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\title{Strongly positive bases in rank-two quantum cluster algebras}
\author{Qiyue Tang}
\date{}

\begin{document}
\maketitle
\begingroup
\renewcommand{\thefootnote}{}
\footnotetext{\textit{2020 Mathematics Subject Classification.}
Primary 13F60; Secondary 16T20.\newline
\textit{Keywords.} Quantum cluster algebra, strongly positive basis,
pointed element, standard monomial.}
\endgroup

\begin{abstract}
We prove that, for arbitrary positive integers $b,c$, every strongly
positive basis of the coefficient-free rank-two quantum cluster algebra
$\mathcal A_v(b,c)$ has a unique element pointed at each integer pair,
and these elements exhaust the basis. This proves Conjecture~12 of
Lee, Li, Rupel, and Zelevinsky. The proof extracts a normalized corner
term from a positive standard-monomial expansion and uses extremal
support to obtain a finite spanning argument.
\end{abstract}

\section{Introduction}

Distinguished bases of cluster algebras are often indexed by integer
vectors that describe the leading terms of their Laurent expansions.
For a basis constructed by prescribing such expansions, this indexing
is part of the construction. For an arbitrary basis specified through
positivity of multiplication and compatibility with cluster monomials,
the existence and uniqueness of leading terms require a separate
argument.

Lee, Li, Rupel, and Zelevinsky introduced strongly positive bases in
their study of rank-two quantum cluster algebras \cite[Section~5]{LLRZ}.
Such a basis is invariant under the bar operation, is independent of
the initial cluster, contains every normalized cluster monomial, and
has nonnegative Laurent polynomials in the quantum parameter as
structure constants. Their Conjecture~12 asks whether every integer
pair labels exactly one pointed basis element. Pointedness specifies
both a support region and a coefficient equal to one at its corner;
the precise conventions appear in Section~\ref{sec:setup}.

We prove this assertion in Theorem~\ref{thm:main}, including the
exhaustion of the basis by the resulting elements. The argument starts
with the standard monomial basis. Each standard monomial has a
prescribed corner and expands positively in any strongly positive
basis. Laurent positivity prevents cancellation in this expansion,
and bar invariance fixes the normalization of the element contributing
at the corner. To pass from existence to uniqueness, we prove a
support estimate for arbitrary, possibly signed, standard-basis
expansions. The estimate gives a finite induction: the standard
monomials outside the strictly positive quadrant are already cluster
monomials, and each remaining reduction decreases the sum of the two
positive coordinates.

The external algebraic inputs are the quantum Laurent phenomenon and
the acyclic standard monomial basis theorem of Berenstein and
Zelevinsky \cite{BZ}. The normalization used here is the one in
\cite{LLRZ}.

\paragraph{AI contribution.}
The mathematical argument and manuscript were generated using OpenAI
Codex. The same AI system performed a separate adversarial review of
the proof and exact finite checks of the normalization formulas.
The user supplied the research question and relevant materials.

\section{Quantum Laurent expansions and standard monomials}
\label{sec:setup}

Fix positive integers $b,c$ and an indeterminate $v$. Set
\[
 R=\mathbb Z[v^{\pm1}],\qquad
 R_+=\mathbb Z_{\geq0}[v^{\pm1}].
\]
Thus $R_+$ includes zero, and membership in $R_+$ means that every
coefficient of every integer power of $v$ is nonnegative.
Let $\mathcal T$ be the quantum torus over $R$ generated by the
invertible elements $X_1,X_2$ with relation $X_2X_1=v^2X_1X_2$.
For $r,s\in\mathbb Z$, define the normalized monomial
\[
 X^{(r,s)}=v^{rs}X_1^rX_2^s.
\]
These monomials form an $R$-basis of $\mathcal T$, and their
multiplication is
\begin{equation}\label{eq:torus}
 X^{(r,s)}X^{(t,u)}
   =v^{st-ru}X^{(r+t,s+u)}
 \qquad(r,s,t,u\in\mathbb Z).
\end{equation}
The bar operation is the additive anti-involution determined by
$\overline v=v^{-1}$, $\overline{X_1}=X_1$,
$\overline{X_2}=X_2$, and $\overline{FG}=\overline G\,\overline F$.
In particular, $\overline{X^{(r,s)}}=X^{(r,s)}$.

