Maximal subgroups of maximal rank in the classical algebraic groups

TL;DR

This paper classifies maximal reductive subgroups of maximal rank in classical algebraic groups over arbitrary fields, providing combinatorial descriptions, realization conditions over various fields, and asymptotic counts as rank increases.

Contribution

It extends the classification of maximal subgroups to classical groups, complementing previous work on exceptional groups, and analyzes their field realizability and asymptotic behavior.

Findings

Classification of maximal reductive subgroups in classical groups

Conditions for realization over finite, real, and p-adic fields

Asymptotic growth of subgroup counts with increasing rank

Abstract

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does the same for the exceptional groups. We determine which of these subgroups may be realised over a finite field, the real numbers, or over a $\mathfrak{p}$-adic field. We also look at the asymptotics of the number of such subgroups as the rank grows large.