Modular Invariant Theory

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''₁, ...,''X''_''n''in the natural way, found a complete set of invariants as follows. Write 'e''₁, ..., ''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 ,1,2of order 3 is

:$\begin x_1 & x_1^q & x_1^\\x_2 & x_2^q & x_2^\\x_3 & x_3^q & x_3^ \end$

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;/  , 1, ..., ''n'' − 1are invariants of GL_''n''(F_''q''), called Dickson invariants. Dickson proved that the full ring of invariants F_''q'' 'X''₁, ...,''X''_''n''sup>GL_''n''(F_''q'') is a polynomial algebra over the ''n'' Dickson invariants , 1, ..., ''i'' − 1, ''i'' + 1, ..., ''n''thinsp;/  , 1, ..., ''n'' − 1for ''i'' = 0, 1, ..., ''n'' − 1.
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 , 1, ..., ''n'' − 1is 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.

See also

* Sanderson's theorem

References

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{{DEFAULTSORT:Modular Invariant Of A Group
Invariant theory