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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:
==> // coefficients: QQ
==> // 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)
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See also:
symmetricBasis.
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