Centers of generic Hecke algebras

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.

“…This section and the next contain a number of results needed for later sections, most of which appear in [12] but not, as far as we can see, in the available literature.…”

Section: Double Cosets Of Maximal Parabolic Subgroupsmentioning

confidence: 99%

“…Some results from [13] have been restated in the context of the Hecke algebra over Z[ξ ], and in the generality of compositions rather than partitions where appropriate.…”

Section: Bases For the Centre Of The Hecke Algebramentioning

confidence: 99%

“…As before, for any multipartition θ of n with derived composition λ of n, we define θ x := λ x . Theorem 4.3 [13]. Let λ n.…”

Section: The Norm Basismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Francis

Jones

2005

The Minimal Basis for the Centre of an Iwahori–Hecke Algebra

Francis

1999

This paper arose out of an attempt to generalize the q q −1 -basis for the centre of an Iwahori-Hecke algebra q found by Jones to q q −1 and to other types. Considering the Iwahori-Hecke algebra over a subring ξ of q 1/2 q −1/2 , where ξ = q 1/2 − q −1/2 , we use a new and natural definition of positivity on to describe the "minimal" ξ -basis for Z in terms of a partial order on the positive part of . The main result is to show that this minimal basis is the set of "primitive" minimal elements of the positive part of the centre, for any Weyl group. In addition, the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize (on setting ξ = 0, the equivalent of setting q = 1 in q 1/2 q −1/2 ) to the sum of elements in a conjugacy class, and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class. A constructive algorithm is provided for obtaining the minimal basis. We use the results of Jones to achieve the result in type A n without the need for character theory, and give the result for all Weyl groups using the character theoretical results of Geck and Rouquier. Finally we discuss the non-crystallographic cases and give some explicit examples.

Algebra of Matrix Arithmetic

Bhowmik

Ramaré

1998

We study the algebra of the arithmetic of integer matrices. A link is established between the divisor classes of matrices and lattices. The algebra of arithmetical functions of integral matrices is then shown to be isomorphic to an extension of the Hecke algebra, also called a Hall algebra in combinatorics. The dictionary helps translate results from one setting to another. One important application is the study of subgroups of a finite abelian group. ᮊ