I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One way of writing down such a cluster structure is to give an initial seed, and indeed the paper gives an algorithm for producing such a seed. The input data for the algorithm is a particular kind of reduced expression for the longest element in a Weyl group.
Let $\Delta$ be a (simply laced) Dynkin diagram, and let $K$ be a subset of its nodes; this data defines a flag variety (in at least two ways, depending on convention, but this will not be important for my question). Denote the full subgraph of $\Delta$ on $K$ by $\Delta_K$, and let the corresponding Weyl groups be $W(\Delta_K)\leq W(\Delta)$.
To run Geiss-Leclerc-Schröer's algorithm for various examples, I want to compute a reduced expression for the longest word of $W(\Delta)$, such that it begins with a reduced expression for the longest word of $W(\Delta_K)$.
There are published tables giving an expression for the longest word of $W(\Delta)$ for each $\Delta$, but when $\Delta$ is not very small (and depending on $K$), it can be somewhat arduous to adjust these expressions to the form I want (and difficult to do this without typos!). Hence my question:
Does anybody know of any software (ideally something like a Java applet) that would make this computation easier to do?
It wouldn't need to actually produce the answer entirely by itself, as I imagine the form I am looking for is too unusual for this to have been implemented. But even something in which the user could click on the letters to apply commutation or braid relations would make it easier to avoid errors.
Sage can (apparently) compute all reduced words for any given element of a Weyl group, but this times out whenever $\Delta$ is non-tiny (and going through the list looking for a word of the right form would be an unenviable task).