Dickson invariant

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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]

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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