Some Consequences of Categorification

TL;DR

This paper demonstrates that categorification through valued quivers can prove several conjectures in acyclic skew-symmetrizable cluster algebras, advancing understanding in algebraic combinatorics.

Contribution

It provides a unified proof of multiple conjectures in cluster algebra theory using categorification techniques.

Findings

Proves conjectures related to d-vectors, g-vectors, and F-polynomials

Establishes categorification as a powerful tool in algebraic combinatorics

Connects categorification with key algebraic conjectures

Abstract

Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to $\mathbf{d}$-vectors, $\mathbf{g}$-vectors, and $F$-polynomials.