Mathematics Department Colloquium: Combinatorics, Categorification, and Crystals

Colloquium | March 1 | 4-5 p.m. | 60 Evans Hall

 Monica Vazirani, UC Davis

 Department of Mathematics

Categorification attempts to replace sets or algebraic and geometric structures with more general categories. It has enjoyed amazing successes, such as Khovanov homology categorifying the Jones polynomial knot invariant, KLR algebras categorifying quantum groups, or Soergel bimodules categorifying Hecke algebras. Many of the algebras we see in categorification can be described diagrammatically, which is in its own way very combinatorial. This is related to an historic motivation for categorification: to construct knot and link invariants. The payoffs to finding these richer, higher categorical structures include not only constructing finer knot invariants, but proving positivity results and producing some fantastic mathematics.

In this talk, I will focus on the second example, that is, on quantum groups. Their crystal bases or canonical bases exhibit the positivity and integrality that is a trademark feature of a decategorified structure. My launch point will be the type $A$ combinatorics of Young diagrams or partitions. Partitions encode the representation theory of the symmetric group, but they also form a crystal–the crystal graph of the basic representation of $\mathfrak {sl}_\infty $. This is not a coincidence. The symmetric groups categorify the basic representation, with induction and restriction functors descending to raising and lowering operators. This phenomenon generalizes to all symmetrizable types replacing the symmetric groups with cyclotomic Khovanov-Lauda-Rouquier (KLR) algebras.