Kostant's problem for permutations of shape $(n-2,1,1)$ and $(n-3,2,1)$

TL;DR

This paper provides a combinatorial solution to Kostant's problem for certain permutations in the symmetric group, specifically those with shapes (n-2,1,1) and (n-3,2,1), within the context of Lie algebra representations.

Contribution

It offers a new combinatorial approach to Kostant's problem for specific permutation shapes and verifies related conjectures in these cases.

Findings

Kostant's problem is solved for permutations with shapes (n-2,1,1) and (n-3,2,1).

Certain conjectures, including the Indecomposability Conjecture, hold for these permutations.

The results connect combinatorial data with representation-theoretic properties.

Abstract

For a permutation $z$ in the symmetric group $\mathrm{S}_{n}$, denote by $L_{z}$ the corresponding simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. In this paper, we provide a combinatorial answer to Kostant's problem for the modules $L_{z}$ when $z$ has shape (associated Young diagram/integer partition via Robinson-Schensted correspondence) equal to $(n-2,1,1)$ or $(n-3,2,1)$. Moreover, we verify that certain closely related conjectures hold for such permutations, including the Indecomposability Conjecture, which states that applying any indecomposable projective functor to the corresponding simple highest weight module outputs either an indecomposable module or zero.