  We will show that there exists a close relationship between quasi-hereditary
algebras of Cline-Parshall-Scott and the rejection theory from the viewpoint of
the approximation theory of Auslander-Smalo. As an application, we will solve
two open problems. One concerns the representation dimension of artin algebras
introduced by M. Auslander about 30 years ago, and another concerns the Solomon
zeta functions of orders introduced by L. Solomon about 25 years ago. It will
turn out that the rejection theory relates these two quite different problems
with each other closely.
