A uniqueness property of {\tau} exceptional sequences

TL;DR

This paper proves a conjecture that two nearly identical complete { au}-exceptional sequences must be identical, extending previous results and revealing linear independence of their modules' dimension vectors.

Contribution

The paper proves Buan and Marsh's conjecture that the uniqueness property of { au}-exceptional sequences holds without the { au}-tilting finite assumption.

Findings

Two complete { au}-exceptional sequences that differ in at most one term are identical.

Dimension vectors of modules in a { au}-exceptional sequence are linearly independent.

Abstract

Recently, Buan and Marsh showed that if two complete $\tau$-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is $\tau$-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a $\tau$-exceptional sequence are linearly independent.