Contemporary Mathematics

Volume 392, 2005

Modular representation theory of Heeke algebras, a survey

Susumu Ariki

ABSTRACT. After reviewing the Kazhdan-Lusztig basis, this surveys recent

advances on Heeke algebras, with stress on classical types. Natural appearance

of two versions of Fock spaces and their crystal bases, and problems arising

from the theory of finite dimensional algebras are explained.

1.

Preliminaries

This paper is based on my minicourse which was delivered in the conference.

The first part is about the Kazhdan-Lusztig basis. In the second part I explain

how Fock spaces and the crystal base theory appear in the modular representation

theory of Heeke algebras. There are two classification results of simple modules of

Heeke algebras now, and there appear two kinds of Fock spaces. One is our version

of higher level Fock spaces in the theory which I developed with A. Mathas. The

other came from B. Leclerc's line of research. Recently, it was found by M. Geck

and N. Jacon that Geck-Rouquier theory explains the appearance of the original

version of higher level Fock spaces. This is the main object of the second part. See

also Jacon's contribution to this volume.

Then in the third part I switch to my old result on the decomposition numbers

and explain its applications. I close this survey with some open problems. They

naturally appear when we view Heeke algebras as finite dimensional algebras which

are akin to group algebras.

Let us fix notation first. Recall that there is a class of finite groups which are

called finite Weyl

groups.

There are 3 infinite series: W(An-d

=

Sn, W(BCn)

=:

Wn, W(Dn)

=:

W~

and some more: W(E6), W(E7), W(EB), W(F4), W(G2). A

finite Weyl group

W

is a product of these groups.

EXAMPLE

1.1. Wn is a semidirect product of C!] and the symmetric group Sn.

Sn acts on { ( c1, ... , en)

E

C!]

I

c1 c2 · · · Cn

=

1} and its semidirect product with Sn

forms a normal subgroup of Wn, which is

W~.

We consider

W

as a Coxeter group. That is, we specify a set of generators

S

c

W

and

W

has a presentation in terms of the generators and the relations

among them.

1991

Mathematics Subject Classification.

20C08, 16G60.

Key words and phrases.

Heeke algebra, Fock space, crystal base, representation type.

©

2005 American Mathematical Society

http://dx.doi.org/10.1090/conm/392/07349