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).
Dickson invariant
When G is the finite group general linear group GLn(Fp) over the finite field Fp acting on the ring Fp[X1, ... ,Xn] in the natural way, Dickson found a complete set of invariants as follows. Write [e1, ... ,en] for the determinant of the matrix whose entries are Xpej
i, where e1, ... ,en are non-negative integers.
Then under the action of an element g of GLn(Fp) these determinants are all multiplied by det(g), so they are all invariants of SLn(Fp) and the ratios of any two are invariants of GLn(Fp), called Dickson invariants. Dickson proved that the full ring of invariants 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.
See also
References
- Dickson, Leonard Eugene (2004) [1914], On invariants and the theory of numbers, Dover Phoenix editions, New York: Dover Publications, ISBN 978-0-486-43828-3, MR0201389
- Rutherford, Daniel Edwin (2007) [1932], Modular invariants, Cambridge Tracts in Mathematics and Mathematical Physics, No. 27, Ramsay Press, ISBN 978-1-4067-3850-6, MR0186665
- Sanderson, Mildred (1913), "Formal Modular Invariants with Application to Binary Modular Covariants", Transactions of the American Mathematical Society, 14 (4), Providence, R.I.: American Mathematical Society: 489–500, ISSN 0002-9947