**Memoirs of the American Mathematical Society**

2017;
219 pp;
Softcover

MSC: Primary 17;

**Print ISBN: 978-1-4704-2679-8
Product Code: MEMO/250/1192**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-4208-8
Product Code: MEMO/250/1192.E**

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# Maximal Abelian Sets of Roots

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*R. Lawther*

In this work the author lets \(\Phi\) be an
irreducible root system, with Coxeter group \(W\). He considers
subsets of \(\Phi\) which are abelian, meaning that no
two roots in the set have sum in \(\Phi \cup \{ 0 \}\). He
classifies 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 the author introduces 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.

#### Table of Contents

# Table of Contents

## Maximal Abelian Sets of Roots

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 110
- Chapter 2. Root systems of classical type 716
- Chapter 3. The strategy for root systems of exceptional type 1120
- Chapter 4. The root system of type πΊβ 1726
- Chapter 5. The root system of type πΉβ 1928
- Chapter 6. The root system of type πΈβ 2332
- Chapter 7. The root system of type πΈβ 3140
- Chapter 8. The root system of type πΈβ 5564
- Chapter 9. Tables of maximal abelian sets 145154
- Appendix A. Root trees for root systems of exceptional type 215224
- Bibliography 219228
- Back Cover Back Cover1234