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# 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)

  Bases: `sage.combinat.sf.classical.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: 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[]

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