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Representation Dimension and Solomon Zeta Function

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.