Categorifications of Non-Integer Quivers: Types $H_4$, $H_3$ and $I_2(2n+1)$

TL;DR

This paper introduces a new framework for categorifying mutations of certain finite type quivers with real weights, extending the theory to types $H_4$, $H_3$, and $I_2(2n+1)$, and establishing compatibility with tropical seed patterns.

Contribution

It defines weighted unfoldings for quivers with real weights and uses this to categorify mutations of specific finite types, linking them to classical types and tropical structures.

Findings

Categorification of mutations for types $H_4$, $H_3$, and $I_2(2n+1)$.

Introduction of weighted unfoldings and their semiring actions.

Compatibility of tropical seed patterns with unfoldings.

Abstract

We define the notion of a weighted unfolding of quivers with real weights, and use this to provide a categorification of mutations of quivers of finite types $H_4$, $H_3$ and $I_2(2n+1)$. In particular, the (un)folding induces a semiring action on the categories associated to the unfolded quivers of types $E_8$, $D_6$ and $A_{2n}$ respectively. We then define the tropical seed pattern on the folded quivers, which includes $c$- and $g$-vectors, and show its compatibility with the unfolding.