Representation type of Hecke algebras (Resume)
Susumu Ariki

Let A be a nite dimensional algebra over an algebraically closed eld F . The
representation type of A is an index for the possibility to classify simple A-modules.
{ If A is of nite type, then we can classify simple A-modules in principle, as
the number is nitely many.
{ If A is of tame type, we can classify simple A-modules of given dimension, as
the modules are parametrized by a nite number of 1 parameter families.
{ If A is of wild type, the classi cation is hopeless, as the problem contains
the classi cation problem of any nite dimensional algebra.
The actual classi cation is a dierent problem, and could be dicult. Our aim
is to explain that the problem can be handled when the algebra is a Hecke algebra.
Let W be a nite Weyl group of classical type, i.e. the irreducible components
are of type A, BC or D, and let q 2 F be invertible. We denote by HW (q) theassociated Hecke algebra. If q = 1 then HW (q) = FW and we are reduced to thecase of group algebras. We are interested in the case where q 6= 1. Before stating
our result, we introduce the Poincare polynomial.
Definition 1. Let `(w) be the length of w 2 W . The Poincare polynomial of
W is dened by
PW (x) =x`(w) :X2w W

Theorem 2 (A). Suppose that W , q and HW (q ) be as above. Then HW (q ) is
{ nite if (x ; q)2 does not divide PW (x).
{ tame if q = ;1 6= 1 and (x + 1)2 jjPW (x).

{ wild in the remaining cases.
A corollary of the proof is the following.
Corollary 3. Suppose that W , q and HW (q ) be as above and let B be a blockalgebra of HW (q).
(1) If B is of nite type then B is a Brauer tree algebra with tree being a straightline without exceptional vertex.
(2) If B is of tame type then B is a special biserial algebra.
12SUSUMU ARIKI

If B is as in (1) then we are essentially in the case of irreducible Weyl groups,and PIM's are of the form

D1DiDrD2  Di;1  Di+1 for 2 i r ; 1 Dr;1 :D1
DiDrThe number of simple B -modules is given by the following formula.
(i) r = o(q) ; 1 if W is type A or if o(q) is odd.
(ii) r = o(2q) or o(2q) + 2 if W is type B and o(q) is even.
(iii) r = o(2q) + 1 if W is type D and o(q) is even.
Here, o(q) is the multiplicative order of q.

Now we remove the assumptions on HW (q). We conjecture that every block
algebra of nite type is a Brauer tree algebra with tree being a straight line without
exceptional vertex, and every block algebra of tame type is a special biserial algebra.
This implies that we can classify all the simple modules if B is not of wild type:{ if B is a Brauer tree algebra of this type, the number of indecomposable
modules is r(r + 1) and we can describe them explicitly.
{ if B is a special biserial algebra and self-injective, it is well known thatthe stable Auslander-Reiten quiver of B is the Auslander-Reiten quiver of
B= Soc B . As the latter is a string algebra, the indecomposable modules
are given by string modules and band modules, which are given by coloredquivers of type A and A~ whose edges are colored by the arrows of the algebra.
Reference: Theorem(A) is proved in math.QA/0302136
S. Ariki, Hecke algebras of classical type and their representation type.
RIMS, Kyoto university, Kyoto 606{8502, Japan

E-mail address : ariki@kurims.kyoto-u.ac.jp