Modular invariant theory

Sub-field of mathematics

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

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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 Xqej
i, where e1, ..., en are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is

| 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 | {\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}}}

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

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* Sanderson's theorem

## References

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* 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, archived from the original (PDF) on 2012-03-05, retrieved 2010-12-02