\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,mathrsfs,indentfirst,xcolor} \usepackage[a4paper,margin=27mm]{geometry} \usepackage[colorlinks=true,allcolors=blue,hyperfootnotes=false]{hyperref} \urlstyle{same} \setlength{\parindent}{1.5em} \setlength{\emergencystretch}{2em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newcommand{\kk}{\Bbbk} \newcommand{\ZZ}{\mathbb Z} \newcommand{\RR}{\mathbb R} \newcommand{\QQ}{\mathbb Q} \newcommand{\NN}{\mathbb N} \newcommand{\LL}{\mathsf L} \newcommand{\D}{\mathfrak D} \newcommand{\tht}{\vartheta} \newcommand{\Aprin}{\mathcal A_{\mathrm{prin}}} \newcommand{\Up}{\operatorname{up}} \newcommand{\Mid}{\operatorname{mid}} \newcommand{\Can}{\operatorname{can}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Proj}{\operatorname{Proj}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\val}{\operatorname{val}} \newcommand{\pair}[2]{\langle #1,#2\rangle} \title{Positive polytopes and the full Fock--Goncharov conjecture} \author{Qiyue Tang} \date{} \begin{document} \maketitle \begingroup \renewcommand{\thefootnote}{} \footnotetext{\textit{2020 Mathematics Subject Classification.} Primary 13F60; Secondary 14M25.\newline \textit{Keywords.} Cluster algebra, theta function, positive polytope, scattering diagram.} \endgroup \begin{abstract} We prove the origin-interior form of the polytope criterion of Gross, Hacking, Keel and Kontsevich: over a field of characteristic zero, full Fock--Goncharov for a principal coefficient cluster variety is equivalent to the existence of a bounded full-dimensional positive polytope containing the origin in its interior. Replacing the interior condition by membership of the origin gives a false criterion. \end{abstract} \begingroup\color{blue} \section*{AI use and scholarly credit} The initial research manuscript was developed using ChatGPT-6-Pro during a research session lasting approximately 270 minutes. A further ChatGPT-6-Pro session of approximately 14 minutes examined the role and necessity of the origin-interior condition. Codex reviewed the arguments, checked the literature, and revised the exposition. A preceding version of the manuscript passed an AI review using Rethlas-OSV \cite{RethlasOSV}, which incorporates the OSVerify framework of Nie, Ye, Liu, Wang, Ermon and Zou \cite{NieEtAl}. By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work. \par\endgroup \section{Introduction} Fomin and Zelevinsky introduced cluster algebras through seeds and their mutations \cite{FZ}. The Laurent phenomenon places all cluster variables in the Laurent polynomial ring of any chosen seed. The intersection of these Laurent rings is the upper cluster algebra. Constructing a basis compatible with mutation requires more than expressing individual cluster variables: its labels and multiplication must also be independent of the initial seed. Fock and Goncharov placed this question in the geometry of cluster ensembles \cite{FG06,FG}. Their two spaces, of types $\mathcal A$ and $\mathcal X$, are obtained by gluing algebraic tori through subtraction-free birational transformations. Duality relates functions on one space to integral tropical points of a dual space. In higher Teichm\"uller theory, laminations and positive coordinates provide models for such a relation. The duality conjectures in \cite[Section~4]{FG06} and \cite[Section~4]{FG} propose canonical functions, positive multiplication, and compatible tropical pairings. The latter paper also constructs canonical pairings in finite type. These formulations distinguish the existence of functions from the additional positivity and compatibility properties expected of them. Gross, Hacking and Keel related cluster varieties to blowups of toric varieties \cite{GHK}. Their examples show that global regular functions can be too scarce to accommodate all tropical labels. For the rank-two skew form with three copies of each of the directions $(1,0)$, $(0,1)$ and $(-1,-1)$, the regular functions on the $\mathcal X$-variety come from its base torus. This example will provide the failure of the positive-polytope criterion when the origin is allowed on the boundary. The original duality question consequently separates into constructing formal functions and determining which of them are regular and