Three meanings of infinite type in rank-two cluster algebras

Definitions and mathematical statement

The source asks to explain cluster algebras of infinite type, but ends with an incomplete "See" and gives no reference or precise classification target. Infinite cluster type at finite rank, an infinite mutation class of exchange matrices, and infinite rank are different notions. The aid supplies a complete comparison only in coefficient-free rank two.

Fix positive integers $b,c$ and initial principal exchange matrix \[ B=\begin{pmatrix}0&b\\-c&0\end{pmatrix}. \] There are no frozen variables here. Let $x_1,x_2$ be algebraically independent and generate the rank-two cluster algebra inside ${\mathbb{Q}}(x_1,x_2)$ by the bi-infinite recurrence \[ x_{m-1}x_{m+1}= \begin{cases}1+x_m^b,&m\text{ odd},\\ 1+x_m^c,&m\text{ even}. \end{cases} \] This convention corresponds to mutating the first initial variable first. The source aid reports that the labeled principal-matrix mutation class is always exactly $\{B,-B\}$, while the number of cluster variables is finite exactly for $bc\le3$. Thus two exchange matrices do not imply finite cluster type.

For $bc\ge4$, write the reduced Laurent expression of a noninitial variable as \[ x_j=\frac{P_j(x_1,x_2)}{x_1^{d_{j,1}}x_2^{d_{j,2}}}, \] with numerator not divisible by either initial variable. For the forward alternating-mutation ray the exact reported denominator vectors are \[ d_{2k+3}=C^k\binom10,\qquad d_{2k+4}=C^k\binom b1,\qquad k\ge0, \quad C=\begin{pmatrix}bc-1&-b\\c&-1\end{pmatrix}. \] The opposite ray has the corresponding exchanged-label description. The index convention matters; these are not asserted to list every initial or reverse-ray variable with the same positive index formula.

At the affine threshold $bc=4$, $(C-I)^2=0$ and $C^k=I+k(C-I)$, so both vector sequences grow linearly in $k$. For $bc>4$ they grow exponentially with per-two-mutation spectral radius \[ \Lambda=\frac{bc-2+\sqrt{bc(bc-4)}}2>1. \] In both infinite regimes the exchange graph is a bi-infinite path, so every radius-$R$ graph ball has exactly $2R+1$ vertices for integer $R\ge0$. Consequently exchange-graph ball growth and principal mutation-class size do not distinguish the affine and indefinite rank-two cases, whereas denominator growth does. This comparison does not supply a theory or classification for all infinite-type or infinite-rank cluster algebras.

Short English statement

Clarify the different meanings of infinite type. In coefficient-free rank two, affine and indefinite cases both have two principal exchange matrices and a path exchange graph, but their denominator vectors grow linearly and exponentially respectively.

Sources

• American Institute of Mathematics workshop Braid groups, clusters and free probability, Problem 5.1, https://aimath.org/WWN/braidgroups/braidgroups.pdf.
• Local UnsolvedMath registry, permanent ID 20002700, source code catalogue entry, canonical location aim-probability-notes.json, notes[141], record dated 14 August 2026. The original statement ends with "See" without a citation; the original literature field is empty.
• Supplied research attempt 1: partial result, problem status at run partially solved, rigor labelled proved, novelty confidence low. Its rank-two three-invariant comparison is machine-generated research aid, not peer reviewed; independently verify before citation. No separate authoritative reference for the classification or formulas is supplied.