Modules determined by their Newton polytopes

Abstract

In the $\ddot{I}„$-tilting theory, there exist two classes of foundamental modules: indecomposable $\ddot{I}„$-rigid modules and left finite bricks. In this paper, we prove the indecomposable $\ddot{I}„$-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.

5 pages. v2: typos corrected. arXiv admin note: text overlap with arXiv:2306.11438