Multiplicative symmetric functions¶
A realization of the ring of symmetric functions is multiplicative if for
a partition
we have
.
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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
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).
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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 ofself
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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