The length and depth of compact Lie groups

Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G = G 0 > G 1 > ⋯ > 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...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2020-04, Vol.294 (3-4), p.1457-1476
Main Authors: Burness, Timothy C., Liebeck, Martin W., Shalev, Aner
Format: Article
Language:English
Subjects:
Chains
Lie groups
Mathematics
Mathematics and Statistics
Subgroups
Upper bounds
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Summary:Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G = G 0 > G 1 > ⋯ > 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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-019-02324-7