Small-dimensional representations of algebraic groups of type $A_l$

TL;DR

This paper classifies small-dimensional irreducible rational representations of algebraic groups of type A_l over algebraically closed fields, providing explicit descriptions for certain cases and bounds on dimensions.

Contribution

It determines all irreducible rational representations of type A_l groups with small dimensions, extending known classifications for large l and specific bounds.

Findings

Classified irreducible representations with dimensions ≤ (l+1)^3 and (l+1)^4 for large l.

Provided explicit descriptions of modules for the case s=3.

Established bounds on dimensions for representations in defining characteristic.

Abstract

For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$, provided that $l > 18$, $l > 35$ respectively. We also give explicit descriptions of the corresponding modules for $s = 3$.