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arXiv · 2602.07929

Newton polytopes in cluster algebras and $\tau$-tilting theory

Peigen Cao
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Abstract

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $\tau$-rigid modules and the left finite multi-semibricks in $\tau$-tilting theory are uniquely determined by the Newton polytopes of these modules. The key tools used in the proofs are the left Bongartz completion, $F$-invariant and partial $F$-invariant in the context of cluster algebras and $\tau$-tilting theory.

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Peigen Cao. 2026-02-08. Newton polytopes in cluster algebras and $\tau$-tilting theory. https://arxiv.org/abs/2602.07929

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