[Mupad-combinat-users] Tensor products

[Nicolas M. T. <Nic...@u-...>]

	Dear MuPAD-Combinat developers and users,

I have made just commited the following change which may be delicate,
but hopefully you will like it. Feedback more than welcome.

operators::tensor now accepts either domain elements (in which case it
returns and element of the tensor product of thedomains), domains, or
morphisms (it does not use the overloaded operator operators::tensor2).

So, now you can do things like:

operators::setTensorSymbol("#"):
S := examples::SymmetricFunctions():

// Building tensor elements:
>> S::s[3] # S::e[1] # S::p[4,2]

                       s[3] # e[1] # p[4, 2]

// Using tensor products of morphisms
>> (S::p::id # S::h::antipode) (S::p[2] # S::h[3])
       - p[2] # h[1, 1, 1] + 2 p[2] # h[2, 1] - p[2] # h[3]

// Note: you can't just do:
>> (id # S::h::antipode) (S::p[2] # S::h[3])
Error: Wrong type of 1. argument (type 'Type::ListOf(DOM_DOMAIN)' expected,
// because # needs to know the source and image set of both morphisms.

// Checking a counit:
>> counit := S::p::mu @ (S::p::id # S::p::antipode) @ S::p::coproduct:
>> counit(S::p::one)
                               p[]
>> counit(S::p[3,2])
                               0

// Checking coassociativity
>> phi := (S::p::coproduct # S::p::id) @ S::p::coproduct
        - (S::p::id # S::p::coproduct) @ S::p::coproduct:
>> phi(S::p[5,4,3])
                                0

// Conversions between tensor spaces
>> (S::s # S::s) (S::p[3] # S::p[4])
- s[1, 1, 1] # s[1, 1, 1, 1] + s[1, 1, 1] # s[2, 1, 1] -
  ...

// The bad news is that, you can't yet do:
>> (S::s # S::s) (S::p[3] # S::h[4])
Error: illegal arguments [(Dom:

// This would indeed require a conversion to be declared from
// S::p # S::h to S::s # S::s

// The good news is that you can nevertheless do:
>> (S::s@S::p::id # S::s@S::h::id) (S::p[3] # S::h[4])
              s[1, 1, 1] # s[4] - s[2, 1] # s[4] + s[3] # s[4]

// In the later case, you are constructing the tensor product
// of two morphisms, and # is able to figure out that
// S::s @ S::p::id is a morphism from S::p to S::s, and
// deduce the source and image set of the result.


Cheers,
				Nicolas
-- 
Nicolas M. Thiéry "Isil" <nt...@us...>
http://Nicolas.Thiery.name/