Homogenous symmetric functions

class sage.combinat.sf.homogeneous.SymmetricFunctionAlgebra_homogeneous(R)

Bases: sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative

class Element(M, x)

Bases: sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical.Element

expand(n, alphabet='x')

Expands the symmetric function as a symmetric polynomial in n variables.

EXAMPLES:

sage: h = SFAHomogeneous(QQ)
sage: h([3]).expand(2)
x0^3 + x0^2*x1 + x0*x1^2 + x1^3
sage: h([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: h([2,1]).expand(3)
x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3
sage: h([3]).expand(2,alphabet='y')
y0^3 + y0^2*y1 + y0*y1^2 + y1^3
sage: h([3]).expand(2,alphabet='x,y')
x^3 + x^2*y + x*y^2 + y^3
sage: h([3]).expand(3,alphabet='x,y,z')
x^3 + x^2*y + x*y^2 + y^3 + x^2*z + x*y*z + y^2*z + x*z^2 + y*z^2 + z^3

omega()

Returns the image of self under the Frobenius / omega automorphism.

EXAMPLES:

sage: h = SFAHomogeneous(QQ)
sage: a = h([2,1]); a
h[2, 1]
sage: a.omega()
h[1, 1, 1] - h[2, 1]
sage: e = SFAElementary(QQ)
sage: e(h([2,1]).omega())
e[2, 1]

SymmetricFunctionAlgebra_homogeneous.dual_basis(scalar=None, prefix=None)

The dual basis of the homogeneous basis with respect to the standard scalar product is the monomial basis.

EXAMPLES:

sage: m = SFAMonomial(QQ)
sage: h = SFAHomogeneous(QQ)
sage: h.dual_basis() == m
True