Consecutive Patterns, Kostant's Problem and Type $A_6$

Abstract

For a permutation $w$ in the symmetric group ${\mathfrak{S}}_n$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra ${\mathfrak{sl}}_n(\mathbb{C})$. We first prove that $L(w)$ is Kostant negative whenever $w$ consecutively contains certain patterns. We then provide a complete answer to Kostant's problem in type $A_6$ and show that the indecomposability conjecture also holds in type $A_6$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.

revised version accepted for publication in Int. J. Alg. Comput