  M. Neunh{\"o}ffer studies in \cite{Ne} a certain basis of $\mathbb{C}[S_n]$
with the origins in \cite{Lu} and shows that this basis is in fact Wedderburn's
basis. In particular, in this basis the right regular representation of $S_n$
decomposes into a direct sum of irreducible representations (i.e. Specht or
cell modules). In the present paper we rediscover essentially the same basis
with a categorical origin coming from projective-injective modules in certain
subcategories of the BGG-category $\mathcal{O}$. An important role in our
arguments is played by the dominant projective module in each of these
categories. As a biproduct of the study of this dominant projective module we
show that {\it Kostant's problem} (\cite{Jo}) has a negative answer for some
simple highest weight module over the Lie algebra $\mathfrak{sl}_4$, which
disproves the general belief that Kostant's problem should have a positive
answer for all simple highest weight modules in type $A$.
