Singular Manual: symmetricBasis

D.3.1.29 symmetricBasis

Procedure from library matrix.lib (see matrix_lib).

Return:: ring, poynomial ring containing the ideal "symBasis", being a basis of the k-th symmetric 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 = symmetricBasis(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
symBasis; // symmetric basis:
==> symBasis[1]=@e(4)^3
==> symBasis[2]=@e(3)*@e(4)^2
==> symBasis[3]=@e(3)^2*@e(4)
==> symBasis[4]=@e(3)^3
==> symBasis[5]=@e(2)*@e(4)^2
==> symBasis[6]=@e(2)*@e(3)*@e(4)
==> symBasis[7]=@e(2)*@e(3)^2
==> symBasis[8]=@e(2)^2*@e(4)
==> symBasis[9]=@e(2)^2*@e(3)
==> symBasis[10]=@e(2)^3
==> symBasis[11]=@e(1)*@e(4)^2
==> symBasis[12]=@e(1)*@e(3)*@e(4)
==> symBasis[13]=@e(1)*@e(3)^2
==> symBasis[14]=@e(1)*@e(2)*@e(4)
==> symBasis[15]=@e(1)*@e(2)*@e(3)
==> symBasis[16]=@e(1)*@e(2)^2
==> symBasis[17]=@e(1)^2*@e(4)
==> symBasis[18]=@e(1)^2*@e(3)
==> symBasis[19]=@e(1)^2*@e(2)
==> symBasis[20]=@e(1)^3