Singular

7.7.12.0. CenCharDec

Usage:

CenCharDec(I, C); I a module, C an ideal

Assume:

C consists of generators of the center of the base ring

Return:

a list L, where each entry consists of three records (if a finite decomposition exists)
L[*][1] ('ideal' type), the central character as a maximal ideal in the center,
L[*][2] ('module' type), the Groebner basis of the weight module, corresponding to the character in L[*][1],
L[*][3] ('int' type) is the vector space dimension of the weight module (-1 in case of infinite dimension);

Purpose:

compute a finite decomposition of C into central characters or determine that there is no finite decomposition

Note:

actual decomposition is the sum of L[i][2] above;
some modules have no finite decomposition (in such case one gets warning message)
The function central in central.lib may be used to obtain C, when needed.

LIB "ncdecomp.lib";
printlevel=0;
option(returnSB);
def a = makeUsl2(); // U(sl_2) in characteristic 0
setring a;
ideal I = e3,f3,h3-4*h;
I = twostd(I);           // two-sided ideal generated by I
vdim(I);                 // it is finite-dimensional
==> 10
ideal Cn = 4*e*f+h^2-2*h; // the only central element
list T = CenCharDec(I,Cn);
T;
==> [1]:
==>    [1]:
==>       _[1]=4ef+h2-2h-8
==>    [2]:
==>       _[1]=h
==>       _[2]=f
==>       _[3]=e
==>    [3]:
==>       1
==> [2]:
==>    [1]:
==>       _[1]=4ef+h2-2h
==>    [2]:
==>       _[1]=4ef+h2-2h-8
==>       _[2]=h3-4h
==>       _[3]=fh2-2fh
==>       _[4]=eh2+2eh
==>       _[5]=f2h-2f2
==>       _[6]=e2h+2e2
==>       _[7]=f3
==>       _[8]=e3
==>    [3]:
==>       9
// consider another example
ideal J = e*f*h;
CenCharDec(J,Cn);
==> There is no finite decomposition
==> 0