\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern,amsmath,amssymb,mathtools,microtype} \usepackage[margin=1.08in]{geometry} \usepackage{indentfirst,xcolor} \definecolor{linkblue}{RGB}{25,65,115} \usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} \setlength{\parindent}{1.5em} \hypersetup{ pdftitle={Quantum Laurent positivity for mutation-acyclic cluster algebras}, pdfsubject={Quantum Laurent positivity} } \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\F}{\mathbb F} \newcommand{\C}{\mathbb C} \newcommand{\bt}{\widetilde B} \newcommand{\gtil}{\widetilde g} \newcommand{\gmod}{\text{-}\mathrm{gmod}} \newcommand{\gMod}{\text{-}\mathrm{gMod}} \newcommand{\qgr}{\mathsf q} \newcommand{\ch}{\operatorname{ch}} \newcommand{\sch}{\operatorname{sch}} \newcommand{\wt}{\operatorname{wt}} \newcommand{\hd}{\operatorname{hd}} \newcommand{\Res}{\operatorname{Res}} \newcommand{\End}{\operatorname{End}} \newcommand{\END}{\operatorname{END}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ext}{\operatorname{Ext}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\Mat}{\operatorname{Mat}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Frac}{\operatorname{Frac}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\id}{\operatorname{id}} \newcommand{\cone}{\operatorname{cone}} \newcommand{\ee}{\mathbf e} \newcommand{\ii}{\mathbf i} \newcommand{\jj}{\mathbf j} \newcommand{\ba}{\mathbf a} \newcommand{\bb}{\mathbf b} \newcommand{\bnu}{\boldsymbol\nu} \newcommand{\ang}[2]{\langle #1,#2\rangle} \title[Mutation-acyclic quantum positivity] {Quantum Laurent positivity for mutation-acyclic cluster algebras} \author{Qiyue Tang} \address{Zhili College, Tsinghua University, Beijing 100084, China} \email{tangqy24@mails.tsinghua.edu.cn} \subjclass[2020]{13F60, 16G20, 17B37} \keywords{Quantum cluster algebra, Laurent positivity, valued quiver, quiver Hecke algebra, quantum shuffle character} % Visible attribution and research disclosure; author metadata remains a name. \makeatletter \renewcommand{\@setauthors}{% \begin{center} \color{linkblue}\normalsize generated by Qiyue Tang\par \medskip \begin{minipage}{0.92\textwidth} \small\centering The main results of this paper were generated jointly using GPT-6-Astra and the Danus system. The manuscript was independently checked by a separate GPT-6-Astra reviewer and by Qiyue Tang's Danus system. \end{minipage} \end{center}} \makeatother \begin{document} \begin{abstract} We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank. For every integral compatible quantization with invertible frozen variables, each cluster variable has a Laurent expansion with coefficients in $\mathbb N[v^{\pm1}]$ in every seed. For an indecomposable rigid representation of an acyclic valued quiver, we identify its Hall shuffle character with the character of a self-dual simple quiver Hecke module. These modules are real and admit affinizations; modules corresponding to the summands of a rigid direct sum strongly commute. A Feigin localization and a comparison of graded composition multiplicities establish positivity at an arbitrary target seed. \end{abstract} \maketitle \section{Introduction} Quantum cluster algebras are generated by variables obtained through iterated mutation. The quantum Laurent phenomenon states that every cluster variable is a finite Laurent polynomial in the variables of any seed \cite[Corollary~5.2]{BZ}. Quantum Laurent positivity asks whether, when this expansion is written in normalized quantum monomials, each coefficient is a Laurent polynomial in the quantum parameter with nonnegative integer coefficients. For classical cluster algebras, Lee and Schiffler proved positivity for skew-symmetric cluster algebras of arbitrary rank \cite[Theorem~1.1]{LS}. Davison established quantum Laurent positivity for skew-symmetric quantum cluster algebras by cohomological methods \cite[Theorem~2.4]{Davison}. We write $n$ for the number of mutable directions. Let $\bt\in\Mat_{m\times n}(\Z)$ have skew-symmetrizable principal part $B$, and let $\Lambda$ be an integral skew-symmetric matrix such that \begin{equation}\label{eq:compatibility} \bt^T\Lambda=(D\ \ 0),\qquad D=\diag(d_1,\ldots,d_n),\quad d_i>0. \end{equation} Let $v$ be an indeterminate, and work over $\Z[v^{\pm1}]$. When using \cite{BZ,Tran}, we identify their formal parameter $q^{1/2}$ with $v$; this is separate from the finite-field parameter $q$ used in Section~\ref{subsec:hall}. We use the based quantum torus convention \begin{equation}\label{eq:torus} M(a)M(b)=v^{a^T\Lambda b}M(a+b). \end{equation} Frozen variables are formal invertible generators of the based quantum torus. A principal matrix is \emph{mutation-acyclic} if some matrix in its mutation class has an acyclic oriented graph. \begin{theorem}\label{thm:B} In every mutation-acyclic quantum cluster pattern of finite rank, each cluster variable has a Laurent expansion with coefficients in $\N[v^{\pm1}]$ in every seed. \end{theorem} The representation-theoretic input for Theorem~\ref{thm:B} is the following character identity. We use the finite-field valued-quiver realization and the quiver Hecke parameters over an algebraically closed field of characteristic zero fixed in Section~\ref{subsec:hall}. Here $\Omega$ is the Hall-to-shuffle map; its value on $[V]^*$ is understood generically, using the counting polynomials of rigid representations, as explained in that section. \begin{theorem}\label{thm:C} For an indecomposable rigid representation $V$ of an acyclic valued quiver, there is a self-dual simple quiver Hecke module $L_V$ such that \[ \Omega([V]^*)=\ch_{t^{-1}}L_V. \] The module $L_V$ is real and admits an affinization. If $V_1,\ldots,V_s$ are indecomposable representations and $V_1\oplus\cdots\oplus V_s$ is rigid, the modules $L_{V_i}$ strongly commute and their ordered convolution is simple. Repeated isomorphism classes among the $V_i$ are allowed. \end{theorem} For an acyclic valued quiver over a finite field, the quantum cluster character identifies isomorphism classes of indecomposable rigid representations with non-initial quantum cluster variables \cite[Theorem~1.1]{RupelCC}; the corresponding Feigin realization is given in \cite[Main Theorem~2.9(b)]{BR}. The character identity in Theorem~\ref{thm:C} is the indecomposable part of Rupel's Conjecture~8.2 in \cite{Feigin}. Rupel proved this conjecture for equal vertex weights \cite[Theorem~8.4]{Feigin}. Here we treat general valuations and obtain the affinization and commutation properties needed for the positivity argument. The categorical passage to Laurent expansions uses the localizations of \cite{K1,K2}; simplicity of convolutions is obtained from the normal-sequence criterion \cite[Lemma~2.10]{K3}. The proof proceeds through two categorical constructions. In Section~\ref{sec:representations}, stability places the Hall character of a rigid indecomposable in a single semicuspidal direction. A support lemma and an extremal word identify the corresponding simple character. Reflection then supplies affinizations, which allow us to turn Hall commutation identities into simple convolutions. Section~\ref{sec:feigin} realizes these objects in a localized category and identifies its Grothendieck ring with a subalgebra of the Feigin torus. In infinite irreducible type, for a variable $X$ and a target cluster $T$, a Coxeter belt supplies an acyclic reference seed relative to which $X$ and every member of $T$ are noninitial. The other types reduce to this case by adjoining and then freezing one vertex. The cluster-character correspondence now lifts $X$ to a simple object and the mutable variables of $T$ to strongly commuting simples. Frozen variables lift to invertible objects. After clearing mutable Laurent denominators, each target monomial is a simple class, while the left-hand product is an object of finite length. Comparing its composition multiplicities proves positivity. Coefficient transport returns the result to the prescribed quantization. Section~\ref{sec:proof} carries out this argument. \section{Quantum seeds and coefficient transport} \label{sec:prelim} \subsection{Quantum seeds and normalization} A toric frame has based monomials $M(a)$, $a\in\Z^m$, satisfying \eqref{eq:torus}. Mutable labels are $1,\ldots,n$. For a column $\bt_k$, write $[\bt_k]_+$ for its componentwise positive part. Mutation at $k$ replaces $X_k=M(\ee_k)$ by \begin{equation}\label{eq:mutation} X_k'=M(-\ee_k+[\bt_k]_+)+M(-\ee_k+[-\bt_k]_+). \end{equation} The other generators are retained, and the matrix and form mutate according to the usual BZ rules \cite{BZ}. The initial torus carries the $\Z$-linear bar anti-involution defined by \[ \overline{v^rM(a)}=v^{-r}M(a),\qquad \overline{AB}=\overline B\,\overline A. \] For the pairwise quasi-commuting generators of any one seed, the normalized monomial is \[ M(a)=v^{-\sum_{i0$ for every simple root. Thus every nonzero positive weight has phase in $(0,\pi)$. A word of total weight $\beta$ is \emph{semicuspidal} if each nonempty proper left prefix of weight $\gamma$ satisfies \[ \arg c(\gamma)\le\arg c(\beta) \quad\Longleftrightarrow\quad \theta(\gamma)h(\beta)-h(\gamma)\theta(\beta)\ge0. \] A module is semicuspidal if every word in its character is semicuspidal. When $\theta(\beta)=0$, this condition is simply $\theta(\gamma)\ge0$; we then also say $\theta$ semicuspidal. This is the left-prefix convention of \cite[Definitions~2.1 and~2.3]{TW}. Equivalently, every nonzero restriction $\Res_{\gamma,\beta-\gamma}L$, with $0<\gamma<\beta$, has $\arg c(\gamma)\le\arg c(\beta)$, since this restriction selects exactly the words with left prefix weight $\gamma$. We use the same phase order and the same left-to-right induction order as \cite[Theorem~2.4]{TW}. That paper works with finite-dimensional nilpotent modules \cite[Definitions~1.41--1.42]{TW}. A finite-dimensional graded module in our setting is nilpotent for the polynomial generators, since their degrees are positive. Its grading does not affect word support, and grading shifts do not affect semicuspidality. For Hall characters, left prefixes in descending flags are quotients. The following Euler-form calculation will make their inequalities strict. \begin{lemma}\label{lem:stability} Let $V$ be indecomposable and rigid, with $\nu=\dim V$. Define \[ \theta_\nu(x)=E(\nu,x)-E(x,\nu). \] Every nonzero proper subrepresentation $U$ satisfies $\theta_\nu(\dim U)<0$. Every nonzero proper quotient has positive $\theta_\nu$ weight. \end{lemma} \begin{proof} By \cite[Theorem~3.4(1)]{RupelCC}, every nonzero endomorphism of $V$ is injective or surjective, hence invertible. Put $T=V/U$. Applying $\Hom(-,V)$ gives \[ \Ext^1(V,V)\longrightarrow\Ext^1(U,V) \longrightarrow\Ext^2(T,V). \] Rigidity and heredity give $\Ext^1(U,V)=0$. A nonzero map $V\to U$ would yield a nonzero noninvertible endomorphism of $V$, so $\Hom(V,U)=0$. Therefore, with dimensions taken over $\F_q$, \[ \theta_\nu(\dim U) =-\dim\Ext^1(V,U)-\dim\Hom(U,V)<0. \] The inclusion $U\hookrightarrow V$ gives strictness. Additivity and $\theta_\nu(\nu)=0$ give the quotient assertion. \end{proof} Every proper nonempty prefix in the support of $F_V$ therefore has positive $\theta_\nu$ weight. This also holds generically: a nonzero counting polynomial is nonzero at all sufficiently large finite field parameters, where Lemma~\ref{lem:stability} applies. Perturb $\theta_\nu$ among real linear functionals vanishing on $\nu$, preserving these finitely many strict inequalities, to obtain \begin{equation}\label{eq:generic-charge} \ker\theta\cap Q_\Q=\Q\nu. \end{equation} Such a choice exists because the excluded loci are countably many proper hyperplanes. Choose a real linear $h$ positive on the simple roots and put $c=\theta+\mathrm{i}h$. Since $h(\nu)>0$, equation \eqref{eq:generic-charge} gives $\ker c\cap Q_\Q=0$. Thus $c$ is injective on the rational root space, and \[ \arg c(\gamma)<\arg c(\nu)=\pi/2 \] for every proper nonempty prefix weight $\gamma$ in $F_V$. The following support lemma passes from virtual characters to semicuspidal simple characters. \begin{lemma}\label{lem:virtual} Let $c$ be injective on the rational root space. A finite virtual KLR character of fixed weight whose nonzero words are all $c$ semicuspidal lies in the linear span of $c$ semicuspidal simple characters. \end{lemma} \begin{proof} Write the virtual character as \[ F=\sum_L a_L\ch_{t^{-1}}L,\qquad a_L\in\Q(u), \] using distinct simple modules up to grading shift and omitting zero coefficients. There is nothing to prove if $F=0$. \emph{The polygon of a simple module.