Modular invariant theory: Difference between revisions

Revision as of 21:26, 11 November 2020

Revision as of 23:12, 7 December 2020

Line 1:

In [[mathematics]], a '''modular invariant''' of a [[group (mathematics)|group]] is an invariant of a [[finite group]] [[Group 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_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}</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'''_''q'') and the ratios [''e''₁, ...,''e''_''n'']&thinsp;/&thinsp;[0,&nbsp;1,&nbsp;..., ''n''&nbsp;−&minus;&nbspthinsp;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''&nbsp;&minus;−&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''&nbsp;&minus;−&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''&nbsp;−&minus;&nbspthinsp;1] is a product of such linear forms, taken over 1&nbsp;+&nbsp;''q''&nbsp;+&nbsp;''q''²&nbsp;+&nbsp;...&nbsp;+&nbsp;''q''^{''n''&nbsp;–&nbsp;1} representatives of (''n''&nbsp;–&nbspthinsp;1)-dimensional [[projective space]] over the field. This factorization is similar to the factorization of the [[Vandermonde determinant]] into linear factors.

==See also==