Ree group

In mathematics, a Ree group is a group of Lie type over a finite field. They are named after Rimhak Ree, who constructed them from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.

Unlike the Steinberg groups, the Ree groups are not given by the points of a connected reductive algebraic group defined over a finite field; in other words, there is no "Ree algebraic group" related to the Ree groups in the same way that (say) unitary groups are related to Steinberg groups. However, there are some exotic pseudo-reductive algebraic groups over non-perfect fields whose construction is related to the construction of Ree groups, as they use the same exotic automorphisms of Dynkin diagrams that change root lengths.

Tits defined Ree groups over infinite fields of characteristics 2 and 3. Tits and Hée introduced Ree groups of infinite-dimensional Kac–Moody algebras.

Construction

If X is a Dynkin diagram, Chevalley constructed split algebraic groups corresponding to X, in particular giving groups X(F) with values in a field F. These groups have the following automorphisms:

Any endomorphism σ of the field F induces an endomorphism ασ of the group X(F)

Any automorphism π of the Dynkin diagram induces an automorphism απ of the group X(F).

The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism of X(F) for F the algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism of F, while for the Steinberg groups the automorphism is the Frobenius endomorphism times an automorphism of the Dynkin diagram.