Group of Lie Type - Steinberg Groups

Steinberg Groups

Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. Steinberg (1959) found a modification of Chevalley's construction that gave these groups and two new families 3D₄, 2E₆, the second of which was discovered at about the same time from a different point of view by Tits (1958). This construction generalizes the usual construction of the unitary group from the general linear group.

The unitary group arises as follows: the general linear group over the complex numbers has a diagram automorphism given by reversing the Dynkin diagram A_n (which corresponds to taking the transpose inverse), and a field automorphism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.

In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.

These gave:

• the unitary groups 2A_n, from the order 2 automorphism of A_n;
• further orthogonal groups 2D_n, from the order 2 automorphism of D_n;
• the new series 2E₆, from the order 2 automorphism of E₆;
• the new series 3D₄, from the order 3 automorphism of D₄.

The groups of type 3D₄ have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the D₄ diagram also give rise to triality.