Modular invariant theory: Difference between revisions

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{{Short description|Sub-field of mathematics}}

In [[mathematics]], a '''modular invariant''' of a [[group (mathematics)|group]] is an invariant of a [[finite group]] [[groupGroup action (mathematics)|acting]] on a [[vector space]] of positive characteristic (usually dividing the [[order (group theory)|order]] of the group). The study of modular invariants was originated in about 1914 by {{harvtxt|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 (mathematics)|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 (mathematics)|matrix]] whose entries are ''X''{{su|b=''i''|p=''q''^''e''_''j''}}, where ''e''₁, ..., ''e''_''n'' are non-negative integers[[integer]]s. For example, the [[Moore determinant over a finite field|Moore determinant]] [0,1,2] of order 3 is

:<math>\begin{vmatrix} x_1 & x_2x_1^q & x_3\\x_1^{q^2}\\x_2 & x_2^q & x_3^q\\x_1x_2^{q^2}\\x_3 & x_2x_3^{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 ratios [''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.

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'''_''pq'') and the ratios [''e''₁, ...,''e''_''n'']&thinsp;/&thinsp;[0,&nbsp;1,&nbsp;..., ''n''&minusnbsp;−&thinsp;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,&nbsp;1,&nbsp;..., ''i''&minusnbsp;−&nbsp;1,&nbsp;''i''&nbsp;+&nbsp;1,&nbsp;...,&nbsp;''n'']&thinsp;/&thinsp;[0, 1, ..., ''n''&minusnbsp;−&thinsp;1] for ''i''&nbsp;=&nbsp;0, 1, ..., ''n''&minusnbsp;−&nbsp;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,&nbsp;1,&nbsp;...,&nbsp;''n''&minusnbsp;−&thinsp;1] is a product of such linear forms, taken over 1&nbsp;+&nbsp;''q''&nbsp;+&nbsp;''q''²&nbsp;+&nbsp;...&nbsp;+&nbsp;''q''^{''n''–1&nbsp;–&nbsp;1} representatives of (''n–1n'' &nbsp;–&thinsp;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 | doi=10.2307/1988736| doi-access=free }}

*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=On invariants and the theory of numbers | origyearorig-date=1914 | url=httphttps://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 | authorlink=Daniel Edwin Rutherford | title=Modular invariants | origyearorig-date=1932 | url=httphttps://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 | authorlink=Mildred Sanderson | 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 | doi=10.2307/1988702| doi-access=free }}

*{{Citation | last1=Steinberg | first1=Robert | authorlink=Robert Steinberg | 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 | access-date=2010-12-02 | archive-url=https://web.archive.org/web/20120305205421/http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/1682/1/jfs340309.pdf | archive-date=2012-03-05 | url-status=dead }}

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