|
In mathematics, a modular invariant of a group is an invariant of a finite group 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 . ==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 F''q''(...,''X''''n'' ) in the natural way, found a complete set of invariants as follows. Write (...,''e''''n'' ) for the determinant of the matrix whose entries are ''X'', where ''e''1, ...,''e''''n'' are non-negative integers. For example, the Moore determinant () of order 3 is : 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''''n'' )/() are invariants of GL''n''(F''q''), called Dickson invariants. Dickson proved that the full ring of invariants F''q''(...,''X''''n'' )GL''n''(F''q'') is a polynomial algebra over the ''n'' Dickson invariants ()/() for ''i'' = 0, 1, ..., ''n'' − 1. gave a shorter proof of Dickson's theorem. The matrices (...,''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 () is a product of such linear forms, taken over 1 + ''q'' + ''q''2 + ... + ''q''''n'' – 1 representatives of (''n'' – 1)-dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Modular invariant theory」の詳細全文を読む スポンサード リンク
|