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Maximal abelian sets of roots
About this Title
R. Lawther, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WB
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 250, Number 1192
ISBNs: 978-1-4704-2679-8 (print); 978-1-4704-4208-8 (online)
DOI: https://doi.org/10.1090/memo/1192
Published electronically: September 7, 2017
Keywords: root system
MSC: Primary 17B22
Table of Contents
Chapters
- 1. Introduction
- 2. Root systems of classical type
- 3. The strategy for root systems of exceptional type
- 4. The root system of type $G_2$
- 5. The root system of type $F_4$
- 6. The root system of type $E_6$
- 7. The root system of type $E_7$
- 8. The root system of type $E_8$
- 9. Tables of maximal abelian sets
- A. Root trees for root systems of exceptional type
Abstract
In this work we let $\Phi$ be an irreducible root system, with Coxeter group $W$. We consider subsets of $\Phi$ which are abelian, meaning that no two roots in the set have sum in $\Phi \cup \{ 0 \}$. We classify all maximal abelian sets (i.e., abelian sets properly contained in no other) up to the action of $W$: for each $W$-orbit of maximal abelian sets we provide an explicit representative $X$, identify the (setwise) stabilizer $W_X$ of $X$ in $W$, and decompose $X$ into $W_X$-orbits.
Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelian $p$-subgroups of finite groups of Lie type over fields of characteristic $p$. Parts of the work presented here have been used to confirm the $p$-rank of $E_8(p^n)$, and (somewhat unexpectedly) to obtain for the first time the $2$-ranks of the Monster and Baby Monster sporadic groups, together with the double cover of the latter.
Root systems of classical type are dealt with quickly here; the vast majority of the present work concerns those of exceptional type. In these root systems we introduce the notion of a radical set; such a set corresponds to a subgroup of a simple algebraic group lying in the unipotent radical of a certain maximal parabolic subgroup. The classification of radical maximal abelian sets for the larger root systems of exceptional type presents an interesting challenge; it is accomplished by converting the problem to that of classifying certain graphs modulo a particular equivalence relation.
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