Speaker: Peter Tingley, Loyola University Chicago
Title: Crystal combinatorics from Lusztig's PBW bases
Abstract: In the early 1990s both Lusztig and Kashiwara developed a theory of canonical bases for certain Kac-Moody algebras (called the canonical basis by Lusztig and the global crystal basis by Kashiwara). The approaches are pretty different, and the work went in different directions: Lusztig used geometry and ultimately categorification, while Kashiwara went in a more combinatorial direction, developing the theory of crystals. In this talk we will discuss an alternate reality where Lusztig's approach, in particular his theory of PBW bases, was used to develop crystal combinatorics (in finite type). This gives new insights into such classical combinatorial objects as Young tableaux, as well as giving rise to more exotic combinatorics. This is joint work with Ben Salisbury and Adam Schultze.