Double centralizers of unipotent elements in simple algebraic groups of typeG2,F4and

Iulian I. Simion

¹

A Counterexample to a Conjugacy Conjecture of Steinberg

“…When G is simple of exceptional type, explicit representatives for the unipotent conjugacy classes in terms of root elements were first given in [Cha68], [Eno70], [Sho74], [Shi74], [Miz77], and [Miz80]. For computations, we have used representatives from the tables in [Sim13], which are based on [LS12] and some computations due to Ross Lawther. For G simple of classical type, representatives in terms of root elements of the form x α (±1) are well known, but do not seem to appear explicitly in the literature.…”

“…For computations, we have used representatives from the tables in [Sim13], which are based on [LS12] and some computations due to Ross Lawther. For G simple of classical type, representatives in terms of root elements of the form x α (±1) are well known, but do not seem to appear explicitly in the literature.…”

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VI. Suzuki and Ree groups

“…Unipotent classes. We see from [18,Table 22.1.5] that G has 5 non-trivial unipotent classes: the regular one O reg , the subregular one O subreg , represented by x β (1)x 3α+β (1), the class labeled by ( Ã1 ) 3 , represented by x 2α+β (1)x 3α+2β (1), and the classes Ã1 and A 1 , represented by x α (1) and x β (1), [26,Table 2,Example 4.3]. The last two classes are interchanged by F , hence they do not intersect G. The dimensions of the remaining ones are all distinct, hence they are all F -stable.…”

Double centralizers in Artin-Tits groups

“…Clearly, for an algebra R and a subalgebra H, the centralizer Z G (H) is also a subalgebra. In this framework, the subalgebra Z G (Z G (H)), called the double centralizer of H, has been considered [10,25]. For instance, a classical result [10] is the socalled Centralizer Theorem, which claims that for a finite dimensional central simple algebra R over a field k and for a simple subalgebra H, one has Z G (Z G (H)) = H. Various generalizations has been obtained leading to applications [26,6].…”