Classification of representation-finite hereditary rings

Question

It is well known that one can associate to a finite dimensional hereditary algebra $A$ a diagram $G(A)$ and then show that $A$ is representation-finite if and only if $G(A)$ is a disjoint union of the Dynkin types $A_n, B_n, C_n , D_n ,F_4, G_2, E_6 ,E_7 , E_8$.

In the article "hereditary artinian rings of finite representation type" (1980) by Dowbor, Ringel and Simson it was shown that the more general class of hereditary artinian rings are classified in the same way buy there also the types $H_3, H_4, I_2(p)$ for $p=5$ or $p \geq 7$ could appear (known from the classification of finite Coxeter groups). It was mentioned that it is not known whether such hereditary artinian rings of types $H_3, H_4, I_2(p)$ for $p=5$ or $p \geq 7$ really do appear.

Question: Has this problem been solved, that is: do also hereditary artinian rings of types $H_3, H_4, I_2(p)$ for $p=5$ or $p \geq 7$ exist?

Answer 1

Aidan Schofield constructed a hereditary artinian ring of type $I_2(5)$. See: A. Schofield, Hereditary Artinian rings of finite representation type and extensions of simple Artinian rings, Math. Proc. Cambridge Philos. Soc. 102 (1987), 411–420, or pp 214-218 of A. Schofield, Representations of rings over skew fields, CUP, 1985.