Low Degree Representations of Simple Lie Groups

TL;DR

This paper establishes upper bounds on the number of low-dimensional irreducible representations of compact simple Lie groups, linking representation counts to group rank and providing insights into subgroup conjugacy classes.

Contribution

It provides new upper bounds for irreducible representations of simple compact Lie groups and relates these bounds to subgroup conjugacy class counts.

Findings

At most n irreducible representations of dimension at most n for a simple compact Lie group.

Number of conjugacy classes of maximal subgroups is O(r), where r is the Lie rank.

Dimensions of weight spaces of a maximal torus are small relative to the module dimension.

Abstract

We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.