On bases of centres of Iwahori–Hecke algebras of the symmetric group

Abstract

In 1990, using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group S n over Q [ ξ ] [Trans. Amer. Math. Soc. 317 (1) (1990) 361–392]. An integral “minimal” basis was later given by the first author in [J. Algebra 221 (1) (1999) 1–28], following [M. Geck, R. Rouquier, Centers and simple modules for Iwahori–Hecke algebras, in: Finite Reductive Groups, Luminy, 1994, Birkhäuser, Boston, MA, 1997, pp. 251–272]. In principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements. In this paper we find an explicit non-recursive expression for the coefficients appearing in these linear combinations. These coefficients are expressed in terms of certain permutation characters of S n . In the process of establishing this main theorem, we prove the following items of independent interest: a result on the projection of the norms onto parabolic subalgebras, the existence of an inner product on the Hecke algebra with some interesting properties, and the existence of a partial ordering on the norms.