Quantum Laurent positivity for mutation-acyclic cluster algebras

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1. Theorem 1.1
   PF:2026.000011/v1/THM-1.1

   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.

2. Theorem 1.2
   PF:2026.000011/v1/THM-1.2

   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.

3. Lemma 2.1
   PF:2026.000011/v1/LEM-2.1

   Fix $B,D$ as in (equation reference), a finite mutation word, and a mutable output label $j\in\{1,\ldots,n\}$. All realizations below start with principal part $B$ and compatible diagonal $D$, and use this same word and label. Set \[ P=\binom{B}{I_n},\qquad \Lambda_P= 0&-D\\D&-DB . \] Let $M_P$ be the initial toric frame of this principal realization, and let $g\in\Z^n$ be the classical $g$-vector of the chosen output variable relative to the initial seed. Write \[ X^P=\sum_{e\in\N^n}C_e(v)M_P((g,0)+Pe). \] Here $C_e(v)\in\Z[v^{\pm1}]$, and only finitely many coefficients are nonzero. For every compatible realization with the same $B,D$, the corresponding variable is exactly X=\sum_e C_e(v)M(\gtil+\bt e), He…

4. Lemma 2.2
   PF:2026.000011/v1/LEM-2.2

   Fix $B,D$ as in (equation reference), a finite mutation word, and a mutable output label. Let $X^D$ and $X^{2D}$ be the corresponding cluster variables in the canonical realizations for $D$ and $2D$, with indeterminates $v$ and $u$, respectively. Write their Laurent expansions in the respective initial based tori as \[ X^D=\sum_{a\in\Z^{2n}}c_a^D(v)M_D(a),\qquad X^{2D}=\sum_{a\in\Z^{2n}}c_a^{2D}(u)M_{2D}(a), \] where $M_D$ and $M_{2D}$ are the initial toric frames. Then \[ c_a^{2D}(u)=c_a^D(u^2)\qquad(a\in\Z^{2n}). \] In particular, one expansion has nonnegative Laurent coefficients if and only if the other does.

5. Lemma 3.1
   PF:2026.000011/v1/LEM-3.1

   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.

6. Lemma 3.2
   PF:2026.000011/v1/LEM-3.2

   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.

7. Proposition 3.3
   PF:2026.000011/v1/PROP-3.3

   Let $V$ be indecomposable and rigid, with $\nu=\dim V$, and choose $\theta$ as in (equation reference). 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 $\lambda_V(t)\in\Q(t)^\times$.

8. Proposition 3.4
   PF:2026.000011/v1/PROP-3.4

   For every indecomposable rigid $V$, the self-dual simple module $L_V$ constructed in Proposition 3.3 satisfies \Omega([V]^*)=\ch_{t^{-1}}L_V. Equivalently, $\lambda_V(t)=1$ in (equation reference).

9. Lemma 3.5
   PF:2026.000011/v1/LEM-3.5

   Let $L$ be a $\theta$ semicuspidal simple of weight $\beta$, where $\theta(\beta)=0$ and $\theta(\alpha_i)\ne0$. Its image under the reflection equivalence on the admissible side is $\theta\circ s_i$ semicuspidal.

10. Proposition 3.6
    PF:2026.000011/v1/PROP-3.6

    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 \[ \Delta/z\Delta\simeq L_V,\qquad \END_{R(\dim V)}\Delta=k[z],\qquad \supp\Delta=\supp L_V. \]

11. Lemma 3.7
    PF:2026.000011/v1/LEM-3.7

    The module $\Delta$ of Proposition 3.6 satisfies the central nonvanishing condition for an affinization.

12. Lemma 3.8
    PF:2026.000011/v1/LEM-3.8

    Let $V,W$ be indecomposable rigid representations such that $V\oplus W$ is rigid. Then $L_V$ and $L_W$ strongly commute.

13. Lemma 4.1
    PF:2026.000011/v1/LEM-4.1

    Let $\ii=(i_1,\ldots,i_\ell)$ be a reduced expression of $w$, and let $\lambda$ be dominant. Put $C_\lambda=\mathsf M(w\lambda,\lambda)$ and $\nu=\lambda-w\lambda$. For every $\ba\in\N^\ell$ with $\sum_k a_k\alpha_{i_k}=\nu$, \[ e(i_1^{a_1},\ldots,i_\ell^{a_\ell})C_\lambda\ne0 \quad\Longleftrightarrow\quad [t^{\ba}]\Psi_{\ii}(\Delta_{w\lambda})\ne0. \] Consequently $\Xi_{\ii}([C_\lambda])$ is a positive formal unit times $t_\lambda^{-1}$.

14. Proposition 4.2
    PF:2026.000011/v1/PROP-4.2

    For a reduced word $\ii$ with full support, $\Xi_\ii$ extends to an injective homomorphism \[ K_0(\mathcal L_w)\otimes_{\Z[\qgr^{\pm1}]}\Z[u^{\pm1}] \lhook\joinrel\longrightarrow\mathcal T_\ii, \qquad \qgr\longmapsto u^{-2}. \] For the doubled word, every frozen Laurent monomial of $\Sigma^-_\ii$ is the image of an invertible simple class times $u^r$. Nonzero localized simples have no nonzero grading period.

15. Lemma 5.1
    PF:2026.000011/v1/LEM-5.1

    Let $B$ be acyclic and suppose that its associated Cartan matrix is irreducible and of infinite type. There is a Coxeter belt of acyclic seeds whose clusters at completed periods have pairwise disjoint sets of coefficient-free classical mutable variables. In particular, for any prescribed finite set of cluster variables, one can choose a completed period whose initial mutable variables avoid all their coefficient-free classical specializations. Along the same mutation words, the prescribed variables are then noninitial in every compatible realization of the principal pattern.

16. Lemma 5.2
    PF:2026.000011/v1/LEM-5.2

    Let $B$ be acyclic, and use the same finite mutation word in realizations with exchange matrices $P^+=\binom BI$ and $P^-=\binom B{-I}$. In the negative principal realization used by the Feigin dictionary, the representations corresponding to the noninitial members of the resulting cluster are the same summands of one locally tilting representation as in the positive principal realization. They are therefore pairwise Ext orthogonal. If all members are noninitial, their sum is a full rigid tilting representation.

17. Lemma 5.3
    PF:2026.000011/v1/LEM-5.3

    In this common realization, $X$ and every based target monomial $M_T(b)$ with $b_i\ge0$ for $1\le i\le n$ lift to simple localized classes times powers of $u$. Distinct exponent vectors $b$ give distinct grading-shift orbits of simple objects.