span the upper algebra. Gross, Hacking, Keel and Kontsevich constructed theta functions by counting broken lines in a consistent scattering diagram \cite{GHKK}. Their wall positivity gives nonnegative integral multiplication coefficients. The middle algebra is spanned by the theta functions with finite Laurent expansions; their results identify it with the upper algebra under geometric conditions on the cluster complex. In particular, a cluster complex not contained in a half-space yields the full basis theorem for principal coefficients. The enough-global-monomials condition connects this geometry with bounded tropical polytopes. Their compactification construction associates a graded theta algebra to a positive polytope and produces toric degenerations. Conjecture~0.10 of \cite{GHKK} asks whether the full Fock--Goncharov conjecture for the principal coefficient variety $\Aprin$ is equivalent to the existence of a bounded full-dimensional polytope convex in their sense. It appears as Conjecture~0.15 in the earlier 2014 author manuscript. The preceding construction uses sets $\{f\ge-1\}$ for continuous homogeneous piecewise linear functions $f$. Each such set contains the origin in its interior. The criterion below explicitly includes this property. Cheung, Magee and N\'ajera Ch\'avez proved that positivity of a closed subset is equivalent to broken-line convexity \cite{CMN}. This identifies the multiplication condition needed for compactification with a condition on paths in the tropical space. Bossinger, Cheung, Magee and N\'ajera Ch\'avez subsequently constructed positive Newton--Okounkov bodies from boundary divisors on partial minimal models, including applications to flag varieties \cite{BCMN}. These results relate tropical convexity to graded rings of sections. More recently, Cheung, Magee, Mandel and Muller proved valuative independence and finite-value theta reciprocity and developed the deformation of minimizing broken lines into taut ones \cite{CMMM}. We use the same local deformation principle and give the finite-witness argument for the cylindrical formal series arising here. The proof of the polytope criterion has two algebraic requirements. Every tropical label must define a finite Laurent theta function, and these functions must span the upper algebra. The first follows from a bound on the complete broken-line trajectory, obtained from the gauge of an absorbing positive polytope. For the second, a compactification and its Poisson structure reduce spanning to the exclusion of missing prime divisors. Their valuations produce a tangent scattering diagram; a minimum-weight argument rules out the resulting vanishing of a theta series. Work over a field $\kk$ of characteristic zero. We use complete principal coefficients: one invertible coefficient variable is added for each seed index, including original frozen indices. Fix a seed and identify the integral tropical points of the dual of $\Aprin$ with a lattice $\LL$. All frozen variables, including principal coefficient variables, are invertible. The upper algebra $\Up(\Aprin)$ is the intersection of the Laurent coordinate rings of the seed tori inside their common function field. For $m\in\LL$, let $\tht_m$ denote the formal theta function. Write $\Theta\subseteq\LL$ for the labels whose theta functions are Laurent polynomials in every seed. Their span is $\Mid(\Aprin)$. When all labels occur and multiplication has finite support, $\Can(\Aprin)$ denotes the algebra with vector space basis $(\tht_m)_{m\in\LL}$. In the principal coefficient setting, \emph{full Fock--Goncharov} means that the theta map is injective and \[ \Theta=\LL,\qquad \Can(\Aprin)=\Mid(\Aprin)=\Up(\Aprin). \] The formal multiplication constants are denoted by $\alpha(p,q,r)$; they are nonnegative integers. A convex polytope $\Xi\subset\LL_\RR:=\LL\otimes_\ZZ\RR$ is \emph{positive} if \begin{equation}\label{eq:positive} p\in a\Xi\cap\LL,\quad q\in b\Xi\cap\LL,\quad \alpha(p,q,r)\ne0 \quad\Longrightarrow\quad r\in(a+b)\Xi \end{equation} for all $a,b\in\NN=\{0,1,\ldots\}$, with $0\Xi=\{0\}$. Cheung, Magee and N\'ajera Ch\'avez identify this condition with broken-line convexity in the principal coefficient setting \cite[Theorem~6.1]{CMN}. \begin{theorem}\label{thm:main} Let $\Aprin$ be a cluster variety with complete principal coefficients over a field of characteristic zero, for skew-symmetrizable seed data. The following conditions are equivalent: \begin{enumerate} \item The full Fock--Goncharov conjecture holds for $\Aprin$. \item There is a bounded full-dimensional positive polytope $\Xi\subset\LL_\RR$ with $0\in\operatorname{int}\Xi$. \end{enumerate} When these conditions hold, $\Xi$ may be chosen rational. \end{theorem} The proof is completed in Section~\ref{sec:conclusion}. \begin{remark}\label{rem:scope} The short statement of \cite[Conjecture~0.10]{GHKK} does not explicitly impose $0\in\operatorname{int}\Xi$. Its preceding sets $\{f\ge-1\}$, for continuous homogeneous $f$, have this property. The dilation argument in \cite[Proposition~8.21]{GHKK}, numbered Proposition~8.17 in the published version, also uses absorption of every tropical label. Theorem~\ref{thm:main} gives the criterion with this condition explicit. With general positive polytopes, the condition cannot be weakened to $0\in\Xi$, as the following proposition shows. \end{remark} \begin{proposition}\label{prop:counterexample} There is an eighteen-dimensional principal coefficient cluster variety for which full Fock--Goncharov fails, although its tropical dual contains a bounded full-dimensional rational positive polytope containing the origin. \end{proposition} \begin{proof} For any seed with $n$ mutable indices and no original frozen indices, in the initial coordinates set \[ P_0=[0,1]^n\times[-1,1]^n\subset\RR^n\oplus\RR^n. \] Every dilate lies in the closed positive cluster chamber. Theta functions with labels in that chamber are cluster monomials, so their products have the single output $p+q$ \cite[Theorem~7.5]{GHKK}. Convexity gives $(p+q)/(a+b)\in P_0$ for $a,b>0$; zero degrees follow from $\tht_0=1$. Thus $P_0$ is positive and $0\in\partial P_0$. Take $N=\ZZ^9$ with basis $e_1,\ldots,e_9$. Set $w_1=w_2=w_3=(1,0)$, $w_4=w_5=w_6=(0,1)$ and $w_7=w_8=w_9=(-1,-1)$, and define $\{e_i,e_j\}=\det(w_i,w_j)$. All indices are mutable and all multipliers are one. This is the seed of \cite[Example~6.3 and Section~7]{GHK}. Let $M=\operatorname{Hom}(N,\ZZ)$, and let $p^*:N\to M$ send $n$ to the functional $\{n,\cdot\}$ of this seed. Put $K=\ker p^*$. The map $e_i\mapsto w_i$ is surjective onto $\ZZ^2$. Nondegeneracy of the determinant form gives \[ K=\left\{\sum_i a_i e_i:\sum_i a_iw_i=0\right\},\qquad \operatorname{rank}K=7. \] For this seed, \cite[Section~7]{GHK} identifies the ring of regular functions on $\mathcal X$ with the group algebra of $K$, that is, the ring of finite sums of characters $z^k$, $k\in K$. The natural pullback identifies \[ \Gamma(\mathcal X,\mathcal O_{\mathcal X})=\kk[K]. \] We may take $\kk=\mathbb C$. The morphism $\widetilde p:\Aprin\to\mathcal X$ is a torsor under the torus $T_N=\Spec\kk[M]$ \cite[Appendix~B]{GHKK}. On the initial chart it pulls $z^n$ back to $z^{(p^*n,n)}$. The weight of $\tht_{(m,n)}$ under this action is $m-p^*n$ \cite[Proposition~7.7]{GHKK}. If $n\notin K$ and $(p^*n,n)\in\Theta$, this theta function would be a regular invariant and hence a finite linear combination of $z^{(0,k)}=\tht_{(0,k)}$, $k\in K$. This contradicts linear independence of distinct principal theta functions. Hence $\Theta\ne\LL$. The above $P_0$ with $n=9$ gives the asserted polytope. \end{proof} The condition also cannot be supplied by translating an arbitrary positive polytope. In rank one, $xx'=1+y$, and \[ \tht_{(m,n)}=\begin{cases}x^my^n&m\ge0,\\(x')^{-m}y^n&m\le0.\end{cases} \] The positive rectangle $[0,1]\times[-1,1]$ translates to $Q=[-1/2,1/2]\times[-1,1]$. But $(1,2),(-1,2)\in2Q$ and their theta product is $y^4+y^5$, whose output $(0,5)$ is outside $4Q$. \section{Principal coefficients and broken lines}\label{sec:setup} For a lattice $L$, write $L^*=\operatorname{Hom}(L,\ZZ)$ and $L_{\QQ}=L\otimes_{\ZZ}\QQ$; the subscript $\RR$ similarly denotes extension to $\RR$. Its group algebra $\kk[L]$ has basis $z^m$, $m\in L$, and product $z^mz^{m'}=z^{m+m'}$. Its spectrum is the torus with character lattice $L$. Pairings between a lattice and its dual are evaluation pairings. Vectors are columns in the bases specified below; a transpose converts a column into a row. Let $N$ be a finite-rank