Symmetric functions in several alphabets

[mantepse]

For several applications (eg., multisort species, playing with the hyperoctahedral group) it would be nice to have symmetric functions in several alphabets. The following is taken verbatim from an email by Nicolas Thiéry:

To emulate symmetric functions in two alphabets, you can take tensor products:

    sage: Sym = SymmetricFunctions(QQ)
    sage: Sym.inject_shorthands()
    sage: ss = tensor([s, s])
    sage: ss.category()
    Category of tensor products of hopf algebras with basis over Rational Field

Basic arithmetic is implemented:

    sage: x = ss.an_element()
    s[] # s[] + 2*s[] # s[1] + 3*s[] # s[2]
    sage: x * x
    s[] # s[] + 4*s[] # s[1] + 4*s[] # s[1, 1] + 10*s[] # s[2] + 12*s[] # s[2, 1] + 9*s[] # s[2, 2] + 12*s[] # s[3] + 9*s[] # s[3, 1] + 9*s[] # s[4]

Now many desirable features are missing:

    sage: x.coproduct()
    ... NotImplemented ...
    sage: pp = tensor([p,p])
    sage: pp(x)
    ... NotImplementedError ...

Those should be implemented generically for tensor products. The first
one should be just a couple lines in:

    Coalgebras.TensorProducts.ParentMethods.coproduct

The other is probably best implemented by adding an appropriate
_coerce_map_from in tensor products. I would need to search for the
best spot for this.

Plethysm (in a stupid way) should not be much more work. To be put in:

     sage.combinat.sf.sfa.SymmetricFunctionsBases.TensorProducts.ElementMethods

or something similar.

Component: combinatorics

Keywords: symmetric functions

Issue created by migration from https://trac.sagemath.org/ticket/13264

[mantepse]

comment:5

plethysm is now available thanks to #23446