\documentclass[11pt]{article} \usepackage[a4paper,margin=29mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{indentfirst} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{xcolor} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,hyperfootnotes=false]{hyperref} \urlstyle{same} \setlength{\parindent}{1.5em} \setlength{\parskip}{0pt} \emergencystretch=1em \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newcommand{\C}{\mathbb C} \newcommand{\Z}{\mathbb Z} \newcommand{\NN}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\fr}{\mathrm{fr}} \newcommand{\mut}{\mathrm{mut}} \newcommand{\op}{\mathrm{op}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\DT}{\operatorname{DT}} \newcommand{\var}{\operatorname{var}} \newcommand{\fg}{\boldsymbol f} \newcommand{\wlong}{w_{\mathrm{long}}} \newcommand{\eps}{\varepsilon} \title{Biperfectness of the theta basis of a maximal unipotent subgroup} \author{Qiyue Tang} \date{} \begin{document} \maketitle \begingroup \renewcommand{\thefootnote}{} \footnotetext{\textit{2020 Mathematics Subject Classification.} Primary 13F60; Secondary 17B10.\newline \textit{Keywords.} Theta basis, perfect basis, maximal unipotent subgroup, cluster algebra.} \endgroup \begin{abstract} We prove that the theta basis of the coordinate ring of a maximal unipotent subgroup of a complex semisimple group is biperfect. This answers affirmatively Question~6.5 in Kamnitzer's ICM~2022 survey. \end{abstract} \begingroup \color{blue} \section*{AI use and scholarly credit} The results were obtained in a 65-minute Codex session using GPT-6-Astra at high reasoning effort. The author and Codex then reviewed and refined the arguments and exposition over several rounds. The manuscript subsequently passed an AI review using Rethlas-OSV~\cite{Rethlas,RethlasOSV,OSVerify}. 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} The search for distinguished bases in representation theory connects the action of Lie algebra generators, tensor product multiplicities, and positivity. For a complex semisimple group, these questions can be studied together in the coordinate ring of a maximal unipotent subgroup. The dual canonical basis of Lusztig and Kashiwara, Lusztig's dual semicanonical basis in simply-laced type, and the Mirkovi\'c--Vilonen basis arise from different constructions. Berenstein and Kazhdan introduced perfect bases to describe the leading action of Lie algebra generators; see \cite[\S1.1]{BKK}. In this normalization, the leading action of each positive Chevalley generator is a single basis vector, with a prescribed coefficient and a remainder of smaller nilpotence degree. The left and right multiplication actions on the unipotent subgroup lead to biperfectness. The common bicrystal of biperfect bases records their leading operators even when the bases themselves differ; see \cite[\S\S1--2]{BKK} and \cite[\S1]{Kam}. Cluster algebras provide a second approach to distinguished bases. Fomin and Zelevinsky introduced them in connection with canonical bases and total positivity~\cite{FZ}. Mutation produces overlapping coordinate systems and their cluster monomials. Extending those monomials to a basis becomes essential when the cluster monomials fail to span the algebra. The cluster structures on double Bruhat cells~\cite{BFZ} and on unipotent coordinate rings place this question inside representation theory. In the latter setting, the dual canonical and, in simply-laced type, dual semicanonical bases contain cluster monomials; Kamnitzer reviews these results in \cite[\S6.1]{Kam}. The theta construction grew from the duality conjectures of Fock and Goncharov~\cite{FG} and the mirror-symmetry construction of Gross, Hacking, and Keel~\cite{GHK}. These predict bases indexed by integral tropical points of a dual space. Gross, Hacking, Keel, and Kontsevich~\cite{GHKK} construct the functions by broken lines: piecewise linear paths whose attached monomials change when they cross walls of a scattering diagram. Consistency of the diagram makes the construction compatible with mutation. Their theorem supplies a theta basis for a middle algebra between the cluster algebra and its upper cluster algebra, together with nonnegative multiplication constants. Under their geometric hypotheses the middle and upper algebras coincide. They also treat partial compactifications obtained by allowing frozen variables to vanish, which select the theta functions regular along the boundary. Quantum positivity developed alongside this construction. Ben Davison proved positivity