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D.4.16.6 torusInvariants
Procedure from library normaliz.lib (see normaliz_lib).
- Usage:
- torusInvariants(intmat A);
- Return:
- Returns an ideal representing the list of monomials generating the ring of
invariants as an algebra over the coefficient field.
@tex
$R^T$.
@end tex
The function returns the ideal given by the input matrix A if one of
the options supp, triang, or hvect has been
activated.
However, in this case some numerical invariants are computed, and
some other data may be contained in files that you can read into
Singular.
- Background:
- @tex
Let $T = (K^*)^r$ be the $r$-dimensional torus acting on the polynomial ring
$R = K[X_1 ,\ldots,X_n]$ diagonally. Such an action can be described as
follows: there are integers $a_{i,j}$, $i=1,\ldots,r$, $j=1,\ldots,n$, such
that $(\lambda_1,\ldots,\lambda_r)\in T$ acts by the substitution
$$ X_j \mapsto \lambda_1^{a_{1,j}} \cdots \lambda_r^{a_{r,j}}X_j,
\quad j=1,\ldots,n.$$
In order to compute the ring of invariants $R^T$ one must specify the matrix
$A=(a_{i,j})$.
@end tex
Example:
| | LIB "normaliz.lib";
ring R=0,(x,y,z,w),dp;
intmat E[2][4] = -1,-1,2,0, 1,1,-2,-1;
torusInvariants(E);
==> _[1]=x2z
==> _[2]=xyz
==> _[3]=y2z
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See also:
diagInvariants;
finiteDiagInvariants;
intersectionValRingIdeals;
intersectionValRings.
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