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 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 Xqeji, where e1, ..., en are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is
- [math]\displaystyle{ \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} }[/math]
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.
See also
- Sanderson's theorem
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): 75–98, doi:10.2307/1988736, 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, https://books.google.com/books?isbn=0486438287
- Rutherford, Daniel Edwin (2007) [1932], Modular invariants, Cambridge Tracts in Mathematics and Mathematical Physics, No. 27, Ramsay Press, ISBN 978-1-4067-3850-6, https://archive.org/details/modularinvariant033204mbp
- 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
- Steinberg, Robert (1987), "On Dickson's theorem on invariants", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 34 (3): 699–707, ISSN 0040-8980, http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/1682/1/jfs340309.pdf, retrieved 2010-12-02
 |  |
