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D.4.16.7 finiteDiagInvariants
Procedure from library normaliz.lib (see normaliz_lib).
- Usage:
- finiteDiagInvariants(intmat U);
- Return:
- @tex
This function computes the ring of invariants of a finite abelian group $G$
acting diagonally on the surrounding polynomial ring $K[X_1,...,X_n]$. The
group is the direct product of cyclic groups generated by finitely many
elements $g_1,...,g_w$. The element $g_i$ acts on the indeterminate $X_j$ by
$g_i(X_j)=\lambda_i^{u_{ij}}X_j$ where $\lambda_i$ is a primitive root of
unity of order equal to $ord(g_i)$. The ring of invariants is generated by all
monomials satisfying the system
$u_{i1}a_1+\ldots+u_{in} a_n \equiv 0$ mod ord$(g_i)$, $i=1,\ldots,w$.
The input to the function is the $w\times(n+1)$ matrix $U$ with rows
$u_{i1}\ldots u_{in}$ ord$(gi)$, $i=1,\ldots,w$. The output is a monomial ideal
listing the algebra generators of the subalgebra of invariants
{$R^G=\{f\in R : g_i f = f$ for all $i=1,\ldots,w\}$}.
@end tex
The function returns the ideal given by the input matrix C 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.
- Note:
Example:
| LIB "normaliz.lib";
ring R = 0,(x,y,z,w),dp;
intmat C[2][5] = 1,1,1,1,5, 1,0,2,0,7;
finiteDiagInvariants(C);
==> _[1]=x35
==> _[2]=x19z
==> _[3]=x14y
==> _[4]=x12y2z
==> _[5]=x14w
==> _[6]=x12yzw
==> _[7]=x12zw2
==> _[8]=x7y3
==> _[9]=x5y4z
==> _[10]=x7y2w
==> _[11]=x5y3zw
==> _[12]=x7yw2
==> _[13]=x5y2zw2
==> _[14]=x7w3
==> _[15]=x5yzw3
==> _[16]=x5zw4
==> _[17]=y5
==> _[18]=y4w
==> _[19]=y3w2
==> _[20]=y2w3
==> _[21]=yw4
==> _[22]=w5
==> _[23]=y3z7
==> _[24]=y2z7w
==> _[25]=yz7w2
==> _[26]=z7w3
==> _[27]=x3z2
==> _[28]=xyz3
==> _[29]=xz3w
==> _[30]=x2z13
==> _[31]=yz14
==> _[32]=z14w
==> _[33]=xz24
==> _[34]=z35
| See also:
diagInvariants;
exportNuminvs;
intersectionValRingIdeals;
intersectionValRings;
showNuminvs;
torusInvariants.
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