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

Let $\tilde{\mathfrak{g}}$ be an affine Lie algebra of type $A_{\mathrm{\ell }}^{(1)}$. 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 $$\tilde{\mathfrak{g}}={\tilde{\mathfrak{g}}}_{-1}\oplus {\tilde{\mathfrak{g}}}_0\oplus {\tilde{\mathfrak{g}}}_1\mathrm{.}$$ Let $L(\mathrm{\Lambda })$ be a standard module of level 1. Feigin-Stoyanovsky’s type subspace $W(\mathrm{\Lambda })$ is the ${\tilde{\mathfrak{g}}}_1$-submodule of $L(\mathrm{\Lambda })$ generated by the highest-weight vector $v_{\mathrm{\Lambda }}$, $$W(\mathrm{\Lambda })=U({\tilde{\mathfrak{g}}}_1)\cdot v_{\mathrm{\Lambda }}\subset L(\mathrm{\Lambda })\mathrm{.}$$ We find a combinatorial basis of $W(\mathrm{\Lambda })$ 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 $L(\mathrm{\Lambda })$ is obtained as an ``inductive limit’’ of the basis of $W(\mathrm{\Lambda })$.