Consecutive Patterns, Kostant's Problem and Type $A_6$

TL;DR

The paper resolves Kostant's problem for type $A_6$ by combining parabolic induction reductions, a novel criterion based on consecutive pattern containment in permutations, and extensive computational work in the Hecke algebra via GAP3/CHEVIE. It shows Kostant's problem holds for exactly 125 involutions and fails for 107 in $\mathfrak{S}_7$, while establishing the Indecomposability Conjecture for $\mathfrak{sl}_7$ by reducing to a finite, tractable set of cases and verifying indecomposability or vanishing for indecomposable projective functors. It also confirms substantial parts of Kåhrström's conjecture for $A_6$, leveraging the connection between Kostant's problem, the right KL order, and KL-cell structure, with fully commutative and consecutive-containment cases treated explicitly. The results illuminate the interaction between category $\mathcal{O}$, Hecke algebra combinatorics, and KL- orders, and provide a blueprint for tackling higher-rank cases with a mix of theoretical reductions and targeted computations.

Abstract

For a permutation $w$ in the symmetric group $\mathfrak{S}_{n}$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. 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 $A_{6}$ and show that the indecomposability conjecture also holds in type $A_{6}$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.

Theorems & Definitions (37)

• Example 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Conjecture 2.4
• Proposition 2.5
• Theorem 2.6
• Remark 2.7
• Corollary 2.8
• proof