Elementary construction of the canonical basis in finite type and connections with crystal combinatorics

Abstract: 

We present an elementary construction of Lusztig's canonical basis. The method, which is essentially Lusztig's original approach, uses the braid group to reduce to rank two calculations. Some of the wonderful properties of the canonical basis are already visible: that it descends to a basis for every highest weight integrable representation, and that it is a crystal basis. We then discuss how to study crystal combinatorics from this approach. Some of the resulting combinatorics is familiar (for instance, Young tableaux appear), but some is less so. This last part includes joint work with John Claxton, Ben Salisbury and Adam Schultze.