Constructing an algebra from combinatorial objects
I want to construct an algebra by first constructing a free module with basis a set of combinatorial objects (permutations, ordered set compositions, parking functions, whatever...) and then specifying a product. How do I do that? The CombinatorialFreeModule class is not sufficient. There used to be a CombinatorialFreeAlgebra class but that may be long deprecated.
What do you mean under "not sufficient"? Please provide a code illustrating the issue.
The CombinatorialFreeModule class lacks a way to specify a product, unless I am missing something.
Is this what you want? https://ask.sagemath.org/question/32064/
Thanks for the link. Unfortunately, no, that is not what I'm looking for. I want to build a ring by first defining it as a free module, then specifying how to multiply basis elements. For example, the symmetric functions could be constructed by first defining a free module on the set of integer partitions (representing, say, Schur functions), then specifying a multiplication on them (in this case the Littlewood-Richardson rule). I'd like to know how to do this more generally. This is quite a common thing to do in algebraic combinatorics, but I haven't been able to figure out how.