Cyclotomic KLR Algebras - Part 1 of 4

Explore the foundations of cyclotomic KLR algebras in this comprehensive lecture, part of the Hausdorff Trimester Program on Symplectic Geometry and Representation Theory. Delve into the significance of these algebras in categorifying highest weight representations of quantum groups. Begin with a broad discussion of these algebras in arbitrary type before focusing on type A, where the Brundan and Kleshchev graded isomorphism theorem connects them to cyclotomic Hecke algebras. Examine the Ariki–Brundan–Kleshchev categorification theorem through the lens of cyclotomic KLR algebras of type A. Cover key topics including quiver Hecke algebras, Q-polynomials, symmetric group actions, diagrammatic presentations, symmetric polynomials, coinvariant algebras, basis theorems, nil Hecke algebras, induction and restriction functors, Grothendieck groups, and canonical bases for integrable highest weight modules.