Saturation rank for nilradical of parabolic subalgebras in Type A

TL;DR

This paper investigates the structure of the nilradical of parabolic subalgebras in type A Lie algebras over fields of positive characteristic, providing a combinatorial approach to understanding centralizers and saturation rank.

Contribution

It introduces a combinatorial characterization of centralizers in the nilradical using quivers and establishes conditions where the saturation rank equals the semisimple rank.

Findings

Centralizer of Richardson element characterized by quivers.

Saturation rank equals semisimple rank under mild conditions.

Focus on cases with one or two simple roots in Levi factor.

Abstract

Let $\mfp(d)$ be a standard parabolic subalgebra of $\mfsl_{n+1}(K)$ and $\mfu$ be the corresponding nilradical defined over an algebraically closed field $K$ of characteristic $p>0$. We construct a finite connected quiver $Q(d)$, through which we provide a combinatorial characterization of the centralizer $c_{\mfu}(x(d))$ of the Richardson element $x(d)$. We specifically focus on the centralizer when the levi factor of $\mfp(d)$ is determined by either one or two simple roots. This allows us to demonstrate that, under certain mild restrictions, the saturation rank of $\mfu$ equals the semisimple rank of the algebraic $K$-group $\SL_{n+1}(K)$.