Singular

D.7.1.10 reynolds_molien

Usage:

reynolds_molien(G1,G2,...[,ringname,flags]);
G1,G2,...: nxn <matrices> generating a finite matrix group, ringname: a <string> giving a name for a new ring of characteristic 0 for the Molien series in case of prime characteristic, flags: an optional <intvec> with three components: if the first element is not equal to 0 characteristic 0 is simulated, i.e. the Molien series is computed as if the base field were characteristic 0 (the user must choose a field of large prime characteristic, e.g. 32003) the second component should give the size of intervals between canceling common factors in the expansion of the Molien series, 0 (the default) means only once after generating all terms, in prime characteristic also a negative number can be given to indicate that common factors should always be canceled when the expansion is simple (the root of the extension field does not occur among the coefficients)

Assume:

n is the number of variables of the basering, G1,G2... are the group elements generated by group_reynolds(), g is the size of the group

Return:

a gxn <matrix> representing the Reynolds operator is the first return value and in case of characteristic 0 a 1x2 <matrix> giving enumerator and denominator of Molien series is the second one; in case of prime characteristic a ring with the name `ringname` of characteristic 0 is created where the same Molien series (named M) is stored

Display:

information if the third component of flags does not equal 0

Theory:

The entire matrix group is generated by getting all left products of the generators with new elements from the last run through the loop (or the generators themselves during the first run). All the ones that have been generated before are thrown out and the program terminates when are no new elements found in one run. Additionally each time a new group element is found the corresponding ring mapping of which the Reynolds operator is made up is generated. They are stored in the rows of the first return value. In characteristic 0 the terms 1/det(1-xE) is computed whenever a new element E is found. In prime characteristic a Brauer lift is involved and the terms are only computed after the entire matrix group is generated (to avoid the modular case). The returned matrix gives enumerator and denominator of the expanded version where common factors have been canceled.

LIB "finvar.lib";
"         note the case of prime characteristic"; 
==>          note the case of prime characteristic
ring R=0,(x,y,z),dp;
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
matrix REY,M=reynolds_molien(A);
print(REY);
==> y, -x,-z,
==> -x,-y,z, 
==> -y,x, -z,
==> x, y, z  
print(M);
==> x3+x2-x+1,-x7+x6+x5-x4+x3-x2-x+1
ring S=3,(x,y,z),dp;
string newring="Qadjoint";
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
matrix REY=reynolds_molien(A,newring);
print(REY);
==> y, -x,-z,
==> -x,-y,z, 
==> -y,x, -z,
==> x, y, z  
setring Finvar::Qadjoint;
print(M);
==> x3+x2-x+1,-x7+x6+x5-x4+x3-x2-x+1
setring S;
kill Finvar::Qadjoint;