In 2004, Iyama introduced a generalisation of some of the central ideas of Auslander-Reiten theory to a “higher dimensional” setting. This has inspired a lot of interesting research over the years. One important concept is that of \(n\)-hereditary algebras, which enjoy some of the key properties of hereditary algebras in the context of higher AR-theory. Their defining characteristic is rather strong, so they form a very small subset of the finite-dimensional algebras of global dimension \(n\). Although many classes of examples of \(n\)-representation-finite and \(n\)-representation-infinite algebras exist, little is known about the different types of restrictions the quiver and relations of an algebra of global dimension \(n\) must satisfy in order to be \(n\)-hereditary. In this talk, we will explore some of these restrictions, with a particular focus on the case of monomial algebras.
This is based on joint work with Mads Hustad Sandøy.