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

In 1990, using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group Sn over [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 Sn. 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.