} By \cite[Theorem~2.4]{TW}, each $L$ is, up to shift, the simple head of $L_1\circ\cdots\circ L_s$, where the $L_j$ are semicuspidal and \[ \beta_j=\wt L_j,\qquad \arg c(\beta_1)>\cdots>\arg c(\beta_s). \] In coordinates $(h,-\theta)$, define $P_L$ by the consecutive vertices \[ p_r=\left(h\left(\sum_{j=1}^r\beta_j\right), -\theta\left(\sum_{j=1}^r\beta_j\right)\right), \qquad 0\le r\le s. \] Its slopes $-\theta(\beta_j)/h(\beta_j)$ strictly decrease, so $P_L$ is concave. A word path joins the images of its consecutive prefix weights. Semicuspidality puts each factor word below its chord. For a prefix of a shuffle, let $x_j$ be the $h$-length taken from the $j$th factor. Its height in these coordinates is at most \[ \sum_{j=1}^s\frac{-\theta(\beta_j)}{h(\beta_j)}x_j, \qquad 0\le x_j\le h(\beta_j),\qquad \sum_jx_j=x. \] At fixed $x$, this sum is maximized by filling the factors in decreasing slope order: transferring available length from a smaller slope to a larger one increases the sum. The maximum is the height of $P_L$ at $x$. Thus every word path of the convolution, and hence of its quotient $L$, lies below $P_L$. The unmixing property \cite[Lemma~2.9 and the proof of Lemma~2.6]{TW} also gives \[ \Res_{\beta_1,\ldots,\beta_s}L \simeq \qgr^{r_L}(L_1\boxtimes\cdots\boxtimes L_s), \qquad r_L\in\Z. \] On characters this restriction cuts words into blocks of the specified weights; extend it linearly to virtual characters. \emph{Eliminating a maximal polygon.} Suppose some term has $s>1$. Choose a pointwise maximal polygon $P$ among such terms, and let $(\beta_1,\ldots,\beta_s)$ be its edge weights. A semicuspidal term has the single chord as its polygon, strictly below $P$ in the interior, so $P$ is maximal among all terms. Since $\arg c(\beta_1)$ exceeds the total phase, the support hypothesis gives \[ \Res_{\beta_1,\ldots,\beta_s}F=0. \] Any contributing simple term has a word path through every vertex of $P$. Its polygon $P'$ dominates those vertices, and concavity then gives $P'\ge P$ on every edge interval: a concave function lies above the chord joining two of its points. Maximality forces $P'=P$. Injectivity of $c$ recovers the block weights from the polygon's edges. The uniqueness in Theorem~2.4 gives distinct ordered tuples of simple factors for distinct terms with this polygon. Their tensor characters are linearly independent, even after extending scalars to $\Q(u)$. The displayed restriction formula therefore prevents their nonzero coefficients from cancelling, a contradiction. \end{proof} \begin{proposition}\label{prop:real-root-character} Let $V$ be indecomposable and rigid, with $\nu=\dim V$, and choose $\theta$ as in \eqref{eq:generic-charge}. Then $\nu$ is a positive real root, and there is a unique $\theta$ semicuspidal simple $L_V$ of weight $\nu$ up to shift. Its self-dual representative satisfies $F_V=\lambda_V(t)\ch_{t^{-1}}L_V$ for some $\lambda_V(t)\in\Q(t)^\times$. \end{proposition} \begin{proof} We first show that $\nu$ is a multiple of a positive real root, then exclude nontrivial multiples. Rigidity gives \[ (\nu,\nu)=2E(\nu,\nu)=2\dim_{\F_q}\End(V)>0. \] Since $F_V\ne0$, Lemma~\ref{lem:virtual} and \cite[Corollary~2.12]{TW} express $\nu$ as a sum of positive roots of phase $\pi/2$. By \eqref{eq:generic-charge} these roots are positive rational multiples of $\nu$, so each has positive norm. Positive imaginary roots have nonpositive norm \cite[Proposition~5.3.1(ii)--(iii)]{Kleshchev}: each is Weyl-conjugate to a positive root $\gamma$ with $(\gamma,\alpha_i)\le0$ for all $i$, whence $ (\gamma,\gamma)=\sum_i\gamma_i(\gamma,\alpha_i)\le0$. Thus the roots in the sum are real. Each real root is a Weyl translate of a simple root and hence is primitive, since the Weyl group acts by integral automorphisms of the root