# 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 GL_{n}(**F**_{q}) over the finite field **F**_{q} of order a prime power *q* acting on the ring **F**_{q}[*X*_{1}, ...,*X*_{n}] in the natural way, Dickson (1911) found a complete set of invariants as follows. Write [*e*_{1}, ...,*e*_{n}] for the determinant of the matrix whose entries are *X*^{qej}_{i}, where *e*_{1}, ...,*e*_{n} 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 GL_{n}(**F**_{q}) these determinants are all multiplied by det(*g*), so they are all invariants of SL_{n}(**F**_{q}) and the ratios [*e*_{1}, ...,*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}[*X*_{1}, ...,*X*_{n}]^{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 [*e*_{1}, ...,*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* + *q*^{2} + ... + *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

## 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, 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, MR 0201389 - Rutherford, Daniel Edwin (2007) [1932],
*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