Symmetric Cyclotomic Hecke Algebras

Abstract

In this paper we prove that the generic cyclotomic Hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters. For this we show that the determinant of the Gram matrix of a certain canonical symmetrizing form introduced by K. Bremke and G. Malle (Indag. Math.8(1997), 453–469) is a unit in any such ring. On the way we show that the Ariki–Koike bases of these algebras are also quasi-symmetric.