Inside the skew field of fractions of $\mathcal T$, the cluster
variables $X_m$, indexed by $m\in\mathbb Z$, are determined from
$X_1,X_2$ by
\begin{equation}\label{eq:exchange}
 X_{m+1}X_{m-1}=
 \begin{cases}
 v^bX_m^b+1,&m\text{ odd},\\
 v^cX_m^c+1,&m\text{ even}.
 \end{cases}
\end{equation}
The coefficient-free quantum cluster algebra $\mathcal A_v(b,c)$ is
the $R$-subalgebra generated by all $X_m$. Its clusters are the adjacent
pairs $(X_m,X_{m+1})$, which satisfy
$X_{m+1}X_m=v^2X_mX_{m+1}$. A normalized cluster monomial is
\[
 v^{ij}X_m^iX_{m+1}^j,
 \qquad i,j\in\mathbb Z_{\geq0},\quad m\in\mathbb Z.
\]
These conventions agree with \cite[Section~2]{LLRZ}.
The quantum Laurent phenomenon \cite[Corollary~5.2]{BZ} implies
$\mathcal A_v(b,c)\subseteq\mathcal T$ and gives a finite Laurent
expansion in every cluster.

For $F\in\mathcal T$, write its finite expansion as
$F=\sum_{r,s\in\mathbb Z}f_{r,s}X^{(r,s)}$, where $f_{r,s}\in R$.
Its support is
\[
 \supp F=\{(r,s)\in\mathbb Z^2:f_{r,s}\ne0\}.
\]
The coefficient at an exponent not in the support is understood to be
zero. We call $F$ Laurent positive in this cluster when all
$f_{r,s}$ belong to $R_+$.

\begin{definition}\label{def:pointed}
An element $F\in\mathcal A_v(b,c)$ is pointed at
$(a_1,a_2)\in\mathbb Z^2$ if it has a finite expansion
\begin{equation}\label{eq:pointed}
 F=\sum_{p,q\geq0}e(p,q)X^{(bp-a_1,cq-a_2)},
 \qquad e(p,q)\in R,\qquad e(0,0)=1,
\end{equation}
where $p,q$ range over nonnegative integers.
\end{definition}

The corner of this expansion is $(-a_1,-a_2)$. In particular,
the minimum of the first coordinate of $\supp F$ is $-a_1$, and
the minimum of the second coordinate is $-a_2$. Thus a pointed
element determines its label uniquely.

\begin{definition}\label{def:strong}
A strongly positive basis is an $R$-basis $\mathcal B$ of
$\mathcal A_v(b,c)$ with the following properties:
\begin{enumerate}
\item every element of $\mathcal B$ is bar invariant;
\item $\mathcal B$ is independent of the choice of initial cluster;
\item $\mathcal B$ contains all normalized cluster monomials;
\item the product of any two elements of $\mathcal B$ is a finite
$R_+$-linear combination of elements of $\mathcal B$.
\end{enumerate}
\end{definition}

Here independence of the initial cluster means that changing the
chosen initial adjacent pair gives the same subset of the ambient
cluster algebra, under its natural identification. This is the
definition in \cite[Section~5]{LLRZ}; the proof below uses properties
(1), (3), and (4) in the fixed initial cluster.

For an integer $h$, let $[h]_+=\max(h,0)$. Define the standard
monomials, with the factors in the displayed order, by
\begin{equation}\label{eq:standard}
 M[a_1,a_2]=v^{a_1a_2}
 X_3^{[a_1]_+}X_1^{[-a_1]_+}
 X_2^{[-a_2]_+}X_0^{[a_2]_+}
 \qquad(a_1,a_2\in\mathbb Z).
\end{equation}

\begin{lemma}\label{lem:standard}
The elements $M[a_1,a_2]$, for $(a_1,a_2)\in\mathbb Z^2$, form an
$R$-basis of $\mathcal A_v(b,c)$. Each $M[a_1,a_2]$ is pointed at
$(a_1,a_2)$. If $a_1\leq0$ or $a_2\leq0$, it is a normalized
cluster monomial.
\end{lemma}

