Dickson invariant

Definition

For an orthogonal group

Suppose $K$ is a field, $n$ is a natural number, and $b$ is a nondegenerate symmetric bilinear form for the $n$-dimensional vector space $K^{n}$. Let $G$ be the orthogonal group corresponding to $b$.

The Dickson invariant is a group homomorphism from $G$ to cyclic group:Z2 (which we think of as a integers modulo 2) defined in the following equivalent ways:

1. The Dickson invariant of $g$ is the parity (i.e., value mod 2) of the rank of the linear transformation $I-g$ where $I$ is the identity transformation. In other words, if $I-g$ has even rank, the Dickson invariant is 0 mod 2 (the trivial element of the group of integers mod 2). If $I-g$ has odd rank, the Dickson invariant is 1 mod 2.
2. In the case that $G$ is generated by reflections, the Dickson invariant is the parity of the number of reflections that need to be multiplied to get $g$. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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If the characteristic of $K$ is not 2, then the determinant map and Dickson invariant are equivalent in the following sense: the Dickson invariant is 0 mod 2 iff the determinant is 1 as an element of $K$, and the Dickson invariant is 1 mod 2 iff the determinant is -1 as an element of $K$.

If the characteristic of $K$ is 2, then the determinant of any element in any orthogonal group is always 1 as an element of $K$, so the Dicksoon invariant carries more information than the determinant.

For a Clifford group

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For a Pin group

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