Top
Back: torusInvariants
Forward: diagInvariants
FastBack:
FastForward:
Up: normaliz_lib
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

D.4.26.7 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]=w5
==> _[2]=z7w3
==> _[3]=z14w
==> _[4]=z35
==> _[5]=yw4
==> _[6]=yz7w2
==> _[7]=yz14
==> _[8]=y2w3
==> _[9]=y2z7w
==> _[10]=y3w2
==> _[11]=y3z7
==> _[12]=y4w
==> _[13]=y5
==> _[14]=xz3w
==> _[15]=xz24
==> _[16]=xyz3
==> _[17]=x2z13
==> _[18]=x3z2
==> _[19]=x5zw4
==> _[20]=x5yzw3
==> _[21]=x5y2zw2
==> _[22]=x5y3zw
==> _[23]=x5y4z
==> _[24]=x7w3
==> _[25]=x7yw2
==> _[26]=x7y2w
==> _[27]=x7y3
==> _[28]=x12zw2
==> _[29]=x12yzw
==> _[30]=x12y2z
==> _[31]=x14w
==> _[32]=x14y
==> _[33]=x19z
==> _[34]=x35
See also: diagInvariants; intersectionValRingIdeals; intersectionValRings; torusInvariants.

Top Back: torusInvariants Forward: diagInvariants FastBack: FastForward: Up: normaliz_lib Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 4.3.2, 2023, generated by texi2html.