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Abstract

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.