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[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

${\displaystyle {\begin{vmatrix}x_{1}&x_{1}^{q}&x_{1}^{q^{2}}\\x_{2}&x_{2}^{q}&x_{2}^{q^{2}}\\x_{3}&x_{3}^{q}&x_{3}^{q^{2}}\end{vmatrix}}}$

Then under the action of an element g of 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 [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, 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, Providence, R.I.: 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 927606