Combinatorics of type \(D\) exceptional sequences (Q2688943)

Exceptional sequences are certain homologically-defined sequences of quiver representations that are useful in understanding the structure of the associated bounded derived category of quiver representations. Crawley-Boevey [\textit{V. Dlab} (ed.) and \textit{H. Lenzing} (ed.), Representations of algebras. Proceedings of the sixth international conference on representations of algebras, Carleton University, Ottawa, Ontario, Canada, August 19-22, 1992. Providence, RI: American Mathematical Society (1993; Zbl 0786.00022)] showed that the braid group acts transitively on the set of exceptional sequences of maximal length. Exceptional sequences are also related to other areas of mathematics, including the combinatorics of Coxeter groups and cluster algebras. In particular, exceptional sequences of representations of Dynkin quivers are in bijection with saturated chains in the corresponding lattice of crossingless partitions that contain the minimal element. Exceptional sequences are important sequences of quiver representations in the study of representation theory of algebras. They are also closely related to the theory of cluster algebras and the combinatorics of Coxeter groups. The authors of the present paper combinatorially classify exceptional sequences of a family of type \(D\) Dynkin quivers; they show how their model for exceptional sequences connects to the combinatorics of type \(D\) crossingless partitions.