}_{(u)}^{\mu}$ where the operator $\mathbf M_{(u)}^{\Xi _v}$ is an invertible matrix whose elements are defined by the coordinates and multiplicities of the shift weights in the folded fans $F\Psi$ and the components of the vector $\mathbf m_{(u)}^\mu$ are the string function coefficients for $L^\mu$ enlisted up to an arbitrary fixed grade $u$. The examples are presented where the string functions for modules of $\mathfrak g=A_2^{(1)}$ are explicitly constructed demonstrating that the set of folded fans provides a compact and effective tool to study the integrable highest weight modules.