Elementary symmetric functions

class sage.combinat.sf.elementary.SymmetricFunctionAlgebra_elementary(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 variables.

EXAMPLES:

sage: e = SFAElementary(QQ)
sage: e([2,1]).expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 3*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: e([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: e([3]).expand(2)
0
sage: e([2]).expand(3)
x0*x1 + x0*x2 + x1*x2
sage: e([3]).expand(4,alphabet='x,y,z,t')
x*y*z + x*y*t + x*z*t + y*z*t
sage: e([3]).expand(4,alphabet='y')
y0*y1*y2 + y0*y1*y3 + y0*y2*y3 + y1*y2*y3

omega()

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

EXAMPLES:

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

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