Clean up demazure_character (?)

[lgoettgens]

Your new code for demazure_character is somehow in conflict (or overlaps) with a polyhedral construction in

Oscar.jl/src/PolyhedralGeometry/Polyhedron/standard_constructions.jl

Line 1070 in 278b79d

demazure_character(lambda::AbstractVector, sigma::PermGroupElem)

This requires a clean up.
While the polyhedral construction was maybe ad hoc, and yours is more general, the interface for the polyhedral one has been fixed for v1.0.
@tbrysiewicz @fingolfin

Originally posted by @micjoswig in #4339 (comment)

[fingolfin]

@lgoettgens and me would be happy to work towards unifying the interface.

However, we had a look at the existing demazure_character method (added by @tbrysiewicz I think) and the paper PS09 it cites, and frankly we did not understand what the existing function does.

We assume the permutation is meant to be an element of a Weyl group of type $A_n$ in its natural permutation representation (i.e. this code only works for that one type). But what exactly is lambda supposed to represent? In the paper in formula 8.4, $\lambda$ occurs but is a dominant weight.

Also, the method of course should produce a "Demazure character". That leaves the question how to encode one. In the root systems implementation, it is represented as a dictionary mapping weight vectors to multiplicities. One could consider adding a dedicated type for it as well, of course... But what exactly is the encoding used in the function based on PS09? It returns a multivariate polynomial, but how should one interpret that?

If @tbrysiewicz or @micjoswig could shed some light on this, that would be helpful.

[tbrysiewicz]

I believe when I wrote this I was looking at
Corollary 15.2 of PS09
for the formula for ch_{lambda,w}(z_1..z_n).
This is claimed to be "The demazure character" as in
Theorem 14.1 of PS09

There is this at the bottom of page 22 of PS09
"The polynomials chλ,w were studied by Lascoux and Sch¨utzenberger [LS1], who called
them essential polynomials, and by Reiner and Shimozono [RS], who called them
key polynomials. To avoid confusion, we will call the polynomials chλ,w simply
Demazure characters."

So we could call them 'key polynomials' if want to free up demazure_character for elsewhere.

[lgoettgens]

We just discussed this and decided that we keep the two different functions for now, but add mentions to both functions' docstrings mentioning the other function.

I'll take care of this when I put together the doc page about characters and stuff (hopefully in January).