Tropicalisations of Quasi-Automorphisms

• Tropicalisations of quasi-automorphisms are procedures that convert algebraic actions into max-plus, piecewise-linear maps, enabling combinatorial analysis of g-vectors.
• The method provides explicit actions for braid group generators and twist maps that redefine dynamics in Grassmannian cluster algebras with stable and unstable fixed points.
• Applications include counting non-real tableaux via Euler’s totient function and interpreting scattering amplitude symbols through tropical geometry.

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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026). 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 (Drummond et al., 27 Jan 2026).

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 (Drummond et al., 27 Jan 2026).