Lectures on cyclotomic Hecke algebras

Abstract

These are lecture notes prepared for London Mathematical Society Symposium "Quantum Groups" held in Durham 19-29 July 1999. We give a survey on cyclotomic Hecke algebras. These algebras have been studied by Dipper, James, Malle, Mathas and the author. In lecture one, we start with definitions. We then go on to Specht module theory of cyclotomic Hecke algebras constructed by Dipper, James and Mathas. In lecture two, we explain Ginzburg and Lusztig's theory on affine Hecke algebras and its application to cyclotomic Hecke algebras due to the author. As Kazhdan-Lusztig bases of Hecke algebras describe decomposition numbers of quantum algebras at roots of unity, canonical bases of quantum affine algebras on integrable modules describe decomposition numbers of cyclotomic Hecke algebras at roots of unity. In particular, it turns out that crystal structure in the sense of Kashiwara exists behind the representation theory of cyclotomic Hecke algebras. This was first observed by Lascoux, Leclerc and Thibon. Its interpretation in terms of socle series was proposed by Rouquier. It is further developed in Vazirani's thesis. The special case of Hecke algebras of type A is extensively studied, and its relation to Lusztig conjecture is now very clear. We explain Varagnolo and Vasserot's work and several algorithms proposed so far. Its application to the decomposition numbers of finite general linear groups due to Dipper and James is also explained. The parametrization of simple modules of cyclotomic Hecke algebras over arbitrary fields in terms of crystal graphs of quantum affine algebras is given by Mathas and the author. In the third lecture, we show that this crystal graph description fits well in the Specht module theory. It uses work of Takemura and Uglov, Varagnolo and Vasserot.