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 .
When G is the finite general linear group GL<sub>n</sub>(F<sub>q</sub>) over the finite field F<sub>q</sub> of order a prime power q acting on the ring F<sub>q</sub>[X<sub>1</sub>, ...,X<sub>n</sub>] in the natural way, found a complete set of invariants as follows. Write [e<sub>1</sub>, ..., e<sub>n</sub>] for the determinant of the matrix whose entries are X, where e<sub>1</sub>, ..., e<sub>n</sub> are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is
Then under the action of an element g of GL<sub>n</sub>(F<sub>q</sub>) these determinants are all multiplied by det(g), so they are all invariants of SL<sub>n</sub>(F<sub>q</sub>) and the ratios [e<sub>1</sub>, ...,e<sub>n</sub>] / [0, 1, ..., n âÂÂ 1] are invariants of GL<sub>n</sub>(F<sub>q</sub>), called Dickson invariants. Dickson proved that the full ring of invariants F<sub>q</sub>[X<sub>1</sub>, ...,X<sub>n</sub>]<sup>GL<sub>n</sub>(F<sub>q</sub>)</sup> is a polynomial algebra over the n Dickson invariants [0, 1, ..., i â 1, i + 1, ..., n] / [0, 1, ..., n âÂÂ 1] for i = 0, 1, ..., n â 1. gave a shorter proof of Dickson's theorem.
The matrices [e<sub>1</sub>, ..., e<sub>n</sub>] are divisible by all non-zero linear forms in the variables X<sub>i</sub> with coefficients in the finite field F<sub>q</sub>. In particular the Moore determinant [0, 1, ..., n âÂÂ 1] is a product of such linear forms, taken over 1 + q + q<sup>2</sup> + ... + q<sup>n â 1</sup> representatives of (n âÂÂ 1)-dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors.