A realization of the ring of symmetric functions is multiplicative if for
a partition
we have
.
Bases: sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical
The class of multiplicative bases of the ring of symmetric functions.
A realization of the ring of symmetric functions is multiplicative if
for a partition
we have
(with
meaning
).
Examples of multiplicative realizations are the elementary symmetric basis,
the complete homogeneous basis, the powersum basis (if the base ring is a
-algebra), and the Witt basis (but not the Schur basis or the
monomial basis).
Return the coproduct on a basis element for multiplicative bases.
INPUT:
OUTPUT:
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[]