lattice. Thus all are the same positive real root $\alpha$, and $\nu=k\alpha$ for some $k\ge1$. Corollary~2.12 gives exactly one semicuspidal simple, up to shift, at each weight $k\alpha$: the only positive root on this ray is $\alpha$, of multiplicity one. Hence $F_V$ is a nonzero scalar multiple of the simple character at $k\alpha$ and has exactly its word support. Let $L_\alpha$ be the semicuspidal simple of weight $\alpha$. Every shuffle of its words has nonnegative $\theta$ weight at each prefix, so all constituents of $L_\alpha^{\circ k}$ are semicuspidal. If $k>1$, concatenating $k$ words of $L_\alpha$ gives a word of this convolution with a proper prefix of weight $\alpha$ and $\theta(\alpha)=0$. Since every constituent is a shift of the unique semicuspidal simple at $k\alpha$, that word occurs in this simple, hence in $F_V$. This contradicts the strict prefix inequality. Therefore $k=1$. Choosing the self-dual representative $L_V$, we obtain \begin{equation}\label{eq:unknown-scalar} F_V=\lambda_V(t)\ch_{t^{-1}}L_V. \end{equation} Here $\lambda_V(t)\in\Q(t)^\times$: compare any nonzero word coefficient on both sides, which belong to $\Q(t)$. \end{proof} \subsection{The extremal word coefficient} To determine the scalar in \eqref{eq:unknown-scalar}, we compare one word coefficient, writing $[\mathbf w]F$ for the coefficient of the word $\mathbf w$ in $F$. We use the factorial conventions \[ (r)!_z=\prod_{a=1}^r(1+z+\cdots+z^{a-1}),\qquad [r]_z!=\prod_{a=1}^r\frac{z^a-z^{-a}}{z-z^{-1}}, \] with both empty products equal to $1$. \begin{proposition}\label{prop:normalization} For every indecomposable rigid $V$, the self-dual simple module $L_V$ constructed above satisfies \begin{equation}\label{eq:hall-simple} \Omega([V]^*)=\ch_{t^{-1}}L_V. \end{equation} Equivalently, $\lambda_V(t)=1$ in \eqref{eq:unknown-scalar}. \end{proposition} \begin{proof} \emph{The Hall coefficient.} Choose a source topological order $i_1,\ldots,i_n$ for $Q$, and consider \[ \mathbf w=i_1^{\nu_{i_1}}\cdots i_n^{\nu_{i_n}}. \] For $0\le r\le n$, let $V^{(r)}$ have zero space at $i_1,\ldots,i_r$ and the full space $V_j$ at every other vertex, with the induced arrow maps. Every arrow goes from an earlier to a later vertex in the chosen order, so $V^{(r)}$ is a subrepresentation. The dimensions force these vertex spaces at the end of each color block; hence any flag of word $\mathbf w$ has this same coarse flag. Inside the $i_r$ block, earlier vertex spaces are zero and later ones are full. Every $\F_{q^{d_{i_r}}}$-subspace $U\subseteq V_{i_r}$ is therefore allowed: incoming maps have zero source, and outgoing maps land in full target spaces. The same zero-source and full-target conditions apply to every species structure map. Successive quotients in the block are $S_{i_r}$ precisely when these subspaces form a complete descending flag over $\F_{q^{d_{i_r}}}$. Thus refinement gives a bijection \[ \mathcal F_{\mathbf w}(V)\simeq \prod_{i\in I}\operatorname{Flag}_{\F_{q^{d_i}}}(V_i), \] where a zero-dimensional space contributes one flag. An $m$-dimensional space over $\F_{q^{d_i}}$ has $(m)!_{q^{d_i}}$ complete flags: choose a hyperplane in $(q^{d_i m}-1)/(q^{d_i}-1)$ ways and continue inside it. Consequently \[ |\mathcal F_{\mathbf w}(V)|=\prod_i(\nu_i)!_{t^{2d_i}}. \] For $r0$. If $N$ is not semicuspidal, write its ordered semicuspidal factors as $L_1,\ldots,L_s$, with $s>1$, and put \[ A=L_1,\qquad U=L_2\circ\cdots\circ L_s,\qquad B=\hd U. \] Then $\theta'(\wt A)<0$. We claim that \begin{equation}\label{eq:two-block-unmixing} \Res_{\wt A,\wt B}N\simeq A\boxtimes B, \end{equation} up to grading. Consider the Mackey filtration of $\Res_{\wt A,\wt U}(A\circ U)$. In a mixed term, a nonzero suffix of $A$ would have the same weight as a sum of prefixes from the factors of $U$. Semicuspidality puts the suffix at a phase at least that of $A$, whereas every nonzero such prefix, and hence their sum, has strictly smaller phase. No mixed term can occur. The restriction is therefore $A\boxtimes U$. The same separation applies to $A\circ B$: every word of $B$, being