Aspects of the conjugacy class structure of simple algebraic groups.

Cook, Martin

Journal of Algebraic Combinatorics, Tome 31 (2010) no. 3, pp. 319-353.

Résumé

Summary: Let $G$ be an adjoint simple algebraic group over an algebraically closed field of characteristic $p$; let $\Phi $be the root system of $G$, and take $t\in \Bbb N$. Lawther has proven that the dimension of the set $G _{[ t]}={ g\in G: g ^{ t }=1}$ depends only on $\Phi $and $t$. In particular the value is independent of the characteristic $p$; this was observed for $t$ small and prime by Liebeck. Since $G _{[ t]}$ is clearly a disjoint union of conjugacy classes the question arises as to whether a similar result holds if we replace $G _{[ t]}$ by one of those classes. This paper provides a partial answer to that question. A special case of what we have proven is the following. Take $p, q$ to be distinct primes and $G _{ p }$ and $G _{ q }$ to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic $p$ and $q$ respectively. If $s\in G _{ p }$ has order $q$ then there exists an element $u\in G _{ q }$ such that $o( u)= o( s)$ and = s^G_p $\dim $u^G_q=$\dim $s^G_p .

Détail

Keywords: keywords algebraic groups, conjugacy classes, characteristic independent