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D.4.19.7 finiteDiagInvariants

Procedure from library normaliz.lib (see normaliz_lib).

Usage:
finiteDiagInvariants(intmat U);
finiteDiagInvariants(intmat U, intvec grading);

Return:
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 $i=1,\ldots,w\}$.


The function returns the ideal given by the input matrix C if one of the options supp, triang, volume, or hseries 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 (see showNuminvs, exportNuminvs).

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]=x3z2
==> _[2]=xz3w
==> _[3]=xyz3
==> _[4]=x2z13
==> _[5]=x19z
==> _[6]=x14w
==> _[7]=z14w
==> _[8]=yz14
==> _[9]=x14y
==> _[10]=xz24
==> _[11]=x12zw2
==> _[12]=x12yzw
==> _[13]=x12y2z
==> _[14]=x7w3
==> _[15]=z7w3
==> _[16]=yz7w2
==> _[17]=x7yw2
==> _[18]=y2z7w
==> _[19]=x7y2w
==> _[20]=y3z7
==> _[21]=x7y3
==> _[22]=x5zw4
==> _[23]=x5yzw3
==> _[24]=x5y2zw2
==> _[25]=x5y3zw
==> _[26]=x5y4z
==> _[27]=x35
==> _[28]=y5
==> _[29]=z35
==> _[30]=w5
==> _[31]=yw4
==> _[32]=y2w3
==> _[33]=y3w2
==> _[34]=y4w
See also: diagInvariants; intersectionValRingIdeals; intersectionValRings; torusInvariants.

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