INTRODUCTION

Let A = RG be a group ring which is given by a complete discrete

valuation ring R and a finite p-group G , such that A is an

isolated singularity in the sense of Auslander [Au 84]. Here, this is

equivalent to the condition that the prime number p is neither a unit

nor zero in R . Consider the problem of classifying all indecomposable

left A-lattices and, what is more, the problem of determining the

Auslander-Reiten quiver of A .

The present state of knowledge in this respect can be summarized

as follows. Denote by C n the cyclic group of order p and by v

the exponential valuation of R . Then A is of finite representation

type if and only if one of the following cases (i)-(iv)occurs:

(i) G = C2 ,

(ii) G = C and v(3) * 3 ,

(iii) G = C and v(p) 2 ,

(iv) G = C 2 and v(p) = 1 [Dr/Ro 67], [Ja 67].

Moreover, A is of tame representation type in each of the

following cases (v)-(vii):

(v) G = C x C and v(2) = 1 [Na 67],

(vi) G = Cg and v(2) = 1 [Ja 72],

(vii) G = C4 and v(2) = 2 [Ko 75].

In each of the cases (i)-(v) the Auslander-Reiten quiver of A is

known [Di 83a]. Furthermore, A is domestic tame in case (v), whereas

it is non-domestic tame in cases (vi) and (vii).