[Paper Review] Suzuki-Ree groups as algebraic groups over $\mathbb{F}_{\sqrt{\smash[b]p}}$

Tom De Medts, Karsten Naert|arXiv (Cornell University)|Apr 24, 2019

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper introduces a framework in which Suzuki–Ree groups and mixed groups arise as rational points of algebraic groups over a 'twisted field' $ℂ_{\sqrt{p}}$, using a novel notion of twisted and mixed categories via an endomorphism of the identity functor (e.g., Frobenius). The key contribution is showing that base change from $ℂ_{\sqrt{p}}$ to $ℂ_p$ transforms twisted groups into mixed groups, and that twisted descent detects which mixed groups arise from this construction.

ABSTRACT

Among the infinite classes of finite simple groups, the most exotic classes are probably the Suzuki groups and the Ree groups. They are "twisted versions" of groups of Lie type, but they cannot be directly obtained as groups of rational points of a suitable linear algebraic group. We provide a framework in which these groups do arise as groups of rational points of algebraic groups over a "twisted field"; in the finite case, such a twisted field can be interpreted as a "field with $\sqrt{\smash[b]p}$ elements". Our framework at once allows for other, perhaps less known, exotic families of groups. Most notably, there is a class of "mixed groups", introduced by J. Tits but also apparent in the work of Steinberg, and we show that they can be obtained as groups of rational points of algebraic groups over a "mixed field". We show that a base change from $\mathbb{F}_{\sqrt{\smash[b]p}}$ to $\mathbb{F}_p$ transforms twisted groups into mixed groups, and we formulate a notion of "twisted descent" that allows to detect which mixed groups arise in this fashion.

Motivation & Objective

• To provide a unified algebraic-geometric framework for Suzuki–Ree groups, which are not rational points of standard algebraic groups over finite fields.
• To resolve the foundational issue that fields with $p^{(2n+1)/2}$ elements do not exist, by introducing 'twisted fields' like $ℂ_{\sqrt{p}}$.
• To extend this framework to mixed groups, as introduced by Tits, showing they arise as rational points of mixed algebraic groups.
• To establish a base change functor from twisted to mixed categories, linking Suzuki–Ree groups to mixed groups.
• To formulate 'twisted descent' as a criterion to detect which mixed groups arise from twisted groups via base change.

Proposed method

• Define a category of twisted objects using an endomorphism $F$ of the identity functor, typically the Frobenius map.
• Construct a category of mixed objects as pairs of objects with a compatible isomorphism and its inverse, forming a descent datum.
• Introduce the notion of 'twisted descent' to identify which mixed objects arise from twisted objects via base change.
• Use the base change functor from $ℂ_{\sqrt{p}}$ to $ℂ_p$ to transform twisted algebraic groups into mixed algebraic groups.
• Show that the resulting groups of rational points recover Suzuki–Ree and mixed groups via the new framework.
• Apply the theory to Dynkin diagrams $ℂ_n$, $ℂ_n$, $ℂ_4$, and $ℂ_2$, where root system duality enables the construction.

Experimental results

Research questions

• RQ1 Can Suzuki–Ree groups be realized as rational points of algebraic groups over a 'twisted field' like $ℂ_{\sqrt{p}}$?
• RQ2 How can mixed groups, introduced by Tits, be systematically constructed as rational points of algebraic groups over a 'mixed field'?
• RQ3 What is the precise relationship between twisted groups over $ℂ_{\sqrt{p}}$ and mixed groups over $ℂ_p$?
• RQ4 Which mixed groups arise as base changes of twisted groups, and what conditions ensure this?
• RQ5 What is the role of descent theory in recovering twisted groups from mixed groups?

Key findings

• The Suzuki–Ree groups ${}^{2}\mathsf{B}_{2}(2^{2n+1})$, ${}^{2}\mathsf{F}_{4}(2^{2n+1})$, and ${}^{2}\mathsf{G}_{2}(3^{2n+1})$ arise as rational points of twisted linear algebraic groups over $ℂ_{\sqrt{p}}$.
• Mixed groups, including those from Tits' mixed buildings, arise as rational points of mixed linear algebraic groups over $ℂ_p$.
• Base change from $ℂ_{\sqrt{p}}$ to $ℂ_p$ transforms twisted groups into mixed groups.
• A mixed group admits twisted descent if and only if it admits an endomorphism $f$ such that $\tau f \circ f = \mathrm{id}$, and this implies $\operatorname{\mathbf{c}}_1 \cong \operatorname{\mathbf{c}}_2$.
• The category of twisted objects embeds fully and essentially faithfully into the category of mixed objects with descent data, via $\tilde{X} \mapsto (\operatorname{twix}\tilde{X}, \mathrm{id})$, establishing a full equivalence.
• The framework applies to all Dynkin types $ℂ_n$, $ℂ_n$, $ℂ_4$, and $ℂ_2$, with the $\mathsf{B}_2$ case being the only one where $\mathsf{B}_2 \cong \mathsf{C}_2$, enabling the Suzuki group construction.