for skew-symmetric quantum cluster algebras using purity in cohomological Donaldson--Thomas theory~\cite{Dav}. Davison and Mandel then combined Donaldson--Thomas theory with scattering diagrams to construct quantum theta functions~\cite{DM}. Their Laurent coefficients and multiplication constants are nonnegative Laurent polynomials in the quantum parameter, and their classical limits recover the GHKK theta functions. These results explain the compatibility between scattering constructions and positivity, while biperfectness asks for a specific leading term under a Lie algebra action. On the representation-theoretic side, Baumann, Kamnitzer, and Knutson construct the Mirkovi\'c--Vilonen basis of the unipotent coordinate ring and prove that it is biperfect~\cite[Theorem~6.2]{BKK}. Their construction uses the geometric Satake correspondence, and the Chevalley action is expressed through intersection multiplicities. Kamnitzer's ICM~2022 report~\cite{Kam} compares this basis with the dual canonical and dual semicanonical bases, then introduces the theta basis in \S6.3. Theorem~6.4 there states that the theta-basis construction applies to the unipotent coordinate ring and that its elements are pointed in each cluster. The resulting tropical parametrization is naturally identified with the bicrystal underlying biperfect bases. This motivates Question~6.5: is the theta basis a biperfect basis? The report separately formulates compatibility of the Mirkovi\'c--Vilonen basis with clusters and its equality with the theta basis as Conjectures~6.6 and~6.9. Subsequent work develops both the crystal and scattering aspects of this question. Fei constructs crystal structures and their algebraic lifts for upper cluster algebras associated with quivers with potential~\cite{Fei}. In the version of June~2026, Theorem~9.24 proves BK-biperfectness of the generic basis for the weak upper normal crystal, and Conjecture~9.25 proposes the corresponding property for theta and triangular bases. Qin's freezing theorem~\cite{Freeze} identifies a selected part of a pointed theta expansion with a theta function for a seed with additional frozen vertices. Cheung, Magee, Mandel, and Muller~\cite{CMMM} prove valuative independence for theta functions of positive scattering diagrams and derive applications to boundary extensions. These results address the control of leading terms that is needed to relate theta functions to crystal operators. Let \(G\) be a connected complex semisimple group and \(N\) its maximal unipotent subgroup for a chosen positive root system. Fix a pinning, and let \(\mathcal I\) index the simple roots. For \(i\in\mathcal I\), let \(e_i\) be the positive Chevalley generator and put \(x_i(t)=\exp(te_i)\), for \(t\in\C\). On \(\C[N]\) define \begin{equation}\label{eq:derivations} D_i^R f(n)=\left.\frac{d}{dt}\right|_{t=0}f(nx_i(t)), \qquad D_i^L f(n)=\left.\frac{d}{dt}\right|_{t=0}f(x_i(t)n). \end{equation} The multiplication maps are algebraic, so substituting \(nx_i(t)\) or \(x_i(t)n\) into a regular function gives an element of \(\C[N][t]\). Differentiating more times than its \(t\)-degree gives zero. Thus both operators are locally nilpotent derivations. For such a derivation \(D\) and \(f\ne0\), set \[ \eps_D(f)=\max\{r\in\NN:D^rf\ne0\}, \qquad \NN=\{0,1,2,\ldots\}. \] We use the weight decomposition for the action of the maximal torus of the pinning \((h\cdot f)(n)=f(h^{-1}nh)\). A basis is \emph{good for \(D\)} if each \(\ker D^r\), \(r\ge0\), is spanned by a subset of that basis, with \(D^0\) the identity. It is \emph{perfect for \(D\)} if it is good and, for every basis element \(b\) with \(\eps_D(b)>0\), there is a basis element \(\widetilde e_D(b)\) such that \begin{equation}\label{eq:perfect} Db=\eps_D(b)\widetilde e_D(b)+r_b, \qquad r_b\in\ker D^{\eps_D(b)-1}. \end{equation} The nonzero arrows \(b\mapsto\widetilde e_D(b)\) are required to be injective. A weight basis of \(\C[N]\) is \emph{biperfect} if it contains \(1\) and is perfect for every \(D_i^L\) and \(D_i^R\). These conventions give the normalization in \cite[\S\S1.2, 1.4]{Kam}. Write \(\mathcal B_\theta(\C[N])\) for the regular theta basis in this cluster structure. \begin{theorem}\label{thm:main} The basis \(\mathcal B_\theta(\C[N])\) is biperfect. \end{theorem} This answers Question~6.5 of \cite{Kam} affirmatively. The proof uses coordinates in which one Chevalley derivation is a monomial times a partial derivative in a frozen variable. The highest degree part of a theta function in that variable