Regions of linearity, lusztig cones, and canonical basis elements for the quantized enveloping algebra of type A(4)

Let U-q be the quantum group associated to a Lie algebra g of rank n. The n egative part U- of U has a canonical basis B with favourable properties (se e M. Kashiwara (1991, Duke Math. J. 63, 465-516) and G. Lusztig (1993. "Int roduction to Quantum Groups," Sect. 14.4.6, Birkhauser, Boston)). The appro aches of Lusztig and Kashiwara lead to a set of alternative parametrization s of the canonical basis, one for each reduced expression for the longest w ord in the Weyl group of g. We show that if g is of type A(4) there are clo se relationships between the Lusztig cones, canonical basis elements, and t he regions of linearity of reparametrization functions arising from the abo ve parametrizations. A graph can be defined on the set of simplicial region s of linearity with respect to adjacency, and we further show that this gra ph is isomorphic to the graph with vertices given by the reduced expression s of the longest word of the Weyl group modulo commutation and edges given by long braid relations, (C) 2000 Academic Press.