\begin{proof}
The basis assertion is the rank-two instance of
\cite[Theorems~7.3 and~7.5]{BZ}, stated in precisely the normalization
\eqref{eq:standard} in \cite[Section~4]{LLRZ}.
For the compatibility convention in \cite{BZ}, take $q^{1/2}=v$,
exchange matrix
$\left(\begin{smallmatrix}0&-b\\c&0\end{smallmatrix}\right)$,
and skew form matrix
$\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$.
The transpose of the former times the latter is
$\operatorname{diag}(c,b)$. There are no frozen variables, and the
exchange matrix is acyclic. Thus the standard monomials form a
basis of the lower bound, the $R$-subalgebra generated by
$X_1,X_2,X_3,X_0$, and the lower bound equals
$\mathcal A_v(b,c)$. In the standard order used in \cite{BZ}, the
factors are $X_1,X_2,X_3,X_0$, with $X_1$ and $X_3$, respectively
$X_2$ and $X_0$, mutually exclusive. Moving $X_3$ past $X_2$ to
obtain \eqref{eq:standard} only multiplies a monomial by a unit
$v^k$, for an integer $k$.

The exchange relations give
\begin{equation}\label{eq:neighbors}
 X_0=X^{(0,-1)}+X^{(b,-1)},\qquad
 X_3=X^{(-1,0)}+X^{(-1,c)}.
\end{equation}
Consequently every exponent in \eqref{eq:standard} has the form
$(bp-a_1,cq-a_2)$ with $p,q\geq0$. The corner is obtained uniquely
by choosing the first summand in each factor of
\eqref{eq:neighbors}. Before multiplication by $v^{a_1a_2}$,
this term is $X_1^{-a_1}X_2^{-a_2}$, so its normalized coefficient
after that multiplication is one. This proves pointedness.

If $a_1,a_2\leq0$, formula \eqref{eq:standard} is the initial
cluster monomial $v^{a_1a_2}X_1^{-a_1}X_2^{-a_2}$.
In the two other boundary regions, adjacent quasi-commutation gives
\begin{align*}
 M[a_1,a_2]&=v^{-a_1a_2}X_0^{a_2}X_1^{-a_1}
   &&(a_1\leq0\leq a_2),\\
 M[a_1,a_2]&=v^{-a_1a_2}X_2^{-a_2}X_3^{a_1}
   &&(a_2\leq0\leq a_1).
\end{align*}
These are normalized cluster monomials as well.
\end{proof}

\section{Positivity and finite spanning}

We first spell out the denominator-clearing argument from
\cite[Section~5]{LLRZ}.

\begin{lemma}\label{lem:positive}
Every element of a strongly positive basis is Laurent positive in
the initial cluster.
\end{lemma}

\begin{proof}
Let $Y\in\mathcal B$, and write
$Y=\sum f_{r,s}X^{(r,s)}$. Since the support is finite, there exist
nonnegative integers $u,w$ such that $r+u,s+w\geq0$ for every
$(r,s)\in\supp Y$. By \eqref{eq:torus},
\begin{equation}\label{eq:clear}
 YX^{(u,w)}=\sum_{r,s}v^{su-rw}f_{r,s}X^{(r+u,s+w)}.
\end{equation}
The monomials on the right are distinct normalized initial cluster
monomials, hence are elements of $\mathcal B$. On the other hand,
$Y$ and $X^{(u,w)}$ belong to $\mathcal B$, so their product has
an $R_+$-linear expansion in this basis. Uniqueness of basis
expansion shows that $v^{su-rw}f_{r,s}\in R_+$ for every $r,s$.
Multiplication by an integer power of $v$ preserves $R_+$, and
therefore $f_{r,s}\in R_+$.
\end{proof}

The next support estimate applies even when the standard-basis
coefficients have negative terms.

