---
title: Tropicalisations of Quasi-Automorphisms
url: https://www.emergentmind.com/topics/tropicalisations-of-quasi-automorphisms
type: topic
---

# Tropicalisations of Quasi-Automorphisms

Tropicalisation of quasi-automorphisms is a procedure that recasts the action of quasi-automorphisms on cluster algebras in terms of piecewise-linear transformations in the tropical semiring. This approach provides a framework for analyzing the induced action on $g$-vectors and tableaux, yielding concrete combinatorial and geometric consequences, notably in the context of Grassmannian cluster algebras and related representation theory and mathematical physics applications [2601.19779].

## 1. Quasi-automorphisms of Cluster Algebras

A quasi-automorphism of a cluster algebra $\mathcal{A}$ of geometric type with frozen-monomial semifield $\mathcal{P}$ is an algebra homomorphism $f:\mathcal{A}\to\mathcal{A}$ characterized as follows. There exist two seeds
$$
\Sigma_0 = (x_1,\ldots,x_n; \hat{y}_1, \ldots, \hat{y}_n; B), \quad
\Sigma_1 = (x_1',\ldots,x_n'; \hat{y}_1',\ldots,\hat{y}_n'; B'),
$$
a permutation $\pi \in S_n$, and a sign $\varepsilon = \pm 1$ such that 
- $f(x_i) \propto x_{\pi(i)}'$ (up to frozen-monomial factor),
- $f(\hat{y}_i) = \hat{y}_{\pi(i)}'^\varepsilon$,
- $B' = \varepsilon B^\pi$.

The $\hat{y}_i$ are defined as the homogeneous $y$-variables of $\Sigma_0$, given by $\hat{y}_i = y_i \prod_j x_j^{b_{ji}}$. This class of maps generalizes automorphisms, permitting label permutations, sign reversal, and nontrivial action on coefficients [2601.19779].

## 2. Tropicalisation and Piecewise-linear $g$-vector Dynamics

Tropicalisation, in this context, refers to replacing usual arithmetic with the tropical operations: addition becomes $\max$, and multiplication becomes $+$. Fixing an initial seed $\Sigma_0$ and letting $\Sigma = f(\Sigma_0)$, one extends $f$ to the ambient field and examines its action on $\hat{y}$-variables:
$$
\hat{y}_{i;0} \mapsto \hat{y}_{\pi(i);t}^{\varepsilon}.
$$
Applying the tropicalisation $Trop^+$ (i.e., interpreting products as sums and sums as maxima), the image of $\hat{y}$-variables yields coordinates $v_i$ satisfying the $g$-vector mutation rule. For a mutation at $k$:
\[
v_k' = -v_k,\qquad 
v_j' = v_j + [b_{jk}]_+ v_k - b_{jk} \max(v_k, 0) \text{ for } j \neq k.
\]
Composing such updates along a mutation path results in a piecewise-linear map $Q^+_{t,t_0}:\mathbb{Z}^n\to\mathbb{Z}^n$ such that for any cluster variable $x$,
$$
g(f \;\triangleright\; x; t_0) = (Q^+_{t,t_0} \circ \pi^{-1})(g(x; t_0))
$$
for $\varepsilon=+1$. This allows the direct computation of the action on $g$-vectors via tropically interpreted quasi-automorphisms [2601.19779].

## 3. Explicit Tropical Actions: Braid Group and Twist Maps

In Grassmannian cluster algebras $\mathbb{C}[\mathrm{Gr}(k,n)]$, the tropicalisation procedure yields explicit combinatorial actions:
- **Braid-group generators $\sigma_i$** as defined by Fraser act on matrices $p \in \mathrm{Gr}(k, n)$ by specific replacements of column vectors, giving rise to rational actions on $\hat{y}$-variables. Tropicalising these rational expressions provides explicit max-plus maps $Q^+_{\sigma_i}$ on $g$-vectors, capturing the braid group symmetry at the piecewise-linear level.
- **Twist map $\tau$** (Marsh–Scott) operates by applying a signed exterior product to consecutive columns of the matrix $p$, again resulting in a rational function of the $\hat{y}$-variables whose tropicalisation defines $Q^+_\tau$. The induced action on $g$-vectors has concrete combinatorial interpretations, seen explicitly in examples such as $\mathrm{Gr}(3,6)$, where the action on Plücker coordinates corresponds directly to the max-plus piecewise-linear update.

These operations generalize to produce tropical analogues of both the classical braid group and twist symmetries, now acting naturally on the discrete combinatorics of $g$-vectors and tableaux [2601.19779].

