Homogeneous symmetric functions

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


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


sage: h = SymmetricFunctions(QQ).e()
sage: h == loads(dumps(h))
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.


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))
expand(n, alphabet='x')

Expands the symmetric function as a symmetric polynomial in \(n\) variables.


  • self – an element of the homogeneous basis of symmetric functions
  • n – a positive integer
  • alphabet – a variable for the expansion (default: \(x\))

OUTPUT: a monomial expansion of an instance of self in \(n\) variables


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

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 sends the power-sum symmetric function \(p_k\) to \((-1)^{k-1} p_k\) for every positive integer \(k\).

The default implementation converts to the Schurs, then performs the automorphism and changes back.


  • the image of self under the omega automorphism


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]

Returns the coproduct on \(h_i\).


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


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


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[]

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