A crystal embedding into Lusztig data of type $A$

Abstract

Let $i$ be a reduced expression of the longest element in the Weyl group of type $A$, which is adapted to a Dynkin quiver with a single sink. We present a simple description of the crystal embedding of Young tableaux of arbitrary shape into $i$-Lusztig data, which also gives an algorithm for the transition matrix between Lusztig data associated to reduced expressions adapted to quivers with a single sink.

22 pages, minor corrections; introduction revised, statements and proofs of Proposition 4.1 and Theorem 4.2 revised, comments added on Remark 4.4, typos corrected