Exceptional sequences of type $B_n/C_n$ and those in the abelian tube

TL;DR

This paper establishes a bijection between exceptional sequences in abelian tubes and those in categories of type Bn/Cn, providing new insights into cluster structures and their combinatorial representations.

Contribution

It introduces a bijection linking exceptional sequences in abelian tubes to those in type Bn/Cn categories, enhancing understanding of cluster combinatorics.

Findings

Bijection between signed exceptional sequences and augmented rooted labeled trees.

Reinterpretation of Buan-Marsh-Vatne formula for clusters of type Bn/Cn.

Comparison of exceptional sequences in abelian tubes and module categories.

Abstract

We examine clusters in the cluster tube of rank $n+1$ using exceptional sequences in the abelian tube of rank $n+1$. Although the abelian tube has more exceptional sequences than the module categories of type $B_{n}/C_{n}$, we obtain a bijection between the set of signed exceptional sequences of any length in these categories. This bijection gives a reinterpretation of the formula of Buan-Marsh-Vatne comparing clusters of type $B_n/C_n$ with maximal rigid objects in the cluster tube of rank $n+1$. The bijection passes through the set of "augmented" rooted labeled trees.