A crystal embedding into Lusztig data of type $A$

TL;DR

This paper provides a straightforward description of how Young tableaux embed into Lusztig data of type A, along with an algorithm for transitioning between different Lusztig data representations related to quivers with a single sink.

Contribution

It introduces a simple description of the crystal embedding of Young tableaux into Lusztig data and an algorithm for transition matrices between Lusztig data for specific reduced expressions.

Findings

Explicit description of crystal embedding into Lusztig data

Algorithm for transition matrices between Lusztig data

Applicable to quivers with a single sink

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.