lattice with basis $(e_i)_{i\in I}$, let $J\subset I$ be the mutable indices, and let $\{\ ,\ \}$ be the rational skew form of the seed. Choose positive integers $d_i$ and let $N^\circ\subset N$ have basis $d_i e_i$. The seed integrality conditions require $d_j\{e_i,e_j\}\in\ZZ$ and $d_i\{e_i,e_j\}\in\ZZ$ for $i\in J$, $j\in I$. Mutation takes place only at indices in $J$. Complete principal coefficients give $2|I|$ invertible variables on each seed chart; the convention adding coefficients only at mutable indices is treated in Section~\ref{sec:conclusion}. Put $M^\circ=\operatorname{Hom}(N^\circ,\ZZ)$, with dual basis $f_i=d_i^{-1}e_i^*$, and set $\LL=M^\circ\oplus N$. Subscript $\bullet J$ below means restriction to columns indexed by $J$; $I$ in a matrix expression denotes the identity matrix. Write \[ \Delta=\operatorname{diag}(d_i),\quad \Omega_{ij}=\{e_i,e_j\},\quad B=-\Delta\Omega,\qquad P=\binom{B_{\bullet J}}{I_{\bullet J}},\quad R=\binom{\Delta^{-1}_{\bullet J}}0. \] Here $B_{ji}=d_j\{e_i,e_j\}$, so the columns of $B$ indexed by $J$ are integral. The lower block $I_{\bullet J}$ embeds $\ZZ^J$ into the second summand of $\LL$. Projection onto these mutable second coordinates is a left inverse of $P$. In particular its image is a direct summand. The map $P:\ZZ^J\to\LL$ is saturated and injective; $R:\QQ^J\to\LL_\QQ^*$ gives rational wall normals. The matrix identity \begin{equation}\label{eq:skew} R^{\mathsf T}P=-\Omega_{JJ} \end{equation} is skew-symmetric. For $n\in\NN^J$, its \emph{charge degree} is $|n|_1=\sum_{i\in J}n_i$. Let $\ell:\LL_\RR\to\RR$ be the sum of the mutable coordinates in the second summand. Then $\ell(Pn)=|n|_1$. Write $e_i$ also for the $i$th standard basis vector of $\ZZ^J$ when it is an input to $P$ or $R$. The initial positive chamber is \[ \{q\in\LL_\RR:\pair{Re_i}{q}>0\text{ for every }i\in J\}. \] The subscript $+$ on $\tht_{m,+}$ denotes its expansion at a general point of this chamber. Introduce independent variables $t_i$, $i\in J$, and write $t^n=\prod_{i\in J}t_i^{n_i}$. A character with exponent $u\in\LL$ is written $z^u$. Use the standard consistent cluster scattering diagram $\D$ in $\LL_\RR$, with positive binomial wall functions, as constructed in \cite[Theorems~1.12--1.13 and Appendix~C]{GHKK}. A wall of charge direction $n\in\NN^J\setminus\{0\}$ has a closed rational polyhedral support in $(Rn)^\perp$ and function $(1+z^{Pn})^c$, $c\in\ZZ_{>0}$. Normals are normalized to primitive integral covectors when acting on characters; positive integer multiples are absorbed into $c$. For $K\ge0$, charge truncation means reduction modulo the monomials $t^n$ with $|n|_1>K$; only walls acting nontrivially in this quotient are retained. It has finitely many walls. At a transverse crossing with primitive normal $\nu$, the wall automorphism sends $z^u$ to $z^u f^{\nu(u)}$, with the sign of $\nu$ chosen according to the crossing direction. A path-ordered product composes these maps in the order in which the path crosses the walls. Consistency means that this product depends only on the endpoints of a path avoiding the joints, at every finite charge order. The only incoming walls are the initial ones: a wall is incoming when its support contains its positive correction direction. A broken line of input $m$ and endpoint $q$ is a continuous piecewise affine map $\gamma:(-\infty,0]\to\LL_\RR$, with $\gamma(0)=q$. On a segment with attached character exponent $e$, its velocity is $-e$; the first exponent is $m$. At a wall crossing, the new attached monomial is a term in the old one times the wall function to the positive integer power given by the absolute normal pairing. Thus a nontrivial bend changes $e$ by $Pn$ with $0\ne n\in\NN^J$. The path avoids wall boundaries and intersections with independent normals, collectively called joints. The function $\tht_{m,q}$ is the sum of its final attached monomials, with their positive coefficients. For formal computations replace each correction $z^{Pn}$ by $z^{Pn}t^n$ and complete in total $t$-degree. For a fixed input, the $z$-exponent determines the charge, so this bookkeeping preserves coefficients and consistency. For a fixed total charge $n$, there are at most $|n|_1$ nontrivial bends. The applicable walls, choices of corrections, and