  Let U be the quantum group associated to a Lie algebra of type A_n. The
negative part U^- of U has a canonical basis B defined by Lusztig and
Kashiwara, with favourable properties. We show how the spanning vectors of the
cones defined by Lusztig when regarded as monomials in Kashiwara's root
operators, can be described using a remarkable rectangle combinatorics. We use
this to calculate the Lusztig parameters of the corresponding canonical basis
elements, conjecturing that translates of these vectors span the simplicial
regions of linearity of Lusztig's piecewise-linear function.
  Keywords: quantum group, canonical basis, Lusztig cone, longest word,
piecewise-linear combinatorics.