a word of the quotient of $U$, has prefixes of phase strictly smaller than that of $A$. Thus $\Res_{\wt A,\wt B}(A\circ B)\simeq A\boxtimes B$. The surjection $A\circ U\twoheadrightarrow A\circ B$ and the unique simple head of $A\circ U$ imply $\hd(A\circ B)\simeq N$. Exact restriction makes $\Res_{\wt A,\wt B}N$ a quotient of the simple module $A\boxtimes B$. This quotient is nonzero by Frobenius reciprocity applied to $A\circ B\twoheadrightarrow N$, proving \eqref{eq:two-block-unmixing}. The phase of $A$ is greater than $\pi/2$, while that of color $i$ is less. Semicuspidality forbids a word of $A$ ending in $i$. The restriction and the corresponding prohibition for $N$ also forbid a word of $B$ ending in $i$. The inverse monoidal equivalence therefore applies to $A$, $B$ and the epimorphism $A\circ B\twoheadrightarrow N$. Frobenius reciprocity gives a nonzero prefix restriction of $L$ with weight $s_i\wt A$. Its $\theta$ value is $\theta'(\wt A)<0$, a contradiction. \emph{Case $\theta(\alpha_i)>0$.} The words of $L$ cannot end in $i$ and the words of its image cannot begin in $i$. Take the last semicuspidal factor $B$, whose $\theta'$ value is positive, and let $A$ be the simple head of the preceding factors. The corresponding Mackey filtration has no mixed term: a nonzero prefix of $B$ has phase at most that of $B$, while a sum of nonzero suffixes from preceding factors has strictly larger phase. The same head and adjunction argument gives $\Res_{\wt A,\wt B}N\simeq A\boxtimes B$, up to grading. Since $\theta'(\alpha_i)<0$, $B$ cannot begin in $i$; the restriction implies the same for $A$. Applying the inverse equivalence to $A\circ B\twoheadrightarrow N$ produces a suffix of $L$ with positive $\theta$ value, contradicting semicuspidality. \end{proof} For a graded module $M=\bigoplus_j M_j$, write \[ \END_R(M)=\bigoplus_{r\in\Z} \{f\in\End_R(M): f(M_j)\subseteq M_{j+r}\text{ for all }j\} \] for its total graded endomorphism ring. \begin{proposition}\label{prop:polynomial-family} The module $L_V$ is real. It has a finitely generated graded family $(\Delta,z)$ with $\deg z>0$, finite free over $k[z]$, such that \[ \Delta/z\Delta\simeq L_V,\qquad \END_{R(\dim V)}\Delta=k[z],\qquad \supp\Delta=\supp L_V. \] \end{proposition} \begin{proof} \emph{Descending to a simple root.} Start with $L=L_V$, $\beta=\dim V$, and the functional $\theta$ used in \eqref{eq:generic-charge}. The preceding argument shows that $\beta$ is a positive real root and $L$ is $\theta$ semicuspidal. If $\beta$ is simple, no reflection is needed. Otherwise retain this functional and proceed as follows. Write $h_i$ for the simple coroot, so that $\ang{h_i}{\beta}=2(\alpha_i,\beta)/(\alpha_i,\alpha_i)$. Since \[ (\beta,\beta)=\sum_i\beta_i(\beta,\alpha_i)>0, \] there is an $i$ with $\ang{h_i}{\beta}>0$. By \cite[Lemma~3.2.2]{Kleshchev}, $s_i\beta$ is positive because $\beta\ne\alpha_i$, and \[ \operatorname{ht}(s_i\beta) =\operatorname{ht}(\beta)-\ang{h_i}{\beta} <\operatorname{ht}(\beta). \] Also $\theta(\alpha_i)\ne0$. Repeated application of Lemma~\ref{lem:reflect-cusp} sends the actual simple module to a simple root module. At the next step one may choose a new height function positive on the simple roots, since semicuspidality at total $\theta$ weight zero does not depend on that choice. Moreover \[ \ker(\theta\circ s_i)\cap Q_\Q=\Q s_i\beta, \] so the genericity condition is preserved. The descent terminates because root height is a positive integer. \emph{Transporting the family.