Embeddings of Rank-2 tori in Algebraic groups

TL;DR

This paper investigates the embedding of rank-2 unitary tori into simple algebraic groups of types A2, G2, and F4 over fields with characteristic not 2 or 3, revealing their role in understanding group invariants and subgroup structures.

Contribution

It provides necessary conditions for embedding rank-2 unitary tori into these groups and explicitly constructs such tori, advancing the understanding of their subgroup and invariant structures.

Findings

Conditions for embedding tori into algebraic groups are established.

Explicit examples of rank-2 unitary tori are constructed.

Mod-2 invariants are shown to be governed by specific subgroups and tori.

Abstract

Let $k$ be a field of characteristic different from $2$ and $3$. In this paper we study connected simple algebraic groups of type $A_2$, $G_2$ and $F_4$ defined over $k$, via their rank-$2$ $k$-tori. Simple, simply connected groups of type $A_2$ play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-$2$ unitary $k$-torus to embed in simple $k$-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-$2$ Galois cohomological invariants attached with these groups. We calculate the number of rank-$2$ $k$-unitary tori generating these algebraic groups (in fact exhibit such tori explicitly). The results in this paper and our earlier work (Invariants mod-2 and subgroups of G_2 and F_4, J. Alg. 411 (2014) 312- 336) show that the mod-$2$ invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their $k$-subgroups of type $A_1$ and $A_2$ as well as the unitary $k$-tori embedded in them.