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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)
Published electronically: September 7, 2017
Keywords:root system

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Table of Contents


  • 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


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