} A finitely generated graded module of fixed KLR weight has degrees bounded below: the PBW spanning set has finitely many permutation factors with bounded negative degrees, and positive polynomial degrees. At each inverse step, write $L$ for the current simple and $\Delta$ for its polynomial family; at the end $L=L_V$ and $\beta=\dim V$. At the endpoint take the regular module $R(\alpha_j)=k[z]$. Transport it back along the equivalences, together with \begin{equation}\label{eq:aff-exact} 0\longrightarrow \qgr^{\deg z}\Delta \xrightarrow{z}\Delta\longrightarrow L\longrightarrow0. \end{equation} At each inverse step, the whole family lies in the required domain. Let $e$ be a forbidden endpoint idempotent. The filtration of $\Delta/z^N\Delta$ by shifts of $L$ implies $e(\Delta/z^N\Delta)=0$, since the functor $M\mapsto eM$ is exact. Consequently \[ e\Delta\subseteq\bigcap_{N\ge1}z^N\Delta=0. \] The intersection is zero because degrees are bounded below and $\deg z>0$. Thus the inverse equivalence applies to \eqref{eq:aff-exact}. Choose homogeneous lifts $b_1,\ldots,b_d$ of a basis of $L$. Reduction modulo $z$ expresses a homogeneous element of $\Delta$ as a linear combination of these lifts plus $z$ times an element of smaller degree. Induction from the lower degree bound proves that the $b_j$ generate over $k[z]$. If \[ \sum_{j=1}^d f_j(z)b_j=0,\qquad f_j(z)\in k[z], \] then reduction modulo $z$ gives $f_j(0)=0$ for every $j$. Injectivity of $z$ permits division of the relation by $z$. Iteration makes each $f_j$ divisible by every power of $z$, so each is zero. Hence $\Delta=\bigoplus_{j=1}^d k[z]b_j$. Full faithfulness gives \begin{equation}\label{eq:end-delta} \END_{R(\beta)}\Delta=k[z]. \end{equation} Finite generation ensures that every ungraded endomorphism is a finite sum of homogeneous endomorphisms, so the same equality holds for the ungraded endomorphism ring. The word support of $\Delta$ equals that of $L$, by reduction modulo $z$ and the lower degree bound. The same monoidal reflection chain sends every $L^{\circ a}$ to a simple single-color nilHecke module up to grading. Hence every $L^{\circ a}$ is simple and $L$ is real; see also \cite[Proposition~2.21]{TW}. \end{proof} The family in Proposition~\ref{prop:polynomial-family} satisfies the remaining affinization condition as follows. \begin{lemma}\label{lem:center} The module $\Delta$ just constructed satisfies the central nonvanishing condition for an affinization. \end{lemma} \begin{proof} \emph{Central elements on the generic fiber.} Put $K=k(z)$, $R_K=R(\beta)\otimes_k K$ and $\Delta_K=\Delta\otimes_{k[z]}K$. Choose a finite $k[z]$-basis of $\Delta$. If $f\in\End_{R_K}\Delta_K$, clearing the denominators of its matrix gives a nonzero $p(z)\in k[z]$ such that $p(z)f$ preserves $\Delta$. It still commutes with $R(\beta)$, and hence \eqref{eq:end-delta} implies $p(z)f=a(z)\id$ for some $a(z)\in k[z]$. Therefore \[ \End_{R_K}\Delta_K=K\id. \] The commutant of the matrices of a finite set of KLR generators is the kernel of a finite system of $K$-linear equations. Extending scalars to an algebraic closure of $K$ preserves this kernel and its dimension, so the endomorphisms there are still scalar. In particular, every central element acts by a scalar. \emph{Excluding mixed spectra.} Consider a joint generalized eigenvalue tuple of the commuting $x_a$ on a nonzero word space. Suppose it contains both zero and nonzero entries. An adjacent zero and nonzero entry can be interchanged invertibly. For distinct colors this follows from $\psi_a^2=Q_{ij}(x_a,x_{a+1})$ and $Q_{ij}(0,b)\ne0$ for $b\ne0$. For equal colors use the nilHecke intertwiner with square one. Thus the zero entries can be moved entirely to the left or entirely to the right. Let $\gamma$ be the weight of the zero entries. Semicuspidal support gives both $\theta(\gamma)\ge0$ and $\theta(\beta-\gamma)\ge0$. Therefore $\theta(\gamma)=0$. By \eqref{eq:generic-charge}, $\gamma=r\beta$ for a rational $0n). \] Then $p_i\ge0$ and $a_i+p_i\ge0$ for every $a\in A$ and $1\le i\le n$. Thus the preceding lifting statement applies both to $M_T(p)$ and to each $M_T(a+p)$. Right multiplication yields \begin{equation}\label{eq:clear} XM_T(p)=\sum_a c_a(u)u^{a^T\Lambda_Tp}M_T(a+p). \end{equation} \emph{Comparing simple classes.