# 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 $$h_\lambda = h_{\lambda_1} h_{\lambda_2} \cdots$$.

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: 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[]

product_on_basis(left, right)

Return the product of left and right.

INPUT:

• left, right – partitions

OUTPUT:

• an element of self

EXAMPLES:

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

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

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

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