Combinatorics of canonical bases revisited: String data in type $A$

Abstract: We give a formula for the crystal structure on the integer points of the string polytopes and the $$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $$-involution on string data for a special choice of reduced word.