Singular

D.7.1.1 invariant_ring

Usage:

invariant_ring(G1,G2,...[,flags]);
G1,G2,...: <matrices> generating a finite matrix group, flags: an optional <intvec> with three entries: if the first one equals 0, the program attempts to compute the Molien series and Reynolds operator, if it equals 1, the program is told that the Molien series should not be computed, if it equals -1 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) and if the first one is anything else, it means that the characteristic of the base field divides the group order (i.e. it will not even be attempted to compute the Reynolds operator or Molien series), the second component should give the size of intervals between canceling common factors in the expansion of 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 occurs not among the coefficients)

Return:

primary and secondary invariants for any matrix representation of a finite group

Display:

information about the various stages of the program if the third flag does not equal 0

Theory:

Bases of homogeneous invariants are generated successively and those are chosen as primary invariants that lower the dimension of the ideal generated by the previously found invariants (see "Generating a Noetherian Normalization of the Invariant Ring of a Finite Group" by Decker, Heydtmann, Schreyer (1998)). In the
non-modular case secondary invariants are calculated by finding a basis (in terms of monomials) of the basering modulo the primary invariants, mapping to invariants with the Reynolds operator and using those or their power products such that they are linearly independent modulo the primary invariants (see "Some Algorithms in Invariant Theory of Finite Groups" by Kemper and Steel (1997)). In the modular case they are generated according to "Generating Invariant Rings of Finite Groups over Arbitrary Fields" by Kemper (1996).

LIB "finvar.lib";
ring R=0,(x,y,z),dp;
matrix A[3][3]=0,1,0,-1,0,0,0,0,-1;
matrix P,S,IS=invariant_ring(A);
print(P);
==> z2,x2+y2,x2y2
print(S);
==> 1,xyz,x2z-y2z,x3y-xy3
print(IS);
==> xyz,x2z-y2z,x3y-xy3