Categorification of Wedderburn’s basis for ${\mathbb{C}}[S_{n}]$

Abstract.

M. Neunhöffer studies in [21] a certain basis of ${\mathbb{C}}[S_{n}]$ with the origins in [14] and shows that this basis is in fact Wedderburn’s basis, hence decomposes the right regular representation of S _n 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 ${\mathcal{O}}$. Inside each of these categories, there is a dominant projective module which plays a crucial role in our arguments and will additionally be used to show that Kostant’s problem ([10]) has a negative answer for some simple highest weight module over the Lie algebra ${\mathfrak{sl}}_{4}$. This disproves the general belief that Kostant’s problem should have a positive answer for all simple highest weight modules in type A.