\begin{lemma}\label{lem:support}
Let $a_1,a_2\in\mathbb Z$, and suppose
$F\in\mathcal A_v(b,c)$ satisfies
\[
 \supp F\subseteq
 \{(r,s)\in\mathbb Z^2:r\geq-a_1,\ s\geq-a_2\}.
\]
In its finite standard-basis expansion
\begin{equation}\label{eq:standardexp}
 F=\sum_{t_1,t_2\in\mathbb Z}d_{t_1,t_2}M[t_1,t_2],
 \qquad d_{t_1,t_2}\in R,
\end{equation}
every nonzero coefficient has $t_1\leq a_1$ and $t_2\leq a_2$.
Moreover, $d_{a_1,a_2}$ is the coefficient of
$X^{(-a_1,-a_2)}$ in $F$.
\end{lemma}

\begin{proof}
The assertion is immediate for $F=0$. Otherwise, among the finitely
many labels with nonzero coefficient in \eqref{eq:standardexp},
choose one with largest first coordinate, and among those choose one
with largest second coordinate. Denote it by $(t_1^*,t_2^*)$.
If another label has $t_1<t_1^*$, every point in the support of
$M[t_1,t_2]$ has first coordinate at least $-t_1>-t_1^*$.
If $t_1=t_1^*$, its second coordinate satisfies $t_2<t_2^*$,
so every point in that support has second coordinate greater than
$-t_2^*$. Thus only $M[t_1^*,t_2^*]$ contributes at
$(-t_1^*,-t_2^*)$, and the coefficient there is
$d_{t_1^*,t_2^*}\ne0$. The assumed support bound gives
$t_1^*\leq a_1$. It follows that all labels occurring in
\eqref{eq:standardexp} have $t_1\leq a_1$.

To bound the second coordinate, choose instead a label with largest
second coordinate, breaking ties by the largest first coordinate.
The same argument, with the roles of the coordinates interchanged,
shows that its corner cannot cancel. The support bound then gives
$t_2\leq a_2$ for every occurring label.

For every such label different from $(a_1,a_2)$, at least one of
these inequalities is strict. Its standard monomial therefore does
not contain $X^{(-a_1,-a_2)}$. Since the corner coefficient of
$M[a_1,a_2]$ is one, the asserted coefficient identity follows.
\end{proof}

\begin{lemma}\label{lem:span}
Suppose that, for every $(a_1,a_2)\in\mathbb Z^2$, an element
$F[a_1,a_2]\in\mathcal A_v(b,c)$ is pointed at $(a_1,a_2)$.
Suppose also that
\[
 F[a_1,a_2]=M[a_1,a_2]
 \quad\text{if }a_1\leq0\text{ or }a_2\leq0.
\]
Then the elements $F[a_1,a_2]$ span $\mathcal A_v(b,c)$ over $R$.
\end{lemma}

\begin{proof}
Lemma~\ref{lem:support} gives a finite expansion
\begin{equation}\label{eq:triangular}
 F[a_1,a_2]=M[a_1,a_2]+
 \sum_{\substack{t_1\leq a_1,\ t_2\leq a_2\\
                  (t_1,t_2)\ne(a_1,a_2)}}
 d_{t_1,t_2}M[t_1,t_2].
\end{equation}
It suffices to put each standard monomial in the span of the
$F[a_1,a_2]$. This holds by assumption if at least one coordinate
is nonpositive. For $a_1,a_2>0$, use induction on $a_1+a_2$.
Each term in the sum in \eqref{eq:triangular} either has a
nonpositive coordinate or has both coordinates positive with
$t_1+t_2<a_1+a_2$. In the former case it is already in the span;
in the latter it is in the span by induction. Solving
\eqref{eq:triangular} for $M[a_1,a_2]$ proves the claim.
All expansions are finite, and the induction parameter decreases
until it reaches the boundary. Lemma~\ref{lem:standard} now gives
the required spanning assertion.
\end{proof}

\section{Pointed parametrization}

\begin{theorem}\label{thm:main}
For arbitrary positive integers $b,c$, let $\mathcal B$ be a strongly
positive basis of $\mathcal A_v(b,c)$. For every
$(a_1,a_2)\in\mathbb Z^2$, there is a unique element
$B[a_1,a_2]\in\mathcal B$ pointed at $(a_1,a_2)$, and
\[
 \mathcal B=\{B[a_1,a_2]:(a_1,a_2)\in\mathbb Z^2\}.
\]
\end{theorem}

This is Conjecture~12 of \cite{LLRZ}, together with its stated
exhaustion consequence. The conjecture is numbered~5.1 in the
arXiv HTML rendering of that article.

