Consecutive Patterns, Kostant's Problem and Type A₆

For a permutation w in the symmetric group Sₙ, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category O for the Lie algebra slₙ(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 A₆ and show that the indecomposability conjecture also holds in type A₆, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.