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
class Element

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

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: 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
sage: e([]).expand(0)
1
sage: (3*e([])).expand(0)
3
omega()

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism ω of the ring of symmetric functions that satisfies ω(ek)=hk for all positive integers k (where ek stands for the k-th elementary symmetric function, and hk 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 pk to (1)k1pk for every positive integer k.

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

ω(eλ)=hλ,ω(hλ)=eλ,ω(pλ)=(1)|λ|(λ)pλ,ω(sλ)=sλ,

where λ is any partition, where (λ) denotes the length (length()) of the partition λ, where λ denotes the conjugate partition (conjugate()) of λ, 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.

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]
omega_involution()

Return the image of self under the omega automorphism.

The omega automorphism is defined to be the unique algebra endomorphism ω of the ring of symmetric functions that satisfies ω(ek)=hk for all positive integers k (where ek stands for the k-th elementary symmetric function, and hk 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 pk to (1)k1pk for every positive integer k.

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

ω(eλ)=hλ,ω(hλ)=eλ,ω(pλ)=(1)|λ|(λ)pλ,ω(sλ)=sλ,

where λ is any partition, where (λ) denotes the length (length()) of the partition λ, where λ denotes the conjugate partition (conjugate()) of λ, 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.

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]
verschiebung(n)

Return the image of the symmetric function self under the n-th Verschiebung operator.

The n-th Verschiebung operator Vn is defined to be the unique algebra endomorphism V of the ring of symmetric functions that satisfies V(hr)=hr/n for every positive integer r divisible by n, and satisfies V(hr)=0 for every positive integer r not divisible by n. This operator Vn is a Hopf algebra endomorphism. For every nonnegative integer r with nr, it satisfies

Vn(hr)=hr/n,Vn(pr)=npr/n,Vn(er)=(1)rr/ner/n

(where h is the complete homogeneous basis, p is the powersum basis, and e is the elementary basis). For every nonnegative integer r with nr, it satisfes

Vn(hr)=Vn(pr)=Vn(er)=0.

The n-th Verschiebung operator is also called the n-th Verschiebung endomorphism. Its name derives from the Verschiebung (German for “shift”) endomorphism of the Witt vectors.

The n-th Verschiebung operator is adjoint to the n-th Frobenius operator (see frobenius() for its definition) with respect to the Hall scalar product (scalar()).

The action of the n-th Verschiebung operator on the Schur basis can also be computed explicitly. The following (probably clumsier than necessary) description can be obtained by solving exercise 7.61 in Stanley [STA].

Let λ be a partition. Let n be a positive integer. If the n-core of λ is nonempty, then Vn(sλ)=0. Otherwise, the following method computes Vn(sλ): Write the partition λ in the form (λ1,λ2,...,λns) for some nonnegative integer s. (If n does not divide the length of λ, then this is achieved by adding trailing zeroes to λ.) Set βi=λi+nsi for every s{1,2,,ns}. Then, (β1,β2,...,βns) is a strictly decreasing sequence of nonnegative integers. Stably sort the list (1,2,,ns) in order of (weakly) increasing remainder of 1βi modulo n. Let ξ be the sign of the permutation that is used for this sorting. Let ψ be the sign of the permutation that is used to stably sort the list (1,2,,ns) in order of (weakly) increasing remainder of i1 modulo n. (Notice that ψ=(1)n(n1)s(s1)/4.) Then, Vn(sλ)=ξψn1i=0sλ(i), where (λ(0),λ(1),,λ(n1)) is the n-quotient of λ.

INPUT:

  • n – a positive integer

OUTPUT:

The result of applying the n-th Verschiebung operator (on the ring of symmetric functions) to self.

EXAMPLES:

sage: Sym = SymmetricFunctions(ZZ)
sage: e = Sym.e()
sage: e[3].verschiebung(2)
0
sage: e[4].verschiebung(4)
-e[1]

The Verschiebung endomorphisms are multiplicative:

sage: all( all( e(lam).verschiebung(2) * e(mu).verschiebung(2)
....:           == (e(lam) * e(mu)).verschiebung(2)
....:           for mu in Partitions(4) )
....:      for lam in Partitions(4) )
True
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[]