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

On crystal operators in Lusztig's parametrizations and string cone defining inequalities