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. The study of modular invariants was originated in about 1914 by.Dickson invariant
When G is the finite general linear group GLn over the finite field Fq of order a prime power q acting on the ring Fq in the natural way, found a complete set of invariants as follows. Write for the determinant of the matrix whose entries are X, where e1,...,en are non-negative integers. For example, the Moore determinant of order 3 is
Then under the action of an element g of GLn these determinants are all multiplied by det, so they are all invariants of SLn and the ratios / are invariants of GLn, called Dickson invariants. Dickson proved that the full ring of invariants FqGLn is a polynomial algebra over the n Dickson invariants / for i = 0, 1,..., n − 1.
gave a shorter proof of Dickson's theorem.
The matrices are divisible by all non-zero linear forms in the variables Xi with coefficients in the finite field Fq. In particular the Moore determinant is a product of such linear forms, taken over 1 + q + q2 + ... + qn – 1 representatives of -dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors.
