The length and depth of compact Lie groups

TL;DR

This paper introduces the concepts of length and depth for connected Lie groups, computes these for compact simple Lie groups, and establishes bounds and characterizations related to these measures.

Contribution

It defines and analyzes the length and depth of connected Lie groups, providing exact values for compact simple groups and bounds for general compact groups.

Findings

Exact length and depth for all compact simple Lie groups

Bounds on length in terms of dimension

Characterization of groups with equal length and depth

Abstract

Let $G$ be a connected Lie group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of $G$ in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on $\dim G'$ in terms of the chain difference of $G$, which is its length minus its depth.