Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra.
Now presumably this algorithm has been implemented by someone somewhere. I am wondering if there are any precomputed tables of dual canonical basis vectors out there (or the software to compute my own). I do know that the quagroup package in GAP computes the canonical basis. I can't see an easy way to extract from that the dual basis.
For immediate purposes, I believe I'd be most interested if the dual canonical basis vectors were given as elements of the quantum shuffle algebra, under the standard embedding of Uq(n)* into the shuffle algebra.
EDIT. Leclerc (personal communication) informs me that he implemented this algorithm, but it is in an obsolete format. And also that one is likely to run up against size issues in trying to perform this task, since elements of a shuffle algebra have a lot of terms.