#### 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

See also: exteriorBasis.

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