  Let $\mathbf{w}_0$ be a reduced expression for the longest element of the
Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and
Kashiwara's (string) parametrizations of the canonical basis associated with
$\mathbf{w}_0$. Crystal operators act in a finite number of patterns in
Lusztig's parametrization, which may be seen as vectors. We show this set gives
the system of defining inequalities of the string cone constructed by Gleizer
and Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a
by-product we get an alternative enumeration of a set of inequalities defining
the string cone, based on hammocks.
