Irreducible modules of $\frak {sl}_{mp}$ in characteristic $p$ with regular or subregular nilpotent $p$-character

TL;DR

This paper investigates the structure of irreducible modules of the Lie algebra rak{sl}_{mp} in characteristic p, focusing on cases where p divides n, and provides a detailed description of simple modules for regular and subregular nilpotent representations.

Contribution

It extends the classification of simple modules of rak{sl}_n to cases where p divides n, covering regular and subregular nilpotent p-characters.

Findings

Explicit descriptions of simple modules for rak{sl}_{mp} when p divides n.

Extension of known classifications to new cases where p divides n.

Clarification of module structures for regular and subregular nilpotent p-characters.

Abstract

Let $\bbk$ be an algebraically closed field of prime characteristic $p$. If $p$ does not divide $n$, irreducible modules over $\frak {sl}_n$ for regular and subregular nilpotent representations have already known(see \cite{Jan2} and \cite{Jan3}). In this article, we investigate the question when $p$ divides $n$, and precisely describe simple modules of $\frak {sl}_n$ for regular and subregular nilpotent representations.