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. 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.  2005 Elsevier Inc. All rights reserved.

Large N Expansion of q-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras

“…Incidentally, (2.37) seems to give a relatively efficient way of caculating the central class elements compared to the ones we are aware of in the mathematical literature. Some interesting papers with explicit formulae for Hecke central elements, which we found useful, are [22,23].…”

On the square root of the centre of the Hecke algebra of type A

“…https://doi.org/10.1017/S1446788700016037 [5] The centre of the Hecke algebra 213 It can be proved by induction on n, but is also a direct consequence of looking at the Hecke algebra as a quotient of the Braid group algebra. In the braid group, the square of the longest word is the 'full twist', and the Murphy element £",, is the braid whose i-th string goes behind strings i -1 , .…”

“…During our research for [5], we were led naturally to the square of the element of the Hecke algebra corresponding to the longest word in the symmetric group. This square, which arises in the definition of certain norms, is well-known to be central, and to have an expression as a product of Murphy elements, which follows from its analogue in the braid group.…”

“…In [5], we determined explicitly, for the centre of the Hecke algebra of type A, how to express each element of the Q[q, ^~']-norm basis in [8] as a linear combination of the elements of the Z[q, ^"'J-minimal basis (see [4]). During our research for [5], we were led naturally to the square of the element of the Hecke algebra corresponding to the longest word in the symmetric group. This square, which arises in the definition of certain norms, is well-known to be central, and to have an expression as a product of Murphy elements, which follows from its analogue in the braid group.…”

Centres of Symmetric Cellular Algebras

“…Then in [5], he used the minimal basis approach to provide an entirely combinatorial way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra. In [7], Francis and Jones found an explicit nonrecursive expression for the coefficients appearing in these linear combinations 512 Y. Li [2] for the Hecke algebras of type A. The relations between the so-called Jucys-Murphy elements and centres of Hecke algebras are also investigated widely.…”