## 4. Stable and Unstable Fixed Points in Tropical Dynamics

For a quasi-automorphism $f$ of rank-$n$ cluster algebra, a $g$-vector $g\in \mathbb{Z}^n$ is a fixed point if $f(g) = g$. It is classified as *stable* if, for any generic $g'$, the iterates $f^m(g')$ eventually converge projectively to $g$; otherwise, it is *unstable*. 

In the context of Grassmannian algebras:
- Every cluster-monomial $g$-vector is unstable under the braid action.
- Distinguished tableaux in $\mathrm{Gr}(4,8)$ such as
  $$
  T_1 = \left\llbracket 1,3; 2,5; 4,7; 6,8 \right\rrbracket,~
  T_2 = \left\llbracket 1,2; 3,4; 5,6; 7,8 \right\rrbracket
  $$
  and in $\mathrm{Gr}(3,9)$
  $$
  \left\llbracket 1,3,4; 2,6,7; 5,8,9 \right\rrbracket,~
  \left\llbracket 1,2,5; 3,4,8; 6,7,9 \right\rrbracket,~
  \left\llbracket 1,2,3; 4,5,6; 7,8,9 \right\rrbracket
  $$
  are stable fixed points for the combined action of $\sigma$ and $\tau$. Iterating these actions on other $g$-vectors results in radial convergence to those fixed points in the tropical (max-plus) geometry. 

This classification underpins further structure in the combinatorics and representation theory of the cluster algebra [2601.19779].

## 5. Enumeration of Non-real Tableaux via Euler's Totient Function

The structure induced by tropicalised quasi-automorphisms on Grassmannian cluster algebras gives rise to counting formulas for prime non-real tableaux. Stable fixed points generate infinite "cones" in the $g$-vector lattice under braid group action. For $\mathrm{Gr}(3,9)$, all braid images of a fixed tableau $T$ yield non-real tableaux in "ranks" $r$ with multiplicities:
$$
N_{3,9}(r) = 
  \begin{cases}
  0, & 3 \nmid r \\
  3\cdot\varphi(r/3), & 3\mid r
  \end{cases}
$$
where $\varphi$ is Euler's totient function. Similarly, for $\mathrm{Gr}(4,8)$,
$$
N_{4,8}(r) = 
  \begin{cases}
  0, & r~\text{odd} \\
  2\cdot\varphi(r/2), & r~\text{even}
  \end{cases}
$$
The proof involves decomposing the lattice of $g$-vectors into integer-basis cones and analyzing the action of the braid group as shear translations, with enumeration reduced to count points of coprime coordinates, hence the appearance of the totient [2601.19779]. This provides a direct combinatorial link between the tropical geometry of quasi-automorphism dynamics and arithmetic functions.

## 6. Applications to Scattering Amplitudes: The Four-mass Box Integral

Tropical fixed point analysis has implications for the symbolic structure of scattering amplitudes in physics. In the two-loop eight-point amplitude, the four-mass box integral contributes a symbol letter
$$
L = z_0 \pm B_z \sqrt{\Delta}, \quad \Delta = A^2 - 4B,
$$
where $A$ and $B$ are (dual-)canonical functions associated to a particular tableau $T = \left\llbracket 1,3;2,5;4,7;6,8 \right\rrbracket$. It is established that this $T$ is the stable fixed point of $\tau^2$ in $\mathrm{Gr}(4,8)$. The discriminant $\Delta$ thus acquires a tropical-geometric interpretation as emerging from the stable fixed point structure of the twist (or Brauer generator) in the cluster algebra. This connection yields a novel rationale for the presence and form of square-root letters in the symbol of such scattering amplitudes [2601.19779].

## 7. Summary Table of Key Structures and Dynamics

| Structure                   | Quasi-Automorphism Action              | Tropicalisation Outcome                        |
|-----------------------------|----------------------------------------|------------------------------------------------|
| Cluster algebra $\mathcal{A}$ | $f:\mathcal{A}\rightarrow\mathcal{A}$ | Label permutation, sign, rational function     |
| $\hat{y}$-variables         | $f(\hat{y}_i) = \hat{y}_{\pi(i)}^\varepsilon$ | Piecewise-linear map on $g$-vectors            |
| Braid group, twist map      | Explicit (matrix/Plücker) rational action | Max-plus piecewise-linear update of $g$-vectors|
| Tableau (Grassmannian)      | Braid/twist iterations                 | Stable/unstable fixed point for $g$-vectors    |

The tropicalisation of quasi-automorphisms thus serves as a unifying language linking algebraic, combinatorial, and physical structures through the geometry of piecewise-linear transformations [2601.19779].

Source: https://www.emergentmind.com/topics/tropicalisations-of-quasi-automorphisms