Regions of Linearity, Lusztig Cones, and Canonical Basis Elements for the Quantized Enveloping Algebra of Type A4

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U− of U has a canonical basis B with favourable properties (see M. Kashiwara (1991, Duke Math. J.63, 465–516) and G. Lusztig (1993. “Introduction to Quantum Groups,” Sect. 14.4.6, Birkhäuser, Boston)). 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 show that if g is of type A4 there are close relationships between the Lusztig cones, canonical basis elements, and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations.