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7.7.9.0. fullSerreRelations
Procedure from library freegb.lib (see freegb_lib).
- Usage:
- fullSerreRelations(A,N,C,P,d); A an intmat, N,C,P ideals, d an int
- Return:
- ring (and ideal)
- Purpose:
- compute the inhomogeneous Serre's relations associated to A in given
variable names
- Assume:
- three ideals in the input are of the same sizes and contain merely
variables which are interpreted as follows: N resp. P stand for negative
resp. positive roots, C stand for Cartan elements. d is the degree bound for
letterplace ring, which will be returned.
The matrix A is a generalized Cartan matrix with integer entries
The result is the ideal called 'fsRel' in the returned ring.
Example:
| | LIB "freegb.lib";
intmat A[2][2] =
2, -1,
-1, 2; // A_2 = sl_3 Cartan matrix
ring r = 0,(f1,f2,h1,h2,e1,e2),dp;
ideal negroots = f1,f2; ideal cartans = h1,h2; ideal posroots = e1,e2;
int uptodeg = 5;
def RS = fullSerreRelations(A,negroots,cartans,posroots,uptodeg);
setring RS; fsRel;
==> fsRel[1]=f1(1)*f2(2)*f2(3)-2*f2(1)*f1(2)*f2(3)+f2(1)*f2(2)*f1(3)
==> fsRel[2]=f1(1)*f1(2)*f2(3)-2*f1(1)*f2(2)*f1(3)+f2(1)*f1(2)*f1(3)
==> fsRel[3]=e1(1)*e2(2)*e2(3)-2*e2(1)*e1(2)*e2(3)+e2(1)*e2(2)*e1(3)
==> fsRel[4]=e1(1)*e1(2)*e2(3)-2*e1(1)*e2(2)*e1(3)+e2(1)*e1(2)*e1(3)
==> fsRel[5]=f2(1)*e1(2)-e1(1)*f2(2)
==> fsRel[6]=f1(1)*e2(2)-e2(1)*f1(2)
==> fsRel[7]=-f1(1)*e1(2)+e1(1)*f1(2)-h1(1)
==> fsRel[8]=-f2(1)*e2(2)+e2(1)*f2(2)-h2(1)
==> fsRel[9]=h1(1)*h2(2)-h2(1)*h1(2)
==> fsRel[10]=h1(1)*e1(2)-e1(1)*h1(2)-2*e1(1)
==> fsRel[11]=f1(1)*h1(2)-h1(1)*f1(2)-2*f1(1)
==> fsRel[12]=h1(1)*e2(2)-e2(1)*h1(2)+e2(1)
==> fsRel[13]=f2(1)*h1(2)-h1(1)*f2(2)+f2(1)
==> fsRel[14]=h2(1)*e1(2)-e1(1)*h2(2)+e1(1)
==> fsRel[15]=f1(1)*h2(2)-h2(1)*f1(2)+f1(1)
==> fsRel[16]=h2(1)*e2(2)-e2(1)*h2(2)-2*e2(1)
==> fsRel[17]=f2(1)*h2(2)-h2(1)*f2(2)-2*f2(1)
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