is another theta function, obtained by freezing certain mutable vertices. Linear independence of these highest parts gives the kernel filtration. Frozen monomial translation then identifies the leading derivative with a unique theta function and gives the coefficient in \eqref{eq:perfect}. The prefix and suffix minor seeds realize the two multiplication actions. Their relevant frozen rows have opposite signs, so we also compare the theta functions for a seed and its opposite. The freezing operation is supplied by Qin~\cite{Freeze}. The comparison with the opposite seed uses reflection of scattering diagrams and the Donaldson--Thomas twist of Kimura, Qin, and Wei~\cite{KQW}. The required minor seeds and their reachability are given by Qin~\cite{QinDC}, using the cluster structures of Goodearl and Yakimov~\cite{GY}. We retain the symmetrizing lattice data in the reflection argument, so these inputs apply to all finite semisimple root systems. \section{Theta functions and freezing}\label{sec:theta} We work with classical cluster seeds of geometric type. A seed \(s\) consists of algebraically independent variables \(x_j\), indexed by a finite set \(I=I_{\mut}\sqcup I_{\fr}\), and an integer exchange matrix \[ B=(b_{jk})_{j\in I,\ k\in I_{\mut}} \] with skew-symmetrizable principal part. The indices in \(I_{\mut}\) are mutable and those in \(I_{\fr}\) are frozen. All frozen variables are inverted in this section. The ambient field is \(K=\C(x_j:j\in I)\). For \(k\in I_{\mut}\), mutation replaces \(x_k\) by \[ x_k'=\frac{\prod_{j\in I}x_j^{[b_{jk}]_+} +\prod_{j\in I}x_j^{[-b_{jk}]_+}}{x_k}, \qquad [a]_+=\max(a,0), \] leaving the other variables fixed. The new matrix has entries \[ b'_{ij}= \begin{cases} -b_{ij},&i=k\text{ or }j=k,\\ b_{ij}+[b_{ik}]_+[b_{kj}]_+-[-b_{ik}]_+[-b_{kj}]_+, &i\ne k,\ j\ne k, \end{cases} \qquad i\in I,\quad j\in I_{\mut}. \] This matrix change preserves rank: add \([b_{ik}]_+\) times row \(k\) to row \(i\ne k\), then add \([-b_{kj}]_+\) times column \(k\) to column \(j\ne k\), and finally negate row \(k\) and column \(k\). Here the coefficients refer to the matrix before mutation; its \(kk\)-entry is zero, so the indicated operations give the displayed formula. The upper cluster algebra is the intersection in \(K\) of the Laurent polynomial rings of all mutation-equivalent seeds. Let \(M^\circ=\Z^I\) be the exponent lattice, with basis \(\fg_j\). For \(g=(g_j)_{j\in I}\in M^\circ\), write \[ x^g=\prod_{j\in I}x_j^{g_j}. \] We use the sign convention of \cite[Appendix~A]{QinTrop}. Throughout the cluster arguments, \(B\) has full column rank. We specify the lattices used in the scattering construction. Choose positive integers \(d_j\), \(j\in I\), such that \(b_{jk}/d_j=-b_{kj}/d_k\) on the principal part. Let \(\mathsf N=\bigoplus_{j\in I}\Z\boldsymbol e_j\) and \(\mathsf N^\circ=\bigoplus_{j\in I}\Z d_j\boldsymbol e_j\). If \(\boldsymbol e_j^*\) is the basis dual to \(\boldsymbol e_j\), identify \[ M^\circ=\operatorname{Hom}(\mathsf N^\circ,\Z), \qquad \fg_j=d_j^{-1}\boldsymbol e_j^*, \qquad \langle d_j\boldsymbol e_j,\fg_k\rangle=\delta_{jk}. \] Here \(\delta_{jk}\) is \(1\) for \(j=k\) and \(0\) otherwise. The skew form \(\omega\) on \(\mathsf N_\mathbb Q=\mathsf N\otimes_\Z\mathbb Q\) is specified whenever the first index is mutable by \(\omega(\boldsymbol e_k,\boldsymbol e_j)=b_{jk}/d_j\). Extend it skew-symmetrically on the remaining pairs. Its restriction defines \[ p^*:\bigoplus_{k\in I_{\mut}}\Z\boldsymbol e_k\longrightarrow M^\circ, \qquad p^*\boldsymbol e_k=\sum_{j\in I}b_{jk}\fg_j. \] A cluster scattering diagram has walls supported on rational codimension-one cones in \(M^\circ_\R=M^\circ\otimes_\Z\R\). For a normal vector \(n\), its orthogonal hyperplane is \(n^\perp=\{m:\langle n,m\rangle=0\}\), and a wall with normal \(n\) has support in \(n^\perp\). A wall with normal \(n\), a primitive nonnegative integral combination of mutable \(\boldsymbol e_k\), carries a formal function \(f=1+\sum_{\ell\ge1}a_\ell z^{\ell p^*n}\). Here \(z^m\) denotes the formal character with exponent \(m\in M^\circ\), and \(z^m z^{m'}=z^{m+m'}\); in the initial chart it is \(x^m\). If \(n_\gamma\) is the primitive normal in \(\mathsf N^\circ\), with sign chosen so that \(\langle n_\gamma,\gamma'\rangle<0\) at a transverse crossing by a path \(\gamma\), its wall-crossing automorphism acts by \begin{equation}\label{eq:wall} z^m\longmapsto z^m f^{\langle n_\gamma,m\rangle}. \end{equation} Write \(p_{\gamma,\mathfrak d}\) for the automorphism in \eqref{eq:wall} at the wall supported on \(\mathfrak d\). The initial walls are \((\boldsymbol e_k^\perp,1+z^{p^*\boldsymbol e_k})\), \(k\in I_{\mut}\). A wall with normal \(n\) is incoming when its support contains \(p^*n\), and outgoing otherwise. An exponent increment \(B\nu\), with \(\nu\in\NN^{I_{\mut}}\), has order \(\sum_k\nu_k\); full column rank makes \(\nu\) unique. For every integer \(r\ge0\), only finitely many walls have a nontrivial crossing automorphism after discarding increments of order greater than \(r\). Consistency means that the ordered product of the crossing automorphisms depends only on the endpoints of a generic path, which crosses walls transversely and avoids their boundaries and intersections. The cluster diagram is determined, up to equality of these products, by its initial walls and the requirement that the additional walls be outgoing~\cite[Theorem~1.12]{GHKK}. A broken line is a piecewise linear path from an unbounded initial segment to a generic point \(Q\), with an attached monomial \(c z^m\) on each segment and velocity \(-m\). At a wall it chooses a term of the wall-crossing image of its previous monomial. Its initial monomial is \(z^g\), and \(g\) is its label. Summing the final monomials for label \(g\) gives the theta function \(\theta^s_{g,Q}\). These sums are interpreted in the completion defined by the order on \(B\NN^{I_{\mut}}\). For label zero the theta function is \(1\). At a generic point in the positive chamber \[ C^+=\{m\in M^\circ_\R:m_k\ge0\text{ for }k\in I_{\mut}\}, \] write \(\theta_g^s=\theta^s_{g,Q}\). Frozen coordinates are unrestricted in this chamber. A Laurent-finite theta function has the pointed expansion \begin{equation}\label{eq:pointed} \theta_g^s=x^g\sum_{\nu\in\NN^{I_{\mut}}}c_{g,\nu}x^{B\nu}, \qquad c_{g,0}=1, \end{equation} with finitely many nonzero coefficients. Theta functions are compatible with mutation and with scattering transport. A seed is \emph{injective-reachable} if a mutation-equivalent seed has mutable \(g\)-vectors equal to the negative mutable unit vectors after projection to \(\Z^{I_{\mut}}\), up to permutation. Here a \(g\)-vector is the exponent of the leading monomial in a pointed expansion. We denote such a seed by \(s[1]\). Every mutation-equivalent seed is then injective-reachable. Moreover, after the indicated permutation, \(s[1]\) and \(s\) have the same principal exchange matrix and symmetrizers; see \cite[\S2.12 and \S3.1]{KQW}. For a full-rank injective-reachable seed, all theta functions in \eqref{eq:pointed} are Laurent polynomials and form a basis of the upper cluster algebra~\cite[Theorem~A.1.5]{QinTrop}. Theta functions with labels in a reachable cluster chamber are the corresponding cluster monomials times frozen Laurent monomials~\cite[Theorems~A.1.3--A.1.4]{QinTrop}. For a subset \(F\subset I_{\mut}\), \emph{freezing \(F\)} makes those vertices frozen, without changing the variables. The new exchange matrix \(B'\) is obtained by deleting the columns in \(F\). On a pointed expansion define \begin{equation}\label{eq:freezing} \mathfrak f_{F,g}\!\left(x^g\sum_\nu c_\nu x^{B\nu}\right) =x^g\sum_{\nu:\,\nu_k=0\ (k\in F)}c_\nu x^{B\nu}. \end{equation} Full column rank makes this definition unambiguous. Qin's freezing theorem~\cite[Theorem~5.2]{Freeze} states that this is the theta function with label \(g\) for the frozen seed. Denote it by \(\theta'_g\). The frozen-factor property is \begin{equation}\label{eq:factor} x^u\theta'_g=\theta'_{g+u} \quad\text{if }u\in\Z^I \text{ is supported on }I_{\fr}\cup F. \end{equation} The \(F\)-polynomial of a pointed theta function is \(\sum_\nu c_{g,\nu}\prod_k t_k^{\nu_k}\), in independent variables \(t_k\) indexed by the mutable vertices. Substituting \(t_k=x^{B_k}\), where \(B_k\) is column \(k\) of the exchange matrix, gives the factor after \(x^g\) in \eqref{eq:pointed}. For the frozen seed it depends only on the remaining mutable coordinates of \(g\) and on the remaining principal exchange matrix; see \cite[Assumption~6.1 and the theta example]{KQW} and \cite[\S6.4]{Freeze}. This identity includes negative frozen exponents. \section{A frozen variable and a perfect filtration}\label{sec:highest} Fix a seed \(s\) and write \(\theta_g=\theta_g^s\) throughout this section. For a finite Laurent polynomial \(f\) in the variables \(x_j\), let \(\deg_{x_a}f\) be the largest exponent of \(x_a\) with nonzero coefficient when the other variables are treated as elements of \(\C(x_j:j\ne a)\). Set \(\deg_{x_a}0=-\infty\). Its part