Modular invariant theory: Difference between revisions

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==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, {{harvtxt|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''{{su|b=''i''|p=''q''^''e''_''j''}}, where ''e''₁, ... ,''e''_''n'' are non-negative integers. For example, the [[Moore determinant over a finite field|Moore determinant]] [0,1,2] of order 3 is

:<math>\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}</math>

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 [''e''₁, ... ,''e''_''n'']/[0,1,...,''n''&minus;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'']^{GL_''n''('''F'''_''q'')} is a polynomial algebra over the ''n'' Dickson invariants [0,1,...,''i''&minus;1,''i''+1,...,''n'']/[0,1,...,''n''&minus;1] for ''i''=0, 1, ..., ''n''&minus;1.

{{harvtxt|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 over a finite field|Moore determinant]] [0,1,...,''n''&minus;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==

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==References==

*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=A Fundamental System of Invariants of the General Modular Linear Group with a Solution of the Form Problem | url=http://www.jstor.org/stable/=1988736 | publisher=[[American Mathematical Society]] | ___location=Providence, R.I. | year=1911 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=12 | issue=1 | pages=75–98}}

*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=On invariants and the theory of numbers | origyear=1914 | url=http://books.google.com/books?isbn=0486438287 | publisher=[[Dover Publications]] | ___location=New York | series=Dover Phoenix editions | isbn=978-0-486-43828-3 | idmr={{MR|0201389}} | year=2004}}

*{{Citation | last1=Rutherford | first1=Daniel Edwin | title=Modular invariants | origyear=1932 | url=http://www.archive.org/details/modularinvariant033204mbp | publisher=Ramsay Press | series=Cambridge Tracts in Mathematics and Mathematical Physics, No. 27 | isbn=978-1-4067-3850-6 | idmr={{MR|0186665}} | year=2007}}

*{{Citation | last1=Sanderson | first1=Mildred | title=Formal Modular Invariants with Application to Binary Modular Covariants | url=http://www.jstor.org/stable/=1988702 | publisher=[[American Mathematical Society]] | ___location=Providence, R.I. | year=1913 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=14 | issue=4 | pages=489–500}}

*{{Citation | last1=Steinberg | first1=Robert | title=On Dickson's theorem on invariants | url=http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/1682/1/jfs340309.pdf | idmr={{MR|927606}} | year=1987 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=34 | issue=3 | pages=699–707}}

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