PBW filtration and bases for irreducible modules in type An

We study the PBW filtration on the highest weight representations V(λ) of $$ \mathfrak{s}{\mathfrak{l}_{n + 1}} $$. This filtration is induced by the standard degree filtration on $$ {\text{U}}\left( {{\mathfrak{n}^{-} }} \right) $$. We give a description of the associated graded $$ S\left( {{\mathfrak{n}^{-} }} \right) $$-module gr V(λ) in terms of generators and relations. We also construct a basis of gr V(λ). As an application we derive a graded combinatorial character formula for V(λ), and we obtain a new class of bases of the modules V(λ) conjectured by Vinberg in 2005.