of degree \(d\) is the sum of all terms with \(x_a\)-exponent \(d\). \begin{lemma}\label{lem:independent} Suppose \(B'\) has full column rank and a family of Laurent polynomials, indexed by all \(g\in\Z^I\), has the form \[ q_g=x^g\left(1+\sum_{0\ne\nu\in\NN^{I_{\mut}\setminus F}} a_{g,\nu}x^{B'\nu}\right). \] Then the \(q_g\) are linearly independent over \(\C\). \end{lemma} \begin{proof} Choose a real linear functional \(\ell\) on \(\R^I\) taking the value \(1\) on each column of \(B'\); its existence follows from column independence. In a nonzero finite relation \(\sum_g a_gq_g=0\), choose an index \(g\) with \(a_g\ne0\) and minimal \(\ell(g)\). A nonleading exponent from any summand with nonzero coefficient has strictly larger \(\ell\)-value than its own label, and hence cannot equal \(g\). The leading exponents of different summands are distinct. The coefficient of \(x^g\) is therefore \(a_g\), a contradiction. If \(B'\) has no columns, the functions are the distinct monomials \(x^g\). \end{proof} \begin{proposition}\label{prop:highest} Suppose all the theta functions of a full-rank seed are Laurent finite. Let \(A\subset K\) be a \(\C\)-subalgebra with basis \(\{\theta_g:g\in S\}\), for a subset \(S\subset\Z^I\). Choose \(a\in I_{\fr}\) such that \(b_{ak}\le0\) for all \(k\in I_{\mut}\), and put \[ F=\{k\in I_{\mut}:b_{ak}<0\}. \] Let \(u\in\Z^I\) satisfy \(u_a=0\) and \(\supp(u)\cap I_{\mut}\subset F\), where \(\supp(u)=\{j:u_j\ne0\}\). Assume that \(D=x^u\partial/\partial x_a\) preserves \(A\) and is locally nilpotent on \(A\). Then the basis is good for \(D\), and \[ \eps_D(\theta_g)=g_a\quad(g\in S). \] For \(g_a>0\), the label \(g+u-\fg_a\) lies in \(S\), and \begin{equation}\label{eq:highest-result} D\theta_g=g_a\theta_{g+u-\fg_a}+r_g, \qquad r_g\in\ker D^{g_a-1}. \end{equation} The map \(g\mapsto g+u-\fg_a\) is injective. \end{proposition} \begin{proof} Every element of \(A\) is a finite Laurent polynomial, since it has a finite expansion in the given basis. Put \(K_0=\C(x_j:j\ne a)\). If a nonzero \(f\in A\) had smallest \(x_a\)-exponent \(r<0\), with coefficient \(a_r\in K_0^\times\), then the coefficient of \(x_a^{r-m}\) in \(D^mf\) would be \[ (x^u)^m a_r\,r(r-1)\cdots(r-m+1)\ne0 \quad(m\ge0), \] where the empty product is \(1\). This contradicts local nilpotence. Thus \(A\subset K_0[x_a]\), and \(\eps_D(f)=\deg_{x_a}f\) for every nonzero \(f\in A\). In \eqref{eq:pointed}, the exponent of \(x_a\) is \(g_a+\sum_k b_{ak}\nu_k\). Its maximal value is \(g_a\), attained precisely when \(\nu_k=0\) for every \(k\in F\). Hence the highest degree part of \(\theta_g\) is \(\theta'_g=\mathfrak f_{F,g}(\theta_g)\). It is nonzero and has degree \(g_a\). The matrix \(B'\) has full column rank because its columns are a subset of those of \(B\). Lemma~\ref{lem:independent} shows that all the \(\theta'_g\), \(g\in\Z^I\), are linearly independent. In a nonzero finite sum \(\sum_{g\in S}a_g\theta_g\), let \(d\) be the largest \(g_a\) with \(a_g\ne0\). Its part of degree \(d\) is the nonzero sum \(\sum_{g_a=d}a_g\theta'_g\). Consequently, for every integer \(r\ge0\), \[ \ker(D^r|_A)= \operatorname{span}_{\C}\{\theta_g:g\in S,\ g_a0\). The part of degree \(g_a-1\) in \(D\theta_g\) is \[ g_a x^{u-\fg_a}\theta'_g =g_a\theta'_{g+u-\fg_a}, \] by \eqref{eq:factor}. Expand \(D\theta_g\) in the basis of \(A\). No basis element of degree greater than \(g_a-1\) can occur, by the preceding independence. Comparing the parts of degree \(g_a-1\) and using independence for all integer labels forces the label \(g+u-\fg_a\) to belong to \(S\), with coefficient \(g_a\). Every remaining basis element has degree at most \(g_a-2\). This gives \eqref{eq:highest-result}. The target has degree \(g_a-1\), since \(u_a=0\). Two distinct possible targets would have a nonzero difference of degree \(g_a-1\), whereas \eqref{eq:highest-result} would put that difference in \(\ker D^{g_a-1}\); hence the target is unique. If \(g_a=1\), the remainder is zero because \(A\subset K_0[x_a]\). If \(g_a=0\), the derivative is zero. Finally, translation by \(u-\fg_a\) is injective. \end{proof} \section{Opposite seeds}\label{sec:opposite} Let \(s^\op\) have the same variables and frozen vertices as \(s\), with the exchange skew form negated. Its extended matrix is \(-B\). We identify the two initial rational function fields by the identity on each \(x_j\). Write \(\Theta_s=\{\theta_g^s:g\in M^\circ\}\), and similarly for \(s^\op\). \begin{lemma}\label{lem:reflection} Let \(Q\) be a generic point of the