Modular invariant theory: Difference between revisions

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==Dickson invariant==

When ''G'' is the finite group 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.

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''−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''−1,''i''+1,...,''n'']/[0,1,...,''n''−1] for ''i''=0, 1, ..., ''n''−1.

{{harvtxt|Steinberg|1987}} gave a shorter proof of Dickson's theorem.