September 28, 2025Open Access

Consecutive Patterns, Kostant's Problem and Type A₆

Key Points

Kostant negativity of L(w) is proven for certain permutation patterns.
A complete solution to Kostant's problem is achieved specifically for type A6.
Indecomposability conjecture is confirmed in type A6, connecting projective functors to module structure.
The work builds on the relationship between permutations and simple highest weight modules in the BGG category.

Abstract

For a permutation w in the symmetric group S₍, let L (w) denote the simple highest weight module in the principal block of the BGG category O for the Lie algebra sl₍ (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.

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Cite This Study

Mazorchuk et al. (Mon,) studied this question.

synapsesocial.com/papers/68d90a0f41e1c178a14f6a2b https://doi.org/https://doi.org/10.48550/arxiv.2503.07809