Combinatorics of canonical bases revisited: Type A
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirkovi\'c conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara's string and Lusztig's parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of $SL_n$ defined by Berenstein-Kazhdan and Gross-Hacking-Keel-Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.