Maximal Abelian Sets of Roots (eBook)

Ross I Lawther (Autor)

eBook Download: PDF

219 Seiten
American Mathematical Society (Verlag)
9781470442088 (ISBN)

Infos

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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• ISBN-13: 9781470442088 / 9781470442088