Finite dimensional Hecke algebras

TL;DR

This paper reviews the structure and categorification of finite-dimensional Hecke algebras, covering canonical basic sets, crystal bases, Fock space theory, and quasihereditary covers, with detailed proofs and applications.

Contribution

It provides a comprehensive overview of recent developments in the categorification and structural understanding of finite-dimensional Hecke algebras.

Findings

Explanation of canonical basic sets by Geck and Jacon

Application of Kashiwara crystal and Fock space theory to cyclotomic Hecke algebras

Detailed exposition of Rouquier's quasihereditary covers

Abstract

These are notes prepared for ICRA workshop at Torun, Poland, August 2007. In the first part, we explain results on canonical basic sets by Geck and Jacon and propose a categorification framework which is suitable for our example of Hecke algebras. In the second part, we review basics of Kashiwara crystal and explain the Fock space theory of cyclotomic Hecke algebras and its applications. In the third part, we explain Rouquier's theory of quasihereditary covers of cyclotomic Hecke algebras. We add detailed explanation of the proofs here. The third part is based on my intensive course given at Nagoya university in January 2007.