Top
Back: primary_invariants_random
Forward: invariant_algebra_perm
FastBack:
FastForward:
Up: finvar_lib
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

D.7.1.5 invariant_algebra_reynolds

Procedure from library finvar.lib (see finvar_lib).

Usage:
invariant_algebra_reynolds(REY[,v]);
REY: a gxn <matrix> representing the Reynolds operator of a finite matrix group, where g ist the group order and n is the number of variables of the basering;
v: an optional <int>

Return:
A minimal homogeneous generating set of the invariant ring, type <matrix>

Assume:
We are in the non-modular case, i.e., the characteristic of the basering does not divide the group order;
REY is the 1st return value of group_reynolds(), reynolds_molien() or the second one of primary_invariants()

Display:
Information on the progress of computations if v does not equal 0

Theory:
We do an incremental search in increasing degree d. Generators of the invariant ring are found among the Reynolds images of monomials of degree d. The generators are chosen by Groebner basis techniques (see S. King: Minimal generating sets of non-modular invariant rings of finite groups).

Note:
invariant_algebra_reynolds should not be used in rings with weighted orders.

Example:
 
LIB "finvar.lib";
ring R=0,(a,b,c,d),dp;
matrix A[4][4]=
0,0,1,0,
0,0,0,1,
1,0,0,0,
0,1,0,0;
list L = group_reynolds(A);
matrix G = invariant_algebra_reynolds(L[1],1);
==>   We have  4  relevant monomials in degree  1
==>     We found generator number  1  in degree  1
==>     We found generator number  2  in degree  1
==> Computing Groebner basis up to the new degree 2
==>   We have  3  relevant monomials in degree  2
==>     We found generator number  3  in degree  2
==>     We found generator number  4  in degree  2
==>     We found generator number  5  in degree  2
==> Computing Groebner basis up to the new degree 3
==> We found the degree bound 2
==> We went beyond the degree bound, so, we are done!
G;
==> G[1,1]=b+d
==> G[1,2]=a+c
==> G[1,3]=b2+d2
==> G[1,4]=ab+cd
==> G[1,5]=a2+c2
See also: invariant_algebra_perm.


Top Back: primary_invariants_random Forward: invariant_algebra_perm FastBack: FastForward: Up: finvar_lib Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 4-0-3, 2016, generated by texi2html.