ordered records are finite at that charge. A record and an endpoint determine all crossing times by the reverse reconstruction below. Hence every fixed charge has finitely many contributions. All formal operations use the inverse limit of these finite charge quotients. Theta transportation is the path-ordered wall product \cite[Theorem~3.5]{GHKK}. We use the near-output multiplication rule \cite[Definition--Lemma~6.2 and Proposition~6.4]{GHKK}. For fixed $p,q,r$, $\alpha(p,q,r)$ is the finite sum of coefficients of pairs of broken lines whose final exponents add to $r$, with endpoint sufficiently close to $r$. In particular, \begin{equation}\label{eq:triangular} \alpha(p,q,r)\ne0\ \Longrightarrow\ r=p+q+Pn\ (n\in\NN^J), \qquad \alpha(p,q,p+q)=1. \end{equation} Coefficient extraction here means taking the coefficient of the character $z^r$ in a fixed finite charge truncation. The same extraction is linear and applies to brackets as well as products. We will choose endpoints outside a countable union of proper affine subspaces. Indeed, a finite rational bend record reconstructs each segment as $Cq-te$, with $C$ invertible and rational. Reversing a bend $\delta$ of normal $\nu$ replaces $C$ by $(I-\delta\nu/\nu(e))C$, an invertible shear since $\nu(\delta)=0$. Meeting a joint contained in a linear space $U$ of codimension at least two forces $q\in C^{-1}(U+\RR e)$. A segment lying in a wall hyperplane $H$ forces $q\in C^{-1}H$. These are proper subspaces. There are countably many records. This construction also allows countably many prescribed translates of the endpoint and finitely many diagrams. Baire's theorem supplies such endpoints in every nonempty open set. \section{A positive polytope bounds theta functions}\label{sec:trajectory} Suppose $\Xi$ satisfies condition (2) of Theorem~\ref{thm:main}. Its Minkowski gauge is \[ g(x)=\inf\{t\ge0:x\in t\Xi\}. \] It is finite, continuous, positively homogeneous, and the maximum of finitely many linear functions. Boundedness and the interior condition give $g(x)\ge c\|x\|$ for some $c>0$ and a fixed norm. Absorption of all lattice points is equivalent to the interior condition: otherwise separation gives a nonzero linear function nonnegative on $\Xi$, and lattice points on the opposite side belong to no nonnegative dilate. \begin{lemma}\label{lem:gauge} If $\alpha(p,q,r)>0$, then $g(r)\le g(p)+g(q)$. On every broken line, $g\circ\gamma$ is convex in its affine time parameter. \end{lemma} \begin{proof} The scaling property \cite[Lemma~8.10]{GHKK} gives $\alpha(kp,kq,kr)>0$ for every positive integer $k$. Apply \eqref{eq:positive} in degrees $\lceil kg(p)\rceil$ and $\lceil kg(q)\rceil$, divide by $k$, and take the limit. To obtain the scaling property, multiply the decorations and the geometry of a contributing pair by $k$. Each old selected term gives a nonzero term with $k$ times its exponent in the corresponding $k$th power. Choosing the original endpoint sufficiently close to $r$ makes the new endpoint sufficiently close to $kr$. Positive coefficients preserve the contribution. The balanced-pair construction of \cite[Proposition~5.4]{CMN} applies this inequality to a rational point of a rational broken-line segment, with the ratio of the two degrees equal to its affine time ratio. It gives the Jensen inequality on those segments. To extend it to real endpoints, fix the finite bend record, including all chosen factors and monomials. Reverse reconstruction expresses the crossing times and segment positions as linear functions of the endpoint divided by the fixed nonzero normal pairings. In the charge truncation of this record, support membership, transversality and time order are finite open conditions. At a nearby rational endpoint, keep this fixed record and interpret it as a nongeneric limit segment in the sense of \cite[Definition~3.3]{CMN}. The strict support and time conditions persist. It is the pointwise limit of real general-endpoint paths with the same decorations and coefficients, obtained by avoiding the countable exceptional sets above. Rational endpoints of this kind are dense in the finite open set of admissible records. For three times $s0$ so that \[ \pair{Re_i}{q+um}\ge\tfrac12\pair{Re_i}{q}>0 \quad(0\le u\le\epsilon,\ i\in J). \] For $-T