Tropical symmetries of cluster algebras

TL;DR

This paper explores tropical symmetries of cluster algebras, especially Grassmannian types, revealing new actions, fixed points, and applications to physics, including scattering amplitudes and combinatorial structures.

Contribution

It introduces a tropical perspective on quasi-automorphisms, derives analogues of braid group actions, and connects these to combinatorics and physics applications.

Findings

Tropical actions on g-vectors can be realized via coordinate changes.

Derived tropical analogues of braid group actions and twist maps.

Connected fixed points of quasi-automorphisms to combinatorial counts and physics interpretations.

Abstract

We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen initial cluster. This perspective allows us to interpret the action on g-vectors as a change of coordinates in the tropical setting. Focusing on Grassmannian cluster algebras, we analyse tropicalisations of quasi-automorphisms in detail. We derive tropical analogues of the braid group action and the twist map on both g-vectors and tableaux. We introduce the notions of unstable and stable fixed points for quasi-automorphisms, which prove useful for constructing cluster monomials and non-real modules, respectively. As an application, we demonstrate that the counts of prime non-real tableaux with a fixed number of columns in $\mathrm{SSYT}(3, [9])$ and $\mathrm{SSYT}(4, [8])$, arising from the braid group action on stable fixed points, are governed by Euler's totient function. Furthermore, we apply our findings to scattering amplitudes in physics, providing a novel interpretation of the square root associated with the four-mass box integral via stable fixed points of quasi-automorphisms of the Grassmannian cluster algebra $\CC[\Gr(4,8)]$.