Aspects of the conjugacy class structure of simple algebraic groups

作者：Cook Martin^*

来源：Journal of Algebraic Combinatorics, 2010, 31(3): 319-353.

DOI：10.1007/s10801-009-0187-y

• Algebraic groups
• Conjugacy classes
• Characteristic independent

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 is an element of N. Lawther has proven that the dimension of the set G([t]) = {g is an element of 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 is an element of G(p) has order q then there exists an element u is an element of G(q) such that o(u) = o(s) and dim u(Gq) = dim s(Gp).

• 出版日期2010-5