Tilting theoretic approach to quasi-hereditary structures

A quasi-hereditary algebra is an algebra equipped with a certain partial order on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module T_ by a classical result due to Ringel. A fundamental question is to determine which tilting modules can be realised as characteristic tilting modules. We answer this question by using the notion of IS-tilting module, which is a pair (T, ) of a tilting module T and a partial order on its direct summands such that iterative idempotent truncation along always reveals a simple direct summand. Specifically, we show that a tilting module T is characteristic if, and only if, there is some so that (T, ) is IS-tilting; in which case, we have T=T_. This result enables us to study quasi-hereditary structures using tilting theory. As an application of the above result, we show that, for an algebra A, all tilting modules are characteristic if, and only if, A is a quadratic linear Nakayama algebra. Furthermore, for such an A, we provide a decomposition of the set of its tilting modules that can be used to derive a recursive formula for enumerating its quasi-hereditary structures. Finally, we describe the quasi-hereditary structures of A via `nodal gluing' and binary tree sequences.