Canonical Bases and Piecewise-linear Combinatorics

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U q of Uq has a canonical basis B with favourable properties (see Kashiwara [3] and Lusztig [6, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We describe the authors’ recent work establishing close relationships between the Lusztig cones, canonical basis elements and the regions of linearity of reparametrization functions arising from the above parametrizations in type A4 and give some speculations for type An.