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.

Intro
Outline of lectures
Quiver Hecke algebras - the Q-polynomials
Quiver Hecke algebras The symmetric group G, acts on/ by place permutations
Diagrammatic presentation for
Symmetric polynomials and the coinvariant algebra
Quiver Hecke algebra basis theorem
Finiteness of cyclotomic quiver Hecke algebras Proposition
The nil Hecke algebra case
Induction and restriction functors
Grothendieck groups
Categorification of highest weight modules
Canonical bases for integrable highest weight modules Corollary Varolo-Vasserot. Brundan-Stroppel