2000/08/07 by Roger Carter, Carter, Roger, Bethany Marsh +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.CO #math.QA #msc:17B37
paper · pdf · doi:10.48550/arxiv.math/0008050
61 pages, 17 figures, uses pictex
arxiv created 2000/08/07 · arxiv updated 2020/12/21
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, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig 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. Keywords: Quantum group, Lie algebra, Canonical basis, Tight monomials, Weyl group, Piecewise-linear functions.