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Isomorphisms of Groups of Kac-Moody Type Over $\mathbb{F}_2$

Sebastian Bischof

TL;DR

The paper addresses the isomorphism problem for RGD-systems of Kac–Moody type over $\mathbb{F}_2$, focusing on $2$-complete and $\tilde{A}_2$-free Coxeter types with centered, finite-rank data. It introduces triangle- and building-based methods to circumvent the lack of a torus in characteristic $2$ and proves that triangles enforce unique chamber configurations, enabling control of group isomorphisms by the root data. The main result shows that any isomorphism between centered $\mathbb{F}_2$-RGD-systems of the specified type induces an isomorphism of the RGD-systems themselves and that the torus is trivial, leading to a clean automorphism decomposition. Consequently, the isomorphism problem is solved for this class, and automorphisms are generated by inner, graph, and sign automorphisms. The work extends Caprace–Mühlherr’s framework to characteristic $2$ by leveraging new geometric rigidity from triangle structures in twin buildings.

Abstract

In \cite{CM06} Caprace and Mühlherr solved the isomorphism problem for Kac-Moody groups of non-spherical type over finite fields of cardinality at least $4$. In this paper we solve the isomorphism problem for RGD-systems (e.g.\ Kac-Moody groups) over $\mathbb{F}_2$ whose type is $2$-complete and $\tilde{A}_2$-free.

Isomorphisms of Groups of Kac-Moody Type Over $\mathbb{F}_2$

TL;DR

The paper addresses the isomorphism problem for RGD-systems of Kac–Moody type over , focusing on -complete and -free Coxeter types with centered, finite-rank data. It introduces triangle- and building-based methods to circumvent the lack of a torus in characteristic and proves that triangles enforce unique chamber configurations, enabling control of group isomorphisms by the root data. The main result shows that any isomorphism between centered -RGD-systems of the specified type induces an isomorphism of the RGD-systems themselves and that the torus is trivial, leading to a clean automorphism decomposition. Consequently, the isomorphism problem is solved for this class, and automorphisms are generated by inner, graph, and sign automorphisms. The work extends Caprace–Mühlherr’s framework to characteristic by leveraging new geometric rigidity from triangle structures in twin buildings.

Abstract

In \cite{CM06} Caprace and Mühlherr solved the isomorphism problem for Kac-Moody groups of non-spherical type over finite fields of cardinality at least . In this paper we solve the isomorphism problem for RGD-systems (e.g.\ Kac-Moody groups) over whose type is -complete and -free.

Paper Structure

This paper contains 6 sections, 37 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter buildings
  4. Triangles in buildings
  5. Groups of Kac-Moody type
  6. The Main result

Key Result

Theorem 1

Suppose that $(W, S)$ and $(W', S')$ are $2$-complete and $\tilde{A}_2$-free Coxeter systems of finite rank at least $3$ and let $\mathcal{D}$ and $\mathcal{D}'$ be two centered RGD-systems over $\mathbb{F}_2$ of type $(W, S)$ and $(W', S')$. Then any isomorphism $G^{\mathcal{D}} \to G^{\mathcal{D}'

Theorems & Definitions (82)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.6
  • ...and 72 more