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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}}.
==Dickson invariant==
When ''G'' is the finite group general linear group GL<sub>''n''</sub>('''F'''<sub>''p''</sub>) over the finite field '''F'''<sub>''p''</sub> acting on the ring '''F'''<sub>''p''</sub>[''X''<sub>1</sub>, ... ,''X''<sub>''n''</sub>] in the natural way, Dickson 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=''p''<sup>''e''<sub>''j''</sub></sup>}}, where ''e''<sub>1</sub>, ... ,''e''<sub>''n''</sub> are non-negative integers.
Then under the action of an element ''g'' of GL<sub>''n''</sub>('''F'''<sub>''p''</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 of any two are invariants of GL<sub>''n''</sub>('''F'''<sub>''p''</sub>), called '''Dickson invariants'''. Dickson proved that the full ring of invariants 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.
==See also==
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