Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

TL;DR

This paper establishes an explicit isomorphism between blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, linking categorification theories and revealing new gradings in algebra blocks.

Contribution

It constructs an explicit isomorphism connecting cyclotomic Hecke algebra blocks with Khovanov-Lauda algebras, advancing categorification understanding.

Findings

Explicit isomorphism between blocks of cyclotomic Hecke and Khovanov-Lauda algebras

Connection of categorification conjecture to Ariki's theorem

Introduction of non-trivial Z-grading on algebra blocks

Abstract

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.