The combinatorics of tensor products of higher Auslander algebras of type $A$

Abstract

We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two drepresentation-finite algebras has a d-precluster-tilting subcategory. Finally we relate mutations of these collections to a form of tilting for these algebras. Contents 1. Introduction 1 Acknowledgements 4 2. Background 4 2.1. Grassmannian cluster algebras 4 2.2. Triangulations of cyclic polytopes 5 3. Non-l-intertwining collections 6 4. Higher precluster-tilting subcategories 10 5. Mutations 15 5.1. Higher APR tilting 15 5.2. Uniterated mutation of non-l-intertwining collections 17 References 22 PRECLUSTER-TILTING FOR TYPE A d n ⊗ A d m