Maximal subgroups of maximal rank in the classical algebraic groups

TL;DR

The paper advances a complete combinatorial classification of index-conjugacy classes for isotropic maximal reductive subgroups of maximal rank in classical absolutely simple $k$-groups, extending the framework of the index and embeddings of indices to the classical Dynkin types. It develops and uses the (Tits) index $\mathcal{I}(G)$, the embedding of indices, and the notion of abstract indices along with almost primitive subsystems to reduce the problem to discrete data, subsequently realized in explicit tables for $A_n$, $B_n$, $C_n$, and $D_n$ (including $D_4$). The authors provide an algorithmic approach (with Julia/Oscar code) to enumerate embeddings and derive consequences such as realizability over fields of cohomological dimension $1$, $\mathbb{R}$, or $\mathfrak{p}$-adic fields, and a subexponential growth bound $N(\Phi)=O(n^{c\log n})$ (sharper $O(n)$ for separably closed $k$). These results complete the parallel program to S for the exceptional and classical groups, offering a detailed map between combinatorial index data and concrete algebraic subgroup realizations with implications for field realizability and asymptotics in 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.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Lemma 2.3: Table 1, Lem. 2.1 of S
• Definition 2.4: Def. 2.4 of S
• Definition 2.5: Def. 1 of S
• Theorem 3.1