Maximal abelian sets of roots
Author(s)
Bibliographic Information
Maximal abelian sets of roots
（Memoirs of the American Mathematical Society, no. 1192）
American Mathematical Society, c2017
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Note
Includes bibliographical references (p. 219)
Description and Table of Contents
Description
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
Introduction
Root systems of classical type
The strategy for root systems of exceptional type
The root system of type $G_2$
The root system of type $F_4$
The root system of type $E_6$
The root system of type $E_7$
The root system of type $E_8$
Tables of maximal abelian sets
Appendix A. Root trees for root systems of exceptional type
Bibliography.
by "Nielsen BookData"