---
abstract: 'Let ${\tilde{\mathfrak{g}}}$ be an affine Lie algebra of type $D_{\ell}^{(1)}$ and $L(\Lambda)$ its standard module with a highest weight vector $v_{\Lambda}$. For a given ${\mathbb{Z}}$-gradation ${\tilde{\mathfrak{g}}}= {\tilde{\mathfrak{g}}}_{-1} + {\tilde{\mathfrak{g}}}_0 + {\tilde{\mathfrak{g}}}_1$, we define Feigin-Stoyanovsky’s type subspace as $$W(\Lambda) = U({\tilde{\mathfrak{g}}}_1) \cdot v_{\Lambda}.$$ By using vertex operator relations for standard modules we reduce the Ponicaré-Brikhoff-Witt spanning set of $W(\Lambda)$ to a basis and prove its linear independence by using Dong-Lepowsky intertwining operators.'
address: 'Faculty of Chemical Engineering and Technology, University of Zagreb, Marulićev trg 19, Zagreb, Croatia'
author:
- Ivana Baranović
title: 'Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level 2 standard modules for $D_4^{(1)}$'
---

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Introduction
============

J. Lepowsky and R.L. Wilson initiated a program whose goal was to study representations of affine Lie algebras by means of vertex operators and to find Rogers-Ramanujan type combinatorial bases of these representations ([@LW1], [@LW2]). Using this approach, Lepowsky and Primc obtained in [@LP] new character formulas for standard $\mathfrak{sl}(2, {\mathbb{C}})^{\widetilde{}}$-modules. In [@FS] Feigin and Stoyanovsky gave another proof and combinatorial interpretation of these formulas. One of the new ingredients introduced in the paper were so-called principal subspaces. These subspaces are generated by the affinization of the nilpotent subalgebra ${\mathfrak{n}}_+$ 