Almost all permutations and involutions are Kostant negative

TL;DR

This paper demonstrates that as the dimension increases, the likelihood of Kostant's problem having a negative answer approaches certainty for most simple modules in a specific category of the Lie algebra sl_n(C).

Contribution

It establishes that for large n, almost all simple highest weight modules in the principal block of category O for sl_n(C) have a negative answer to Kostant's problem, revealing a broad negative trend.

Findings

Negative answer for Kostant's problem in almost all cases as n→∞

Most simple modules in the principal block are Kostant negative at large n

Asymptotic behavior of Kostant's problem in high-dimensional Lie algebra representations

Abstract

We prove that, when $n$ goes to infinity, Kostant's problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.