Covering Numbers of Unipotent Conjugacy Classes in Simple Algebraic Groups

Abstract

For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.

Covering Numbers for Simple Algebraic Groups

“…Continuing with the discussion of the case where G is a finite simple group and C a nontrivial conjugacy class, the main result of [19] shows that there is an absolute constant c such that cn(G, C) ≤ c log |G| log |C| for all such G, C. (Obviously log |G| log |C| is a lower bound for the covering number.) In [22], it is suggested that an analogous result should hold for algebraic groups: namely, there should be a constant c such that cn(G, C) ≤ c dim G dim C for all non-central conjugacy classes C of simple algebraic groups G. Such a bound is established in [22, Theorem C] for the case where C is a unipotent conjugacy class in good characteristic. In this paper we prove such a result in general.…”

Products of conjugacy classes in simple algebraic groups in terms of diagrams

“…Moreover, if G is a Chevalley group defined over an algebraically closed field, they show that ecn(G) ≤ 4 • rk(G). Extending [17,Theorem 1.1], upper bounds on covering numbers of unipotent conjugacy classes are given in terms of their dimension and in terms of their (co)ranks in [25] for G a simple algebraic group over an algebraically closed field k of good characteristic. Recall that the characteristic p of k is good for G if p = 2 when G is not of type A, p = 3 if G is an exceptional group and p = 5 if G is of type E 8 .…”

“…Recall that the characteristic p of k is good for G if p = 2 when G is not of type A, p = 3 if G is an exceptional group and p = 5 if G is of type E 8 . Theorem C in [25] is extended and improved by [18, Theorem 1] where it is shown that there exists an absolute constant c ≤ 120 such that whenever C is a non-central conjugacy class of a simple algebraic group G then C k = G for any integer k at least c • (dim(G)/ dim(C)).…”