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

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

2007-08-29

arxiv.org/abs/0708.3949v3

Arch. Math. (Basel) 91 (2008), no. 1, 1--11.

Mathematics

Representation Theory

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

Scientific paper

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 $A$.

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