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

TL;DR

This paper categorifies Wedderburn's basis for the group algebra of the symmetric group, revealing new insights into its structure and disproving a long-standing conjecture about Kostant's problem in type A Lie algebras.

Contribution

It provides a categorical construction of Wedderburn's basis using projective-injective modules in category , connecting representation theory of symmetric groups with Lie algebra modules.

Findings

Categorification of Wedderburn's basis via category  modules.

Disproof of Kostant's problem for some simple modules in .

New link between symmetric group representations and Lie algebra modules.

Abstract

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 $A$.