positive chamber, and let \(\nu\) be the involution of the Laurent ring defined by \(\nu(z^m)=z^{-m}\). Reflection of scattering diagrams gives \begin{equation}\label{eq:reflection} \nu\bigl(\theta^{s^\op}_{m,Q}\bigr)=\theta^s_{-m,-Q}. \end{equation} This identity holds for skew-symmetrizable classical seeds, with their original symmetrizing data. \end{lemma} \begin{proof} Retain the lattices \(\mathsf N^\circ\subset\mathsf N\), the integers \(d_j\), and the pairing defined in Section~\ref{sec:theta}. Negating \(\omega\) replaces \(p^*\) by \(-p^*\). Send each wall \((\mathfrak d,f)\) to \((-\mathfrak d,\nu(f))\) and each transverse path \(\gamma\) to \(-\gamma\). The oriented primitive normal becomes \(-n_\gamma\), and \(\langle-n_\gamma,-m\rangle=\langle n_\gamma,m\rangle\). Formula~\eqref{eq:wall} consequently gives \[ p_{-\gamma,-\mathfrak d}\nu=\nu p_{\gamma,\mathfrak d}. \] The reflected diagram is consistent. Its initial walls are \((\boldsymbol e_k^\perp,1+z^{-p^*\boldsymbol e_k})\), with the same primitive normal \(d_k\boldsymbol e_k\) up to orientation. The incoming condition is preserved because both \(p^*\boldsymbol e_k\) and the support are negated. Uniqueness of the consistent completion therefore identifies it with the opposite cluster diagram. Reflecting each segment of a broken line and negating its monomial exponent preserves the rule that the velocity is the negative exponent. The preceding wall identity preserves the possible bending terms and their coefficients. It gives a bijection of broken lines with label and endpoint \((m,Q)\) and \((-m,-Q)\), proving \eqref{eq:reflection}. The map on formal series goes from the completion for the original positive monoid to the completion for its negative image. \end{proof} We now use the twist to compare the two diagrams in the same positive chamber. Write \(C^-=-C^+\). For chambers with generic points \(Q_1,Q_2\), denote their scattering transport by \(p^s_{Q_2\leftarrow Q_1}\). \begin{proposition}\label{prop:opposite} For a full-rank injective-reachable classical seed, \[ \Theta_{s^\op}=\Theta_s \] as sets of rational functions in the initial field. \end{proposition} \begin{proof} All theta functions are Laurent finite in \(C^+\) by \cite[Theorem~A.1.5]{QinTrop}. The negative chamber is the chamber of the reachable seed \(s[1]\); see \cite[Appendix~A.1]{QinTrop}. Mutation compatibility and the same theorem for this seed give Laurent finiteness in its chamber chart. The linear identification of this chart with the chamber in \(M^\circ_\R\) changes exponents by an invertible integral map, so it preserves finite support. Lemma~\ref{lem:reflection} then gives Laurent finiteness for the opposite seed in \(C^+\). Consider the composite \begin{equation}\label{eq:DT} \DT_s=p^s_{Q\leftarrow -Q}\nu. \end{equation} Chamber monomials give \begin{equation}\label{eq:DTgenerators} \DT_s(x_j)=\theta^s_{-\fg_j,Q}\quad(j\in I), \end{equation} Indeed, \(\fg_j\in C^+\) for every \(j\), including frozen \(j\), so the positive-chamber monomial theorem gives \(\theta^{s^\op}_{\fg_j,Q}=x_j\). Reflection gives \(\theta^s_{-\fg_j,-Q}=x_j^{-1}\). Transporting this last identity from \(-Q\) to \(Q\) yields \eqref{eq:DTgenerators}. Relabel the mutable vertices of \(s[1]\) by the permutation in injective reachability. Let \(H\) be the integer matrix with rows indexed by \(I_{\fr}\) and columns indexed by \(I_{\mut}\), whose \(j\)-th column records the frozen components of the \(j\)-th mutable \(g\)-vector of \(s[1]\). The lattice identification \(\psi^M\) from its monomial lattice to that of \(s\), and the variation \(\var^M\) from the latter lattice to the former, have matrices \begin{equation}\label{eq:blocks} \psi^M= \begin{pmatrix}-\operatorname{Id}_{I_{\mut}}&0\\ H&\operatorname{Id}_{I_{\fr}}\end{pmatrix}, \qquad \var^M= \begin{pmatrix}\operatorname{Id}_{I_{\mut}}&0\\ -H&-\operatorname{Id}_{I_{\fr}}\end{pmatrix}. \end{equation} These are the maps constructed in \cite[Proposition~5.1]{KQW}, and their product is \(-\operatorname{Id}_{M^\circ}\). Write \(\var\) for the map of Laurent rings that substitutes the monomials specified by \(\var^M\). Composing with the mutation pullback \(\mu^*\), which expresses functions in the \(s[1]\) chart as rational functions in the \(s\) chart, gives a twist automorphism \(T=\mu^*\var\). For mutable \(j\), \(T(x_j)\) is one cluster variable of \(s[1]\) times a frozen Laurent monomial. It is a theta function, and its full label is \(\psi^M\var^M\fg_j=-\fg_j\). For frozen \(j\), it is \(x_j^{-1}=\theta^s_{-\fg_j,Q}\). Thus \(T\) and \(\DT_s\) agree on every field generator by \eqref{eq:DTgenerators}. In particular \(\DT_s\) is the invertible twist in \cite[Theorem~5.2]{KQW}. The theta \(F\)-polynomials satisfy the mutation and variation hypotheses there, so \cite[Theorem~6.2(2)]{KQW} gives \begin{equation}\label{eq:DTpermutes} \DT_s(\Theta_s)=\Theta_s. \end{equation} Combining reflection with transport yields, for every \(m\in M^\circ\), \[ \DT_s(\theta^{s^\op}_{m,Q}) =p^s_{Q\leftarrow -Q}\theta^s_{-m,-Q} =\theta^s_{-m,Q}. \] As \(m\) ranges over \(M^\circ\), the right side ranges over \(\Theta_s\). Together with \eqref{eq:DTpermutes} and injectivity of \(\DT_s\), this proves the asserted equality. \end{proof} \begin{lemma}\label{lem:intersection} Let \(R\subset K\) be a \(\C\)-subalgebra and suppose a subset \(\mathcal B\subset\Theta_s\) is a basis of \(R\). Then \(\mathcal B=\Theta_s\cap R\). Under the assumptions of Proposition~\ref{prop:opposite}, it also equals \(\Theta_{s^\op}\cap R\). \end{lemma} \begin{proof} An element of \(\Theta_s\cap R\) has a finite expansion in \(\mathcal B\). Linear independence of \(\Theta_s\) forces that element to belong to \(\mathcal B\). The second assertion follows from Proposition~\ref{prop:opposite}. \end{proof} \section{Minor seeds and Chevalley derivations}\label{sec:minors} The simply connected covering of \(G\) restricts to an isomorphism on \(N\): its kernel is finite and central, its restriction to the maximal unipotent subgroup is surjective, and a complex unipotent group has no nontrivial finite subgroup. This isomorphism identifies the pinned root subgroups and both actions in \eqref{eq:derivations}. We may therefore assume \(G\) simply connected. Let \(W\) be its Weyl group, with simple reflections \(s_i\), simple roots \(\alpha_i\), simple coroots \(\alpha_i^\vee\), and fundamental weights \(\omega_i\), \(i\in\mathcal I\). Let \(\wlong\) be the longest element. In the irreducible representation of highest weight \(\omega_i\), choose a highest weight vector \(v_{\omega_i}\) and the functional \(\phi_{\omega_i}\) which is \(1\) on that vector and zero on all other weight spaces. Let \(f_i\) be the negative Chevalley generator, normalized by \([e_i,f_i]=\alpha_i^\vee\), and put \(y_i(t)=\exp(tf_i)\). Use the Weyl lifts \(\overline s_i=x_i(-1)y_i(1)x_i(-1)\). Their products along reduced words define \(\overline w\) for \(w\in W\). The generalized minors are \begin{equation}\label{eq:minors} \Delta_{u\omega_i,v\omega_i}(n) =\phi_{\omega_i}\bigl(\overline u^{-1}n\overline v v_{\omega_i}\bigr) \quad(u,v\in W,\ n\in N). \end{equation} These are the pinned normalization of \cite{BFZ}. Fix a reduced word \((j_1,\ldots,j_L)\) for \(\wlong\), and define \(w_k=s_{j_1}\cdots s_{j_k}\), with \(w_0=1\). For \(1\le k\le L\), let \(p(k)\) and \(s(k)\) be the previous and next positions of \(j_k\) in this word. Set \(p(k)=-\infty\) or \(s(k)=+\infty\) when the corresponding position is absent. The two coordinate families are \begin{equation}\label{eq:XY} X_k=\Delta_{\omega_{j_k},w_k\omega_{j_k}}, \qquad Y_k=\Delta_{w_{k-1}\omega_{j_k},\wlong\omega_{j_k}}. \end{equation} The \(X_k\) form a full-rank injective-reachable seed with the last occurrence of each letter frozen. The \(Y_k\), indexed by their starting position in \eqref{eq:XY}, form a mutation-equivalent seed with the first occurrence of each letter frozen. Denote their extended exchange matrices by \(B^X=(b^X_{rk})\) and \(B^Y=(b^Y_{rk})\). In Qin's sign convention their endpoint rows are \begin{equation}\label{eq:rows} b^X_{L,k}=-\delta_{k,p(L)},\qquad b^Y_{1,k}=\delta_{k,s(1)} \end{equation} for mutable columns \(k\); a nonexistent predecessor or successor gives the zero row. These inputs come from \cite[Theorems~8.1.2, 8.2.2 and Lemma~9.1.3]{QinDC}. The interval minors there become \eqref{eq:XY} under the pinned transpose from the lower to the upper unipotent subgroup. The integral identification in \cite[Theorem~9.2.3]{QinDC}, together with \cite[\S6.3, (6.14), and Theorem~7.3(a)]{GY}, specializes to the ordinary coordinate ring and the normalized classical minors when the quantum deformation parameter \(q\) is set to \(1\). The rescaling denoted by \(\mathrm{cor}\) in \cite{QinDC} is a power of \(q\), so its specialization is \(1\). The compatible skew matrix \(\Lambda\) in \cite[Theorem~8.1.2(1)]{QinDC} satisfies \[ (B^X)^T\Lambda= \bigl(\operatorname{diag}(\lambda_k:k\in I_{\mut})\ \ 0\bigr), \qquad \lambda_k\in\Z\setminus\{0\}, \] with mutable