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Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions are closely related by studying sufficiently generic representations of the preprojective algebra.This research was supported by the Graduiertenkolleg "Global Structures in Geometry und Analysis" and the DFG Priority program Darstellungstheorie 1388.The author is deeply indebted to Volker Genz and Markus Reineke for several valuable and clarifying discussion and comments. Further a big thanks goes to Michael Ehrig for careful proofreading and helpful advise and Yoshiyuki Kimura for his comments on an earlier version of this paper.

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" for the action 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-Mirković 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 a Borel subgroup of SLn defined by Berenstein-Kazhdan and Gross-Hacking-Keel-Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

We prove a conjecture of Naito–Sagaki about a branching rule for the restriction of irreducible representations of sl(2n,C) to sp(2n,C). The conjecture is in terms of certain Littelmann paths, with the embedding given by the folding of the type A2n−1 Dynkin diagram. So far, the only known non‐Levi branching rules in terms of Littelmann paths are the diagonal embeddings of Lie algebras in their product yielding the tensor product multiplicities.