Consecutive patterns, Kostant's Problem and type A ₆

For a permutation [Formula: see text] in the symmetric group [Formula: see text], let [Formula: see text] denote the simple highest weight module in the principal block of the BGG category [Formula: see text] for the Lie algebra [Formula: see text]. We first prove that [Formula: see text] is Kostant negative whenever [Formula: see text] consecutively contains certain patterns. We then provide a complete answer to Kostant’s problem in type [Formula: see text] and show that the indecomposability conjecture also holds in type [Formula: see text], that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.