# Modular invariant theory

In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by Dickson (2004).

## 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, ...,Xn] in the natural way, Dickson (1911) found a complete set of invariants as follows. Write [e1, ...,en] for the determinant of the matrix whose entries are Xqej
i
, where e1, ...,en 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, ...,en]/[0, 1, ...,n  1] are invariants of GLn(Fq), called Dickson invariants. Dickson proved that the full ring of invariants Fq[X1, ...,Xn]GLn(Fq) is a polynomial algebra over the n Dickson invariants [0, 1, ...,i  1, i + 1, ..., n]/[0,1,...,n1] for i = 0, 1, ..., n  1. Steinberg (1987) gave a shorter proof of Dickson's theorem.

The matrices [e1, ...,en] are divisible by all non-zero linear forms in the variables Xi 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 + ... + qn  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.