of ${\mathfrak{g}}$ from the triangular decomposition ${\mathfrak{g}}= {\mathfrak{n}}_{-} \oplus {\mathfrak{h}}\oplus {\mathfrak{n}}_+$. Using the intertwining operators, G. Georgiev ([@G]) proved the linear independence of quasi-particle bases of Feigin-Stoyanovsky level $k$ principal subspaces for $\mathfrak{sl}(\ell + 1, {\mathbb{C}})$.

Similar approach was taken by S. Caparelli, Lepowsky and A. Milas in [@CLM1] and [@CLM2]. They obtained Rogers-Selberg recursions for characters of level $k$ principal subspaces for $\mathfrak{sl}(2, {\mathbb{C}})$ using the so-called Dong-Lepowsky intertwining operators ([@DL]). As a continuation of this work, C. Calinescu obtained reccurence system for the characters of principal subspaces of level 1 standard modules for $\mathfrak{sl}(\ell + 1, {\mathbb{C}})^{\widetilde{}}$ ([@C1]) and of certain higher-level standard modules for $\mathfrak{sl}(3, {\mathbb{C}})^{\widetilde{}}$ ([@C2]).

In [@P1] M. Primc studied an object similar to the principal supbspace, the so-called Feigin-Stoyanovsky’s type subspace $W(\Lambda)$ for ${\mathfrak{g}}= \mathfrak{sl}(\ell + 1, {\mathbb{C}})$. He constructed a combinatorial base for this space using Schur functions. A basis of the whole $L(\Lambda)$ could then be obtained as an inductive limit of translations of the basis of $W(\Lambda)$ by a certain Weyl group element. This was done for a particular choice of gradation of $L(\Lambda)$ and for any dominant $\Lambda$. It remained unclear how to extend this approach to other algebras. In 