Elementary symmetric functions

class sage.combinat.sf.elementary.SymmetricFunctionAlgebra_elementary(Sym)

Bases: sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative

A class for methods for the elementary basis of the symmetric functions.

INPUT:

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

TESTS:

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

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

INPUT:

  • self – an element of the elementary basis of the 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

EXAMPLES:

sage: e = SymmetricFunctions(QQ).e()
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
sage: e([]).expand(2)
1
omega()

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

INPUT:

  • self – an element of the elementary basis of the symmetric functions

OUTPUT:

  • returns the image of the element under the omega automorphism

EXAMPLES:

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

Returns the coproduct on self[i].

INPUT:

  • self – an elementary basis of the symmetric functions
  • i – a nonnegative integer

OUTPUT:

  • returns the coproduct on the elementary generator e(i)

EXAMPLES:

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

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