Modules determined by their Newton polytopes

TL;DR

This paper proves that in $ au$-tilting theory, indecomposable $ au$-rigid modules and left finite bricks are uniquely identified by their Newton polytopes, generalizing Gabriel's classical result.

Contribution

It establishes that modules in $ au$-tilting theory are uniquely determined by their Newton polytopes, extending Gabriel's theorem to a broader class of modules.

Findings

Indecomposable $ au$-rigid modules are uniquely determined by their Newton polytopes.

Left finite bricks are uniquely determined by their Newton polytopes.

Generalization of Gabriel's result to $ au$-tilting modules.

Abstract

In the $\tau$-tilting theory, there exist two classes of foundamental modules: indecomposable $\tau$-rigid modules and left finite bricks. In this paper, we prove the indecomposable $\tau$-rigid modules and the left finite bricks are uniquely determined by their Newton polytopes spanned by the dimensional vectors of their quotient modules. This is a kind of generalization of Gabriel's result that the indecomposable modules over path algebras of Dynkin quivers are uniquely determined by their dimensional vectors.