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).
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
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,...,n−1] 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.
- Dickson, Leonard Eugene (1911), "A Fundamental System of Invariants of the General Modular Linear Group with a Solution of the Form Problem", Transactions of the American Mathematical Society, 12 (1): 75–98, doi:10.2307/1988736, ISSN 0002-9947, JSTOR 1988736
- Dickson, Leonard Eugene (2004) , On invariants and the theory of numbers, Dover Phoenix editions, New York: Dover Publications, ISBN 978-0-486-43828-3, MR 0201389
- Rutherford, Daniel Edwin (2007) , Modular invariants, Cambridge Tracts in Mathematics and Mathematical Physics, No. 27, Ramsay Press, ISBN 978-1-4067-3850-6, MR 0186665
- Sanderson, Mildred (1913), "Formal Modular Invariants with Application to Binary Modular Covariants", Transactions of the American Mathematical Society, 14 (4): 489–500, doi:10.2307/1988702, ISSN 0002-9947, JSTOR 1988702
- Steinberg, Robert (1987), "On Dickson's theorem on invariants" (PDF), Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 34 (3): 699–707, ISSN 0040-8980, MR 0927606