Multiplicative symmetric functions

A realization \(h\) of the ring of symmetric functions is multiplicative if for a partition \(\lambda = (\lambda_1,\lambda_2,\ldots)\) we have \(h_\lambda = h_{\lambda_1} h_{\lambda_2} \cdots\).

class sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative(Sym, basis_name=None, prefix=None, graded=True)

Bases: SymmetricFunctionAlgebra_classical

The class of multiplicative bases of the ring of symmetric functions.

A realization \(q\) of the ring of symmetric functions is multiplicative if for a partition \(\lambda = (\lambda_1,\lambda_2,\ldots)\) we have \(q_\lambda = q_{\lambda_1} q_{\lambda_2} \cdots\) (with \(q_0\) meaning \(1\)).

Examples of multiplicative realizations are the elementary symmetric basis, the complete homogeneous basis, the powersum basis (if the base ring is a \(\QQ\)-algebra), and the Witt basis (but not the Schur basis or the monomial basis).

coproduct_on_basis(mu)

Return the coproduct on a basis element for multiplicative bases.

INPUT:

• mu – a partition

OUTPUT:

• the image of self[mu] under comultiplication; this is an element of the tensor square of self

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(QQ)
sage: p = Sym.powersum()
sage: p.coproduct_on_basis([2,1])
p[] # p[2, 1] + p[1] # p[2] + p[2] # p[1] + p[2, 1] # p[]

sage: e = Sym.elementary()
sage: e.coproduct_on_basis([3,1])
e[] # e[3, 1] + e[1] # e[2, 1] + e[1] # e[3] + e[1, 1] # e[2] + e[2] # e[1, 1] + e[2, 1] # e[1] + e[3] # e[1] + e[3, 1] # e[]

sage: h = Sym.homogeneous()
sage: h.coproduct_on_basis([3,1])
h[] # h[3, 1] + h[1] # h[2, 1] + h[1] # h[3] + h[1, 1] # h[2] + h[2] # h[1, 1] + h[2, 1] # h[1] + h[3] # h[1] + h[3, 1] # h[]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(QQ)
>>> p = Sym.powersum()
>>> p.coproduct_on_basis([Integer(2),Integer(1)])
p[] # p[2, 1] + p[1] # p[2] + p[2] # p[1] + p[2, 1] # p[]

>>> e = Sym.elementary()
>>> e.coproduct_on_basis([Integer(3),Integer(1)])
e[] # e[3, 1] + e[1] # e[2, 1] + e[1] # e[3] + e[1, 1] # e[2] + e[2] # e[1, 1] + e[2, 1] # e[1] + e[3] # e[1] + e[3, 1] # e[]

>>> h = Sym.homogeneous()
>>> h.coproduct_on_basis([Integer(3),Integer(1)])
h[] # h[3, 1] + h[1] # h[2, 1] + h[1] # h[3] + h[1, 1] # h[2] + h[2] # h[1, 1] + h[2, 1] # h[1] + h[3] # h[1] + h[3, 1] # h[]

product_on_basis(left, right)

Return the product of left and right.

INPUT:

• left, right – partitions

OUTPUT: an element of self

EXAMPLES:

Sage

sage: e = SymmetricFunctions(QQ).e()
sage: e([2,1])^2  # indirect doctest
e[2, 2, 1, 1]

Python

>>> from sage.all import *
>>> e = SymmetricFunctions(QQ).e()
>>> e([Integer(2),Integer(1)])**Integer(2)  # indirect doctest
e[2, 2, 1, 1]

Sage

sage: h = SymmetricFunctions(QQ).h()
sage: h([2,1])^2
h[2, 2, 1, 1]

Python

>>> from sage.all import *
>>> h = SymmetricFunctions(QQ).h()
>>> h([Integer(2),Integer(1)])**Integer(2)
h[2, 2, 1, 1]

Sage

sage: p = SymmetricFunctions(QQ).p()
sage: p([2,1])^2
p[2, 2, 1, 1]

Python

>>> from sage.all import *
>>> p = SymmetricFunctions(QQ).p()
>>> p([Integer(2),Integer(1)])**Integer(2)
p[2, 2, 1, 1]

Sage

sage: QQx.<x> = QQ[]
sage: p = SymmetricFunctions(QQx).p()
sage: (x*p([2]))^2
x^2*p[2, 2]

sage: TestSuite(p).run() # to silence sage -coverage

Python

>>> from sage.all import *
>>> QQx = QQ['x']; (x,) = QQx._first_ngens(1)
>>> p = SymmetricFunctions(QQx).p()
>>> (x*p([Integer(2)]))**Integer(2)
x^2*p[2, 2]

>>> TestSuite(p).run() # to silence sage -coverage