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

Details On bases of centres of Iwahori-Hecke algebras of the symmetric group On bases of centres of Iwahori-Hecke algebras of the symmetric group

2004-01-05

arxiv.org/abs/math/0401036v3

Journal of Algebra (2005), vol. 289 (1), p42-69

Mathematics

Quantum Algebra

29 pages. To appear J. Algebra. Original version January 2004

Scientific paper

Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over $\Q[\xi]$ in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and Rouquier. 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 readily computable numbers involving orders of symmetric groups and conjugacy classes. 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.

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