Generic dual bases symmetric functions

class sage.combinat.sf.dual.SymmetricFunctionAlgebra_dual(dual_basis, scalar, scalar_name='', basis_name=None, prefix=None)

Bases: sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical

Generic dual basis of a basis of symmetric functions.

INPUT:

  • dual_basis – a basis of the ring of symmetric functions

  • scalar – A function \(z\) on partitions which determines the scalar product on the power sum basis by \(\langle p_{\mu}, p_{\mu} \rangle = z(\mu)\). (Independently on the function chosen, the power sum basis will always be orthogonal; the function scalar only determines the norms of the basis elements.) This defaults to the function zee defined in sage.combinat.sf.sfa, that is, the function is defined by:

    \[\lambda \mapsto \prod_{i = 1}^\infty m_i(\lambda)! i^{m_i(\lambda)}`,\]

    where \(m_i(\lambda)\) means the number of times \(i\) appears in \(\lambda\). This default function gives the standard Hall scalar product on the ring of symmetric functions.

  • scalar_name – (default: the empty string) a string giving a description of the scalar product specified by the parameter scalar

  • basis_name – (optional) a string to serve as name for the basis to be generated (such as “forgotten” in “the forgotten basis”); don’t set it to any of the already existing basis names (such as homogeneous, monomial, forgotten, etc.).

  • prefix – (default: 'd' and the prefix for dual_basis) a string to use as the symbol for the basis

OUTPUT:

The basis of the ring of symmetric functions dual to the basis dual_basis with respect to the scalar product determined by scalar.

EXAMPLES:

sage: e = SymmetricFunctions(QQ).e()
sage: f = e.dual_basis(prefix = "m", basis_name="Forgotten symmetric functions"); f
Symmetric Functions over Rational Field in the Forgotten symmetric functions basis
sage: TestSuite(f).run(elements = [f[1,1]+2*f[2], f[1]+3*f[1,1]])
sage: TestSuite(f).run() # long time (11s on sage.math, 2011)

This class defines canonical coercions between self and self^*, as follow:

Lookup for the canonical isomorphism from self to \(P\) (=powersum), and build the adjoint isomorphism from \(P^*\) to self^*. Since \(P\) is self-adjoint for this scalar product, derive an isomorphism from \(P\) to self^*, and by composition with the above get an isomorphism from self to self^* (and similarly for the isomorphism self^* to self).

This should be striped down to just (auto?) defining canonical isomorphism by adjunction (as in MuPAD-Combinat), and let the coercion handle the rest.

Inversions may not be possible if the base ring is not a field:

sage: m = SymmetricFunctions(ZZ).m()
sage: h = m.dual_basis(lambda x: 1)
sage: h[2,1]
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

By transitivity, this defines indirect coercions to and from all other bases:

sage: s = SymmetricFunctions(QQ['t'].fraction_field()).s()
sage: t = QQ['t'].fraction_field().gen()
sage: zee_hl = lambda x: x.centralizer_size(t=t)
sage: S = s.dual_basis(zee_hl)
sage: S(s([2,1]))
(-t/(t^5-2*t^4+t^3-t^2+2*t-1))*d_s[1, 1, 1] + ((-t^2-1)/(t^5-2*t^4+t^3-t^2+2*t-1))*d_s[2, 1] + (-t/(t^5-2*t^4+t^3-t^2+2*t-1))*d_s[3]

TESTS:

Regression test for trac ticket #12489. This ticket improving equality test revealed that the conversion back from the dual basis did not strip cancelled terms from the dictionary:

sage: y = e[1, 1, 1, 1] - 2*e[2, 1, 1] + e[2, 2]
sage: sorted(f.element_class(f, dual = y))
[([1, 1, 1, 1], 6), ([2, 1, 1], 2), ([2, 2], 1)]
class Element(A, dictionary=None, dual=None)

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

An element in the dual basis.

INPUT:

At least one of the following must be specified. The one (if any) which is not provided will be computed.

  • dictionary – an internal dictionary for the monomials and coefficients of self
  • dual – self as an element of the dual basis.
dual()

Return self in the dual basis.

OUTPUT:

  • the element self expanded in the dual basis to self.parent()

EXAMPLES:

sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(scalar=zee)
sage: a = h([2,1])
sage: a.parent()
Dual basis to Symmetric Functions over Rational Field in the monomial basis
sage: a.dual()
3*m[1, 1, 1] + 2*m[2, 1] + m[3]
expand(n, alphabet='x')

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

INPUT:

  • 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: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(zee)
sage: a = h([2,1])+h([3])
sage: a.expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.dual().expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.expand(2,alphabet='y')
2*y0^3 + 3*y0^2*y1 + 3*y0*y1^2 + 2*y1^3
sage: a.expand(2,alphabet='x,y')
2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^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 :meth`omega()` method.

OUTPUT:

  • the result of applying omega to self

EXAMPLES:

sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(zee)
sage: hh = SymmetricFunctions(QQ).homogeneous()
sage: hh([2,1]).omega()
h[1, 1, 1] - h[2, 1]
sage: h([2,1]).omega()
d_m[1, 1, 1] - d_m[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 :meth`omega()` method.

OUTPUT:

  • the result of applying omega to self

EXAMPLES:

sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(zee)
sage: hh = SymmetricFunctions(QQ).homogeneous()
sage: hh([2,1]).omega()
h[1, 1, 1] - h[2, 1]
sage: h([2,1]).omega()
d_m[1, 1, 1] - d_m[2, 1]
scalar(x)

Return the standard scalar product of self and x.

INPUT:

  • x – element of the symmetric functions

OUTPUT:

  • the scalar product between x and self

EXAMPLES:

sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(scalar=zee)
sage: a = h([2,1])
sage: a.scalar(a)
2
scalar_hl(x)

Return the Hall-Littlewood scalar product of self and x.

INPUT:

  • x – element of the same dual basis as self

OUTPUT:

  • the Hall-Littlewood scalar product between x and self

EXAMPLES:

sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(scalar=zee)
sage: a = h([2,1])
sage: a.scalar_hl(a)
(t + 2)/(-t^4 + 2*t^3 - 2*t + 1)
SymmetricFunctionAlgebra_dual.transition_matrix(basis, n)

Returns the transition matrix between the \(n^{th}\) homogeneous components of self and basis.

INPUT:

  • basis – a target basis of the ring of symmetric functions
  • n – nonnegative integer

OUTPUT:

  • A transition matrix from self to basis for the elements of degree n. The indexing order of the rows and columns is the order of Partitions(n).

EXAMPLES:

sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym.schur()
sage: e = Sym.elementary()
sage: f = e.dual_basis()
sage: f.transition_matrix(s, 5)
[ 1 -1  0  1  0 -1  1]
[-2  1  1 -1 -1  1  0]
[-2  2 -1 -1  1  0  0]
[ 3 -1 -1  1  0  0  0]
[ 3 -2  1  0  0  0  0]
[-4  1  0  0  0  0  0]
[ 1  0  0  0  0  0  0]
sage: Partitions(5).list()
[[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
sage: s(f[2,2,1])
s[3, 2] - 2*s[4, 1] + 3*s[5]
sage: e.transition_matrix(s, 5).inverse().transpose()
[ 1 -1  0  1  0 -1  1]
[-2  1  1 -1 -1  1  0]
[-2  2 -1 -1  1  0  0]
[ 3 -1 -1  1  0  0  0]
[ 3 -2  1  0  0  0  0]
[-4  1  0  0  0  0  0]
[ 1  0  0  0  0  0  0]

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