columns ordered first on the right. The right side has rank \(|I_{\mut}|\), so \(B^X\) has full column rank. Matrix mutation preserves that rank, and hence gives the same conclusion for the suffix seed. Injective reachability is preserved by mutation as recalled in Section~\ref{sec:theta}. To verify the endpoint rows, write \(C_{ab}=\langle\alpha_a^\vee,\alpha_b\rangle\) for the Cartan matrix, with \(a,b\in\mathcal I\). The formulas in \cite[Remarks~8.1.4 and A.3.1]{QinDC} are \[ b^X_{rk}= \begin{cases} 1&r=p(k),\\ -1&r=s(k),\\ C_{j_rj_k}&r1\), the root \(w_{k-1}^{-1}\alpha_i\) is negative because the inversion set already contains \(\alpha_i\). In the derivative \[ D_i^LY_k(n) =\phi_{\omega_{j_k}} \bigl(\overline w_{k-1}^{-1}e_i n \overline{\wlong}v_{\omega_{j_k}}\bigr), \] conjugate \(e_i\) by \(\overline w_{k-1}\). The resulting negative-root operator has zero highest weight component after application to any vector in the representation. Thus \(D_i^LY_k=0\). For \(k=1\), the functionals \(\phi_{\omega_i}e_i\) and \(\phi_{\omega_i}\overline s_i^{-1}\) are supported on the same extremal weight space of weight \(s_i\omega_i\). They both take the value \(1\) on \(f_iv_{\omega_i}\), by the length-one \(i\)-string calculation. They are therefore equal. If \(s(1)\) exists, all letters in positions \(2,\ldots,s(1)-1\) differ from \(i\). Their inverse lifts fix \(\phi_{\omega_i}\). Using the order of factors in \(\overline w_{s(1)-1}^{-1}\) gives \[ \phi_{\omega_i}\overline w_{s(1)-1}^{-1} =\phi_{\omega_i}\overline s_i^{-1}. \] It follows that \(D_i^LY_1=Y_{s(1)}\). If no next occurrence exists, the same calculation gives \(\phi_{\omega_i}\overline{\wlong}^{-1} =\phi_{\omega_i}\overline s_i^{-1}\). The derivative is then the diagonal extremal minor \[ \Delta_{\wlong\omega_i,\wlong\omega_i}(n)=1. \] Indeed, the action of \(n\in N\) on an extremal weight vector changes it only by terms of strictly higher weight, which the functional defining the same extremal minor annihilates. This proves the formula on every coordinate generator. \end{proof} \begin{example}\label{ex:slthree} For \(G=\mathrm{SL}_3\), write \[ n=\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}. \] For the word \((1,2,1)\), the two coordinate lists are \[ (X_1,X_2,X_3)=(a,ac-b,b),\qquad (Y_1,Y_2,Y_3)=(b,ac-b,c). \] The right derivation \(D_2^R\) has values \(0,0,a\) on the \(X\)-coordinates, while \(D_1^L\) has values \(c,0,0\) on the \(Y\)-coordinates. The exchange relation is \(ac=b+(ac-b)\). In the prefix convention its frozen row at \(X_3\) is \(-1\), whereas the suffix row at \(Y_1\) is \(+1\). \end{example} \section{Proof of biperfectness} \begin{proof}[Proof of Theorem~\ref{thm:main}] Fix \(i\in\mathcal I\). Choose a reduced word ending in \(i^*\) and use its prefix seed. The coordinate \(X_L\) is frozen, and its row is nonpositive by \eqref{eq:rows}. If \(p(L)\) exists, it is mutable and is the unique negative entry in that row. Lemma~\ref{lem:right} and Proposition~\ref{prop:highest}, applied to \(A=\C[N]\), therefore prove goodness and \eqref{eq:perfect} for \(D_i^R\), with the required coefficient and injective arrows. If \(p(L)\) is absent, the monomial is \(1\), the row is zero, and the same proposition applies with \(F=\varnothing\). For \(D_i^L\), choose a reduced word beginning with \(i\) and use its suffix seed. The coordinate \(Y_1\) is frozen. Its row is nonnegative, and its only possible nonzero mutable entry is at \(s(1)\). Proposition~\ref{prop:opposite} and Lemma~\ref{lem:intersection} identify the regular theta basis for this seed with that for its opposite, in the same rational function field with the same coordinate functions. The opposite row at \(Y_1\) is nonpositive. By Lemma~\ref{lem:left}, the monomial in the derivation is \(Y_{s(1)}\), or \(1\) when no successor exists. Proposition~\ref{prop:highest} now gives goodness, \eqref{eq:perfect}, and injective arrows for \(D_i^L\). All the coordinate choices belong to the same cluster structure, so these statements concern the same basis \(\mathcal B_\theta(\C[N])\). It contains \(1=\theta_0\). The minor \(\Delta_{u\omega_i,v\omega_i}\) has weight \(v\omega_i-u\omega_i\): in the torus action, \(h^{-1}\) acts on the extremal functional and \(h\) on the extremal vector in \eqref{eq:minors}. Each column of the exchange matrix has total weight zero, by \cite[Theorem~8.1.2(1)]{QinDC}. Every monomial in a pointed expansion therefore has the weight of its leading monomial. 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