} Perform this comparison in \[ K_0(\mathcal L_w)\otimes_{\Z[\qgr^{\pm1}]}\Z[u^{\pm1}], \qquad \qgr\longmapsto u^{-2}, \] where $\mathcal L_w$ is the localized finite-length category. Choose one representative $U$ in each orbit of simple objects under genuine grading shift. There are no nonzero grading periods, as proved in Section~\ref{sec:feigin}. The Jordan--H\"older theorem therefore gives \[ K_0(\mathcal L_w)=\bigoplus_U\Z[\qgr^{\pm1}][U]. \] The extension is free, hence flat: $\Z[u^{\pm1}]$ has basis $1,u$ over $\Z[u^{\pm2}]$. Consequently the scalar-extended group is $\bigoplus_U\Z[u^{\pm1}][U]$. An actual object $H$ has graded composition multiplicities $m_{H,U}(\qgr)\in\N[\qgr^{\pm1}]$. For every integer $r$, \[ u^r[H]=\sum_U u^rm_{H,U}(u^{-2})[U],\qquad u^rm_{H,U}(u^{-2})\in\N[u^{\pm1}]. \] Thus multiplication by every $u^r$ preserves the scalar-extended positive cone. Each monomial $M_T(a+p)$ on the right of \eqref{eq:clear} lifts to $u^{r_a}[U_a]$, where the $U_a$ are pairwise distinct shift-orbit representatives. For the left side, choose simple localized objects $L_X,L_p$ and integers $r,s$ such that \[ X=\Xi_{\ii}(u^r[L_X]),\qquad M_T(p)=\Xi_{\ii}(u^s[L_p]). \] Multiplicativity gives \[ XM_T(p)=\Xi_{\ii}\bigl(u^{r+s}[L_X\circ L_p]\bigr). \] Here $L_X\circ L_p$ is an actual object of the finite-length category $\mathcal L_w$; it need not be simple. Its composition multiplicities, multiplied by the positive formal unit $u^{r+s}$, are nonnegative as above. Injectivity of $\Xi_{\ii}$ on the scalar-extended localized Grothendieck ring identifies this class with the lifted right side of \eqref{eq:clear}. Set $H=L_X\circ L_p$. Comparison of the coefficient of $[U_a]$ gives \[ u^{r+s}m_{H,U_a}(u^{-2})=c_a(u)u^{a^T\Lambda_Tp+r_a}. \] The left side belongs to $\N[u^{\pm1}]$. Multiplication by a Laurent monomial only shifts degrees, so $c_a(u)\in\N[u^{\pm1}]$. \emph{Returning to the prescribed quantization.} In the Feigin coordinates, based multiplication is $t^{\Lambda_{\ii}/2}$. If the seed frame is $M_\Sigma(a)=t^{\Phi a}$, its form is \[ \Lambda_\Sigma=\Phi^T\Lambda_{\ii}\Phi. \] For the doubled source-adapted word $\ii=(1,\ldots,n,1,\ldots,n)$, Section~7.1 of \cite{BR} gives $\Phi=\varphi_\ii^{-1}\rho_\ii^-$ and $\bt_\Sigma=\binom{B}{-I_n}$. Its exchangeable positions are exactly $1,\ldots,n$, as required in \cite[Proposition~4.9]{BR}. Writing $b_k^-$ for the $k$th column of $\bt_\Sigma$, that proposition gives \[ \Lambda_\ii(\Phi b_k^-,\Phi\ee_\ell)=2d_k\delta_{k\ell} \quad(1\le k\le n,\ 1\le\ell\le2n), \qquad \bt_\Sigma^T\Lambda_\Sigma=(2D\ \ 0). \] After $t=u^2$, this is an ordinary integral BZ realization with compatible diagonal $2D$. Apply Lemma~\ref{lem:transport} at the actual target to obtain canonical $2D$ positivity, then Lemma~\ref{lem:descent} to obtain canonical $D$ positivity, and finally Lemma~\ref{lem:transport} again to recover any prescribed realization. \subsection{Finite type, reducible matrices and zero edges} The case of no mutable directions is vacuous. Let $B,D$ be acyclic compatible principal data with $n\ge1$. Put $L=\operatorname{lcm}(d_1,\ldots,d_n)$ and adjoin a new source vertex $0$ with \[ \widehat D=\diag(L,d_1,\ldots,d_n),\qquad \widehat B_{0i}=2d_i,\quad \widehat B_{i0}=-2L. \] Retain the old principal block $B$. The new data are integral, acyclic, connected and skew-symmetrizable. The Cartan product on every new edge is $4Ld_i\ge4$, so the Cartan matrix is of infinite type. The preceding argument applies. Take an integral compatible realization of the enlarged data. Freeze direction $0$: delete its mutable column and retain its torus generator among the frozen variables. After reordering the remaining mutable directions first, compatibility restricts to \[ \widetilde B_{\mathrm{fr}}^T\widehat\Lambda=(D\ \ 0). \] Every mutation word in the original directions gives the same seeds in this frozen pattern as in the enlarged pattern. The established positivity therefore applies at each of its targets. Lemma~\ref{lem:transport}, rooted at that target, gives positivity for every realization of the original principal data. A mutation-acyclic pattern can be rooted at an acyclic seed and every specified source and target retained. This proves Theorem~\ref{thm:B}. In particular, rank-two quantum cluster patterns satisfy the same positivity statement, with arbitrary additional coordinates treated as frozen. \input{bibliography_mutation_acyclic.tex} \end{document}