Modular representations of the special linear groups with small weight multiplicities

TL;DR

This paper classifies irreducible representations of special linear groups with small weight multiplicities, provides bounds for these multiplicities, and extends the Steinberg tensor product theorem to certain inductive systems.

Contribution

It introduces a classification of small weight multiplicity representations and generalizes the Steinberg tensor product theorem for inductive systems.

Findings

Classified irreducible representations with small weight multiplicities.

Provided estimates for maximal weight multiplicities.

Established an analogue of the Steinberg tensor product theorem.

Abstract

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural embeddings of the classical groups, inductive systems of representations with totally bounded weight multiplicities are classified. An analogue of the Steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved.