Tropical symmetries of cluster algebras

TL;DR

This work develops a tropical framework for quasi-automorphisms of cluster algebras, showing that the action on g-vectors is realized by tropicalising the dual $\hat{y}$-variables, effectively interpreting g-vector transformations as coordinate changes in the tropical setting. Specialising to Grassmannian cluster algebras, the authors construct tropical analogues of braid-group actions and twist maps, relate g-vectors to semistandard Young tableaux via the dual canonical basis, and connect these to representations of quantum affine algebras. They introduce unstable and stable fixed points of quasi-automorphisms and demonstrate how stable fixed points yield prime non-real basis elements, while unstable fixed points give cluster monomials. In concrete cases ${\mathbb{C}}[\mathrm{Gr}(3,9)]$ and ${\mathbb{C}}[\mathrm{Gr}(4,8)]$, they prove that fixed-point data is governed by Euler’s totient function and reveal a novel link to scattering amplitudes through the interpretation of square-root structures as stable fixed points of braid actions. The results provide a unifying, combinatorial and algebro-geometric toolkit for exploring cluster structures, tableaux, and physical applications in a tropical, braid-theoretic setting.

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 $\CC[\Gr(4,8)]$.

Figures (6)

• Figure 1: The relevant seeds in $\mathcal{A}$ and $\mathcal{A}'$ and the action of the quasi-isomorphism $f$.
• Figure 2: The quiver diagram for the initial seed for ${\mathbb {C}}[\mathop{\mathrm{Gr}}\nolimits(5,10)]$, where $(a,b)$'s in the vertices are used to denote the positions of cluster variables and frozen variables.
• Figure 3: The g-vectors of the form $b{\bf g}_3 + c{\bf g}_4$ obtained by the braid group action for $\mathop{\mathrm{Gr}}\nolimits(4,8)$. In the case that the greatest common factor of $b,c$ is $1$, the number at coordinate $(b,c)$ is the degree of the polynomial in Plücker coordinates corresponding to the g-vector $b{\bf g}_3 + c{\bf g}_4$. If the greatest common factor of $b,c$ is greater than $1$, then we put a "$\bullet$" at the position $(b,c)$.
• Figure 4: The g-vectors of the form $b{\bf g}_3 + c{\bf g}_4$ obtained by the braid group action for $\mathop{\mathrm{Gr}}\nolimits(4,8)$. We draw arrows which correspond to the braid group action. The underlying undirected graph of the quiver in the figure is symmetric.
• Figure 5: The g-vectors of the form $b {\bf g}_1 + c{\bf g}_4$ obtained by the braid group action for $\mathop{\mathrm{Gr}}\nolimits(3,9)$. In the case that the greatest common factor of $b,c$ is $1$, the number at coordinate $(b,c)$ is the degree of the polynomial in Plücker coordinates corresponding to the g-vector $b{\bf g}_1 + c{\bf g}_4$. If the greatest common factor of $b,c$ is greater than $1$, then we put a "$\bullet$" at the position $(b,c)$.

Theorems & Definitions (56)

• Remark 2.1
• Lemma 2.2
• proof
• Example 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• Example 2.7
• Example 2.8
• Example 2.9