What question did this study set out to answer?

This research aims to address Kostant's problem in type A6 and explore the properties of highest weight modules.

December 21, 2025

Consecutive patterns, Kostant's Problem and type A 6

Key Points

This research aims to address Kostant's problem in type A6 and explore the properties of highest weight modules.
Proved Kostant negativity with respect to consecutive patterns.
Analyzed simple highest weight modules within the BGG category for type A6.
Examined the implications of applying indecomposable projective functors.
Established that certain patterns lead to Kostant negativity.
Provided a complete solution to Kostant's problem in type A6.
Confirmed the indecomposability conjecture holds in type A6.

Abstract

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.

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