\begin{proof}
Fix $(a_1,a_2)\in\mathbb Z^2$. The standard monomial
\eqref{eq:standard} is an integer power of $v$ times a finite
product of cluster variables. Every factor belongs to
$\mathcal B$, so repeated use of multiplication positivity gives
a finite expansion
\begin{equation}\label{eq:positiveexp}
 M[a_1,a_2]=\sum_{Y\in\mathcal B}h_Y Y,
 \qquad h_Y\in R_+.
\end{equation}
By Lemma~\ref{lem:positive}, all Laurent coefficients of every
$Y$ are in $R_+$. A product of two nonzero elements of $R_+$
is nonzero, and a sum of nonnegative Laurent polynomials cannot
cancel. Consequently, for every $Y$ with $h_Y\ne0$,
\begin{equation}\label{eq:supportcontain}
 \supp Y\subseteq\supp M[a_1,a_2]
 \subseteq\{(bp-a_1,cq-a_2):p,q\in\mathbb Z_{\geq0}\}.
\end{equation}

Let $k_Y\in R_+$ be the coefficient of
$X^{(-a_1,-a_2)}$ in $Y$. Taking that coefficient in
\eqref{eq:positiveexp} yields
\begin{equation}\label{eq:corner}
 1=\sum_{Y\in\mathcal B}h_Yk_Y.
\end{equation}
There is exactly one nonzero summand in this identity, and it equals
one. For the corresponding element $Y$, we have $h_Yk_Y=1$.
Evaluating these two nonnegative Laurent polynomials at $v=1$
shows that each has coefficient sum one. Hence, for some integer
$j$, they are $k_Y=v^j$ and $h_Y=v^{-j}$.

Since $Y$ is bar invariant and every normalized Laurent monomial is
fixed by bar, each Laurent coefficient of $Y$ is fixed by the
substitution $v\mapsto v^{-1}$. In particular $v^j=v^{-j}$.
As $v$ is indeterminate, $j=0$. Thus $k_Y=1$, and
\eqref{eq:supportcontain} proves that this element is pointed at
$(a_1,a_2)$. Denote the selected element by $B[a_1,a_2]$.

When $a_1\leq0$ or $a_2\leq0$, Lemma~\ref{lem:standard} shows
that $M[a_1,a_2]$ is itself a normalized cluster monomial, hence
belongs to $\mathcal B$. Its expansion
\eqref{eq:positiveexp} therefore consists of itself with coefficient
one. It follows that
\[
 B[a_1,a_2]=M[a_1,a_2]
 \quad\text{if }a_1\leq0\text{ or }a_2\leq0.
\]
Lemma~\ref{lem:span} now implies that the selected elements
$B[a_1,a_2]$ span the entire algebra. They are a subset of the
basis $\mathcal B$, so this subset equals $\mathcal B$:
otherwise an omitted basis element would be a finite linear
combination of the selected ones, contradicting linear independence.

Finally, the label of a pointed element is determined by the two
coordinate minima of its support. Thus different pairs give
different selected elements. If $Y\in\mathcal B$ is pointed at
$(a_1,a_2)$, exhaustion gives $Y=B[t_1,t_2]$ for some pair
$(t_1,t_2)$. Comparing the two coordinate minima gives
$(t_1,t_2)=(a_1,a_2)$, which proves uniqueness.
\end{proof}

\begin{thebibliography}{9}

\bibitem{BZ}
A.~Berenstein and A.~Zelevinsky,
\textit{Quantum cluster algebras},
Adv. Math. \textbf{195} (2005), no.~2, 405--455.
\href{https://arxiv.org/abs/math/0404446}{arXiv:math/0404446}.

\bibitem{LLRZ}
K.~Lee, L.~Li, D.~Rupel, and A.~Zelevinsky,
\textit{Greedy bases in rank 2 quantum cluster algebras},
Proc. Natl. Acad. Sci. USA \textbf{111} (2014), no.~27, 9712--9716.
\href{https://arxiv.org/abs/1405.2311}{arXiv:1405.2311}.

\end{thebibliography}

\bigskip
\noindent\textsc{Zhili College, Tsinghua University, Beijing 100084, China}\par
\noindent\textit{Email address:}
\href{mailto:tangqy24@mails.tsinghua.edu.cn}{\texttt{tangqy24@mails.tsinghua.edu.cn}}

\end{document}