k-Involutions of SL(n,k) over fields of characteristic 2

Abstract

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n, k) in the case that k is any field of characteristic 2; we also describe fixed point groups of each type of involution.

Fixed point groups of involutions of type $\mathrm{O}(\mathrm{q},k)$ over a field of characteristic two

“…In particular we want to extend Helminck's study of k-involutions and symmetric k-varieties [15] to include fields of characteristic 2. This has been studied for groups of type G 2 and A n in [22,25] and over fields of characteristic not 2 in [6,3,2,4,19,20,21]. We also extend the results of Aschbacher and Seitz [1] who studied similar structures for finite fields of characteristic 2.…”

On involutions of type O(q,k) over a field of characteristic two

“…The primary motivation is to extend Helminck's [14] study of k-involutions and symmetric k-varieties to include fields of characterstic 2. This has been studied for groups of type G 2 and A n in [19,22] and over fields of characteristic not 2 in [6,3,2,4,16,17,18]. We also extend the results of Aschbacher and Seitz [1] who studied similar structures for finite fields of characteristic 2.…”