Combinatorics of Exceptional Sequences in Type A

Abstract

Exceptional sequences are certain ordered sequences of quiver representations. We introduce a class of objects called strand diagrams and use this model to classify exceptional sequences of representations of a quiver whose underlying graph is a type An Dynkin diagram. We also use variations of this model to classify c-matrices of such quivers, to interpret exceptional sequences as linear extensions of posets, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This work is an extension of the classification of exceptional sequences for the linearly-ordered quiver by the first and third authors. ‚ chains in the lattice of noncrossing partitions [Bes03, HK13, IT09], ‚ c-matrices and cluster algebras [ST13], ‚ factorizations of Coxeter elements [IS10], and ‚ t-structures and derived categories [Bez03, BK89, Rud90].Despite their ubiquity, very little work has been done to concretely describe exceptional sequences, even for path algebras of Dynkin quivers [Ara13, GM15]. In this paper, we give a concrete description of exceptional sequences

Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions

“…We conclude the paper with a classification of c-matrices of quivers defined by triangulations of polygons (see Theorem 9.1). This classification is similar to the classification obtained in [53] for acyclic quivers and to the classification found in [29] for type A Dynkin quivers.…”

Oriented Flip Graphs, Noncrossing Tree Partitions, and Representation Theory of Tiling Algebras

“…An important application of the latter is a classification of c-matrices of quivers that are mutation-equivalent to type A Dynkin quivers (see Theorem 9.1). This classification is similar to one obtained in [36,Theorem 1.1] for acyclic quivers and to the classification found in [18,Theorem 3.15] for type A Dynkin quivers.…”

Combinatorics of Exceptional Sequences of Type $\tilde{\mathbb{A}}_n$

“…In this paper, we will study exceptional sequences of type Ãn first by using a combinatorial object introduced in [7] known as strand diagrams. Our first result will be a bijection between exceptional collections and strand diagrams.…”