I too needed some code to compute with Demazure characters using the
weight ring.
I agree that this is useful independent of the Demazure characters as
implemented
in the crystal code.
What we did was just to make a standalone program that did what we
needed, but it would make a good enhancement to sage.
I think the right way would be as a method of the weight ring.
So the syntax would be
ch.demazure_character(w)
where ch is a weight ring element and w is an element of the Weyl
group.
There could be a helper method
ch._demazure_character(i)
that would produce the operator corresponding to the i-th simple
reflection.
It would do something like this:
def _demazure(self, i):
a = self._parent.space().simple_coroots()[i]
ret = self._parent(0)
d = self._mdict
for t in d.keys():
k = t.inner_product(a)
if k >= 0:
ret += d[t]*sum(self._parent(t-j*a) for j in range(k
+1))
else:
ret += -d[t]*sum(self._parent(t+j*a) for j in
range(1,k))
return ret
This helper method would then be called by the method demazure,
which would use the reduced_word method of Weyl group elements
to to produce a sequence of called to the helper method.
The plot method for WeightRingElements, just for rank two, would
also be very useful.
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