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In mathematics, a '''modular invariant''' of a [[group (mathematics)|group]] is an invariant of a [[finite group]] [[group action|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 {{harvtxt|Dickson|2004}}.
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==Dickson invariant==
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When ''G'' is the finite general linear group GL<sub>''n''</sub>('''F'''<sub>''q''</sub>) over the finite field '''F'''<sub>''q''</sub> of order a prime power ''q'' acting on the ring '''F'''<sub>''q''</sub>[''X''<sub>1</sub>, ...,''X''<sub>''n''</sub>] in the natural way, {{harvtxt|Dickson|1911}} found a complete set of invariants as follows. Write [''e''<sub>1</sub>, ...,''e''<sub>''n''</sub>] for the determinant of the matrix whose entries are ''X''{{su|b=''i''|p=''q''<sup>''e''<sub>''j''</sub></sup>}}, where ''e''<sub>1</sub>, ...,''e''<sub>''n''</sub> are non-negative integers. For example, the [[Moore determinant over a finite field|Moore determinant]] [0,1,2] of order 3 is
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:<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>
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Then under the action of an element ''g'' of GL<sub>''n''</sub>('''F'''<sub>''q''</sub>) these determinants are all multiplied by det(''g''), so they are all invariants of SL<sub>''n''</sub>('''F'''<sub>''p''</sub>) and the ratios [''e''<sub>1</sub>, ...,''e''<sub>''n''</sub>]/[0,&nbsp;1,&nbsp;...,''n''&nbsp;&minus;&nbsp;1] are invariants of GL<sub>''n''</sub>('''F'''<sub>''q''</sub>), called '''Dickson invariants'''. Dickson proved that the full ring of invariants '''F'''<sub>''q''</sub>[''X''<sub>1</sub>, ...,''X''<sub>''n''</sub>]<sup>GL<sub>''n''</sub>('''F'''<sub>''q''</sub>)</sup> is a polynomial algebra over the ''n'' Dickson invariants [0,&nbsp;1,&nbsp;...,''i''&nbsp;&minus;&nbsp;1,&nbsp;''i''&nbsp;+&nbsp;1,&nbsp;...,&nbsp;''n'']/[0,1,...,''n''&minus;1] for ''i''&nbsp;=&nbsp;0, 1, ..., ''n''&nbsp;&minus;&nbsp;1.
{{harvtxt|Steinberg|1987}} gave a shorter proof of Dickson's theorem.
 
The matrices [''e''<sub>1</sub>, ...,''e''<sub>''n''</sub>] are divisible by all non-zero linear forms in the variables ''X''<sub>''i''</sub> with coefficients in the finite field '''F'''<sub>''q''</sub>. In particular the [[Moore determinant over a finite field|Moore determinant]] [0,&nbsp;1,&nbsp;...,&nbsp;''n''&nbsp;&minus;&nbsp;1] is a product of such linear forms, taken over  1&nbsp;+&nbsp;''q''&nbsp;+&nbsp;''q''<sup>2</sup>&nbsp;+&nbsp;...&nbsp;+&nbsp;''q''<sup>''n''&nbsp;–&nbsp;1</sup> representatives of (''n''&nbsp;–&nbsp;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==
*{{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 | jstor=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 | 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 | mr=0186665 | year=2007}}
*{{Citation | last1=Sanderson | first1=Mildred | title=Formal Modular Invariants with Application to Binary Modular Covariants | jstor=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 | 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}}
 
{{DEFAULTSORT:Modular Invariant Of A Group}}
[[Category:Invariant theory]]

Latest revision as of 19:32, 7 August 2014

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