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 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, found a complete set of invariants as follows. Write [e1, ..., en] for the determinant of the matrix whose entries are X$q^{e_{j}}|undefined i$, 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]&thinsp;/&thinsp;[0, 1, ..., n −&thinsp;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]&thinsp;/&thinsp;[0, 1, ..., n −&thinsp;1] for i = 0, 1, ..., n − 1. 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 −&thinsp;1] is a product of such linear forms, taken over 1 + q + q2 + ... + qn – 1 representatives of (n –&thinsp;1)-dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors.