Tropical Duality in $(d+2)$-angulated categories

TL;DR

This paper generalizes the concept of tropical duality from 2-Calabi-Yau triangulated categories to higher-dimensional $(d+2)$-angulated categories, establishing new properties of cluster tilting objects and $c$-vectors.

Contribution

It extends tropical duality to $(d+2)$-angulated categories using higher cluster tilting objects, demonstrating sign coherence and module-theoretic recoverability of $c$-vectors.

Findings

Generalization of tropical duality to higher dimensions

Sign coherence of $c$-vectors in $(d+2)$-angulated categories

Formulas for computing $c$-vectors

Abstract

Let $\mathscr{C}$ be a $2$-Calabi-Yau triangulated category with two cluster tilting subcategories $\mathscr{T}$ and $\mathscr{U}$. Results by Demonet-Iyama-Jasso and J{\o}rgensen-Yakimov known as tropical duality says that the index with respect to $\mathscr{T}$ provides an isomorphism between the split Grothendieck groups of $\mathscr{U}$ and $\mathscr{T}$. We also have the notion of $c$-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, $c$-vectors in the $(d+2)$-angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the $c$-vectors are recoverable as dimension vectors of modules in a module category.