Orthogonal Group - The Dickson Invariant

The Dickson Invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (positive integers modulo 1), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.

Algebraically, the Dickson invariant can be defined as D(f)= rank (I-f) modulo 2, where I is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant and usually has index 2 in O(n,F). When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n,F) is commonly defined to be the elements of O(n,F) with determinant 1. Each element in O(n,F) has determinant ±1. Thus in characteristic 2, the determinant is always 1.

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).