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

${\displaystyle {\begin{vmatrix}x_{1}&x_{1}^{q}&x_{1}^{q^{2))\\x_{2}&x_{2}^{q}&x_{2}^{q^{2))\\x_{3}&x_{3}^{q}&x_{3}^{q^{2))\end{vmatrix))}$

Then under the action of an element g of GL_n(F_q) these determinants are all multiplied by det(g), so they are all invariants of SL_n(F_q) and the ratios [e₁, ...,e_n] / [0, 1, ..., n − 1] are invariants of GL_n(F_q), called Dickson invariants. Dickson proved that the full ring of invariants F_q[X₁, ...,X_n]^GL_n(F_q) 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. Steinberg (1987) 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 F_q. 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.