Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\tilde{\mathfrak sl}(\ell+1,\C)$

Abstract

Let $\tilde{\mathfrak g}$ be an affine Lie algebra of type $A_\ell^{(1)}$. Suppose we're given a $\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus {\mathfrak g}_1$; then we also have the induced $\mathbb Z$-gradation of the affine Lie algebra $$\tilde{\mathfrak g}=\tilde{\mathfrak g}_{-1} \oplus \tilde{\mathfrak g}_0 \oplus \tilde{\mathfrak g}_1.$$ Let $L(\Lambda)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(\Lambda)$ is the $\tilde{\mathfrak g}_1$-submodule of $L(\Lambda)$ generated by the highest-weight vector $v_\Lambda$, $$W(\Lambda)=U(\tilde{\mathfrak g}_1)\cdot v_\Lambda\subset L(\Lambda).$$ We find a combinatorial basis of $W(\Lambda)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(\Lambda)$ is obtained as an ``inductive limit'' of the basis of $W(\Lambda)$.