# Maximal abelian sets of roots

### About this Title

**R. Lawther**

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

### Table of Contents

**Chapters**

- Chapter 1. Introduction
- Chapter 2. Root systems of classical type
- Chapter 3. The strategy for root systems of exceptional type
- Chapter 4. The root system of type $G_2$
- Chapter 5. The root system of type $F_4$
- Chapter 6. The root system of type $E_6$
- Chapter 7. The root system of type $E_7$
- Chapter 8. The root system of type $E_8$
- Chapter 9. Tables of maximal abelian sets
- Appendix A. Root trees for root systems of exceptional type

### Abstract

In this work we let be an irreducible root system, with Coxeter group . We consider subsets of which are

*a*belian, meaning that no two roots in the set have sum in . We classify all maximal abelian sets (i.e., abelian sets properly contained in no other) up to the action of : for each -orbit of maximal abelian sets we provide an explicit representative , identify the (setwise) stabilizer of in , and decompose into -orbits.Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelian -subgroups of finite groups of Lie type over fields of characteristic . Parts of the work presented here have been used to confirm the -rank of , and (somewhat unexpectedly) to obtain for the first time the -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

*r*adical 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.

**[1]**N. Bourbaki,*Groupes et algèbres de Lie, Chapitres 4, 5 et 6*, Hermann, Paris, 1975.**[2]**R. W. Carter,*Conjugacy classes in the Weyl group*, Compositio Math.**25**(1972), 1–59. MR**0318337****[3]**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,*Atlas of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219****[4]**Daniel Gorenstein, Richard Lyons, and Ronald Solomon,*The classification of the finite simple groups*, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR**1303592****[5]**Robert M. Guralnick and Gunter Malle,*Classification of $2F$-modules. I*, J. Algebra**257**(2002), no. 2, 348–372. MR**1947326****[6]**Robert M. Guralnick and Gunter Malle,*Classification of $2F$-modules. II*, Finite groups 2003, Walter de Gruyter, Berlin, 2004, pp. 117–183. MR**2125071****[7]**Robert M. Guralnick, Ross Lawther, and Gunter Malle,*The 2F-modules for nearly simple groups*, J. Algebra**307**(2007), no. 2, 643–676. MR**2275366****[8]**R. Lawther,*$2F$-modules, abelian sets of roots and 2-ranks*, J. Algebra**307**(2007), no. 2, 614–642. MR**2275365****[9]**Ross Lawther, Martin W. Liebeck, and Gary M. Seitz,*Fixed point ratios in actions of finite exceptional groups of Lie type*, Pacific J. Math.**205**(2002), no. 2, 393–464. MR**1922740****[10]**A. Malcev,*Commutative subalgebras of semi-simple Lie algebras*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**9**(1945), 291–300 (Russian, with English summary). MR**0015053****[11]**J. Pevtsova and J. Stark, âĂĲVarieties of elementary subalgebras of maxiaml dimension for modular Lie algebrasâĂİ, arXiv:1503.01043v1 [math.RT].**[12]**G. Royle, {verb!http://staffhome.ecm.uwa.edu.au/ 00013890/remote/graphs/!.