Modular invariant of a group

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 GL_n(F_q) over the finite field F_q of order a prime power q acting on the ring F_q[X₁, ...,X_n] in the natural way, Dickson (1911) found a complete set of invariants as follows. Write [e₁, ...,e_n] for the determinant of the matrix whose entries are Xq^e_j
i, where e₁, ...,e_n are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is

$\begin{vmatrix} x_1 & x_2 & x_3\\x_1^q & x_2^q & x_3^q\\x_1^{q^2} & x_2^{q^2} & x_3^{q^2} \end{vmatrix}$

Then under the action of an element g of GL_n(F_q) these determinants are all multiplied by det(g), so they are all invariants of SL_n(F_p) and the ratio [e₁, ...,e_n]/[0,1,...,n−1] are invariants of GL_n(F_q), called Dickson invariants. Dickson proved that the full ring of invariants F_q[X₁, ...,X_n]^GL_n(F_q) 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, ...,e_n] are divisible by all non-zero linear forms in the variables X_i with coefficients in the finite field F_q. In particular the Moore determinant [0,1,...,n−1] is a product of such linear forms, taken over 1+q+q²+...+q^n–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

• Miss 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 (Providence, R.I.: American Mathematical Society) 12 (1): 75–98, ISSN 0002-9947, JSTOR 1988736 
• 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, http://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, MR0186665, http://www.archive.org/details/modularinvariant033204mbp 
• Sanderson, Mildred (1913), "Formal Modular Invariants with Application to Binary Modular Covariants", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 14 (4): 489–500, ISSN 0002-9947, JSTOR 1988702 
• 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, MR927606, http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/1682/1/jfs340309.pdf