Singular Manual: exteriorBasis

D.3.1.30 exteriorBasis

Procedure from library matrix.lib (see matrix_lib).

Return:: qring, an exterior algebra containing the ideal "extBasis", being a basis of the k-th exterior power of an n-dim vector space.
Note:: The output polynomial ring has characteristics 0 and n variables named "S(i)", where the base variable name S is either given by the optional string argument(which must not contain brackets) or equal to "e" by default.

Example:

LIB "matrix.lib";
// basis of the 3-rd symmetricPower of a 4-dim vector space:
def r = exteriorBasis(4, 3, "@e"); setring r;
r; // container ring:
==> //   characteristic : 0
==> //   number of vars : 4
==> //        block   1 : ordering dp
==> //                  : names    @e(1) @e(2) @e(3) @e(4)
==> //        block   2 : ordering C
==> //   noncommutative relations:
==> //    @e(2)@e(1)=-@e(1)*@e(2)
==> //    @e(3)@e(1)=-@e(1)*@e(3)
==> //    @e(4)@e(1)=-@e(1)*@e(4)
==> //    @e(3)@e(2)=-@e(2)*@e(3)
==> //    @e(4)@e(2)=-@e(2)*@e(4)
==> //    @e(4)@e(3)=-@e(3)*@e(4)
==> // quotient ring from ideal
==> _[1]=@e(4)^2
==> _[2]=@e(3)^2
==> _[3]=@e(2)^2
==> _[4]=@e(1)^2
extBasis; // exterior basis:
==> extBasis[1]=@e(2)*@e(3)*@e(4)
==> extBasis[2]=@e(1)*@e(3)*@e(4)
==> extBasis[3]=@e(1)*@e(2)*@e(4)
==> extBasis[4]=@e(1)*@e(2)*@e(3)