$\tau$-exceptional sequences and the shard intersection order in type A

TL;DR

This paper develops a combinatorial model for $ au$-exceptional sequences in type A, linking shard intersection orders, wide subcategories, and arc diagrams, and proves EL-shellability using this framework.

Contribution

It introduces a novel combinatorial model for $ au$-exceptional sequences in type A, connecting shard intersection orders with arc diagrams and providing a new proof of EL-shellability.

Findings

Established a correspondence between shard intersection orders and wide subcategories.

Developed a combinatorial arc diagram model for $ au$-exceptional sequences.

Proved EL-shellability of the shard intersection order in type A.

Abstract

Reading's "shard intersection order" on the symmetric group can be realized as the "lattice of wide subcategories" of the corresponding preprojective algebra. In this paper, we first use Bancroft's combinatorial model for the shard intersection order to associate a unique shard to each downward cover relation. We then show that, under the correspondence with wide subcategories, this process coincides with Jasso's "$\tau$-tilting reduction". In particular, this yields a combinatorial model for this algebras's $\tau$-exceptional sequences" (defined by Buan and Marsh). We formulate this model using the combinatorics of arc diagrams. Finally, we use our model to give a new representation-theoretic proof that the shard intersection order is EL-shellable in type A.