Homogeneous symmetric functions

By this we mean the basis formed of the complete homogeneous symmetric functions h_\lambda, not an arbitrary graded basis.

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

Bases: sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative

A class of methods specific to the homogeneous basis of symmetric functions.

INPUT:

  • self – a homogeneous basis of symmetric functions
  • Sym – an instance of the ring of symmetric functions

TESTS:

sage: h = SymmetricFunctions(QQ).e()
sage: h == loads(dumps(h))
True
sage: TestSuite(h).run(skip=['_test_associativity', '_test_distributivity', '_test_prod'])
sage: TestSuite(h).run(elements = [h[1,1]+h[2], h[1]+2*h[1,1]])
class Element(M, x)

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

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
sage: f == loads(dumps(f))
True
expand(n, alphabet='x')

Expand the symmetric function self as a symmetric polynomial in n variables.

INPUT:

  • n – a nonnegative integer
  • alphabet – (default: 'x') a variable for the expansion

OUTPUT:

A monomial expansion of self in the n variables labelled by alphabet.

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
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
sage: (h([]) + 2*h([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
omega()

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism \omega of the ring of symmetric functions that satisfies \omega(e_k) = h_k for all positive integers k (where e_k stands for the k-th elementary symmetric function, and h_k stands for the k-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function p_k to (-1)^{k-1} p_k for every positive integer k.

The images of some bases under the omega automorphism are given by

\omega(e_{\lambda}) = h_{\lambda}, \qquad
\omega(h_{\lambda}) = e_{\lambda}, \qquad
\omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
p_{\lambda}, \qquad
\omega(s_{\lambda}) = s_{\lambda^{\prime}},

where \lambda is any partition, where \ell(\lambda) denotes the length (length()) of the partition \lambda, where \lambda^{\prime} denotes the conjugate partition (conjugate()) of \lambda, and where the usual notations for bases are used (e = elementary, h = complete homogeneous, p = powersum, s = Schur).

omega_involution() is a synonym for the omega() method.

OUTPUT:

  • the image of self under the omega automorphism

EXAMPLES:

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

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism \omega of the ring of symmetric functions that satisfies \omega(e_k) = h_k for all positive integers k (where e_k stands for the k-th elementary symmetric function, and h_k stands for the k-th complete homogeneous symmetric function). It furthermore is a Hopf algebra endomorphism and an involution, and it is also known as the omega involution. It sends the power-sum symmetric function p_k to (-1)^{k-1} p_k for every positive integer k.

The images of some bases under the omega automorphism are given by

\omega(e_{\lambda}) = h_{\lambda}, \qquad
\omega(h_{\lambda}) = e_{\lambda}, \qquad
\omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
p_{\lambda}, \qquad
\omega(s_{\lambda}) = s_{\lambda^{\prime}},

where \lambda is any partition, where \ell(\lambda) denotes the length (length()) of the partition \lambda, where \lambda^{\prime} denotes the conjugate partition (conjugate()) of \lambda, and where the usual notations for bases are used (e = elementary, h = complete homogeneous, p = powersum, s = Schur).

omega_involution() is a synonym for the omega() method.

OUTPUT:

  • the image of self under the omega automorphism

EXAMPLES:

sage: h = SymmetricFunctions(QQ).h()
sage: a = h([2,1]); a
h[2, 1]
sage: a.omega()
h[1, 1, 1] - h[2, 1]
sage: e = SymmetricFunctions(QQ).e()
sage: e(h([2,1]).omega())
e[2, 1]
SymmetricFunctionAlgebra_homogeneous.coproduct_on_generators(i)

Returns the coproduct on h_i.

INPUT:

  • self – a homogeneous basis of symmetric functions
  • i – a nonnegative integer

OUTPUT:

  • the sum \sum_{r=0}^i h_r \otimes h_{i-r}

EXAMPLES:

sage: Sym = SymmetricFunctions(QQ)
sage: h = Sym.homogeneous()
sage: h.coproduct_on_generators(2)
h[] # h[2] + h[1] # h[1] + h[2] # h[]
sage: h.coproduct_on_generators(0)
h[] # h[]