Categorification of Wedderburn's basis for \mathbb{C}[S_n]

Abstract

M. Neunh{ö}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 $A$.

11 pages, some corrections, to appear in Arch. Math

References in corpus (1)

• Categorification of (induced) cell modules and the rough structure of generalized Verma modules

Cited by in corpus (1)

• A new approach to Kostant's problem