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About Maximal Abelian Sets of Roots (Memoirs of the American Mathematical Society)
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 $WX$ of $X$ in $W$, and decompose $X$ into $WX$-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 $E8(pn)$, 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.
Detailed Information
| Author: | R Lawther |
|---|---|
| Publication Year: | 2017 |
| ISBN: | 9781470426798 |
| Pages: | 234 |
| Language: | English |
| File Size: | 1.9 |
| Format: | |
| Price: | FREE |
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