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
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