Modular invariant theory

Dickson invariant

When G is the finite general linear group GLn(Fq) over the finite field Fq of order a prime power q acting on the ring Fq[X1, ...,X**n] in the natural way, found a complete set of invariants as follows. Write [e1, ..., e**n] for the determinant of the matrix whose entries are X, where e1, ..., e**n are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is

:\begin{vmatrix} x_1 & x_1^q & x_1^{q^2}\x_2 & x_2^q & x_2^{q^2}\x_3 & x_3^q & x_3^{q^2} \end{vmatrix}

Then under the action of an element g of GLn(Fq) these determinants are all multiplied by det(g), so they are all invariants of SLn(Fq) and the ratios [e1, ...,e**n] / [0, 1, ..., n − 1] are invariants of GLn(Fq), called Dickson invariants. Dickson proved that the full ring of invariants Fq[X1, ...,X**n]GLn(Fq) is a polynomial algebra over the n Dickson invariants [0, 1, ..., i − 1, i + 1, ..., n] / [0, 1, ..., n − 1] for i = 0, 1, ..., n − 1. gave a shorter proof of Dickson's theorem.

The matrices [e1, ..., e**n] are divisible by all non-zero linear forms in the variables X**i with coefficients in the finite field Fq. In particular the Moore determinant [0, 1, ..., n − 1] is a product of such linear forms, taken over 1 + q + q2 + ... + q**n – 1 representatives of (n – 1)-dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors.

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