A lower bound for the dimension of a highest weight module

TL;DR

This paper establishes the minimum possible dimension for irreducible highest weight modules of complex simple Lie algebras with a given highest weight height, and classifies modules with prime product dimensions.

Contribution

It provides a lower bound for the dimension of highest weight modules and classifies modules with dimensions as products of two primes.

Findings

Determined the least dimension for modules with given highest weight height.

Classified modules with dimensions equal to a product of two primes.

Abstract

For each integer $t>0$ and each complex simple Lie algebra $\mathfrak{g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak{g}$ whose highest weight has height $t$. As a corollary, we classify all irreducible modules whose dimension equals a product of two primes.