Quasi-Hereditary Orderings of Nakayama Algebras

To determine an algebra is quasi-hereditary is a difficult problem. An effective method, Green-Schroll set, is introduced in this paper to tackle this problem. It is well known that an algebra is quasi-hereditary if and only if it admits a quasi-hereditary ordering of simple modules. Let A be a Nakayama algebra. We prove a necessary and sufficient criterion to determine whether an ordering of simple modules is quasi-hereditary on A, and A is quasi-hereditary if and only if its Green-Schroll set is nonempty. This seems to be the simplest characterization currently known, since it does not involve any algebraic concepts. A general iteration formula for the number q (A) of all quasi-hereditary orderings of A is given via Green-Schroll set. The q-ordering conjecture is proved to be true for A.