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 (1911) 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 ratio [e1, ... ,en]/[0,1,...,n−1] 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 (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), Providence, R.I.: American Mathematical Society: 75–98, ISSN 0002-9947
- 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