Content deleted Content added
m R.e.b. moved page Modular invariant of a group to Modular invariant theory: Better title |
No edit summary |
||
Line 4:
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
:<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<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, 1, ...,''n'' − 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, 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.
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, 1, ..., ''n'' − 1] is a product of such linear forms, taken over 1 + ''q'' + ''q''<sup>2</sup> + ... + ''q''<sup>''n''
[[pt:Invariante modular]]
| |||