Let $W$ a Weyl group and $W_I$ the parabolic subgroup associated to the subset $I$ of simple roots.
Currently I am facing with the problem of computing explicitly the set of minimal coset representatives of $W/W_I$ in an efficient way.
With "in an efficient way" I mean "avoiding to compute the mcr of every element of W".
The reason is essentially computational: let consider $A_3=W_{{1,2,3,4}} <A_4$. C Compute the mcr of each element in $A_4$ need $5!$ iterations. A direct computation looking at the quotient will need only 5 iterations...
I am familiar with SageMath but it seems that does not exist a package for dealing with quotients by parabolic subgroups and compute their set of mcr.
Moreover I know that there exists some ad hoc functions in Lie, but I am not very familiar with this program.
Does anyone know if there is a way to make this computation using sage? Are there some other even intuitive programs that can I use (and that support polynomial computations too)?