Dugas’ Restricted Quiver Mutation Conjecture
Problem B
The restricted mutation classes were computed through a modified sage. A general pattern that I noted while playing with examples is that quivers with a higher density of oriented $3$-cycles seem to have larger restricted mutation classes (e.g. the $j \to j - 2 \pmod n$ and triangular quivers below). Those that don’t tend to be have relatively small restricted mutation classes in comparison.
Below, RMCS is short for $|[Q]_\text{res}|$.
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ of multiplicity $2$
[proof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows exponentially in cardinality in $n$. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $3$ | $1$ |
| $4$ | $2$ |
| $5$ | $3$ |
| $6$ | $5$ |
| $7$ | $5$ |
| $8$ | $8$ |
| $9$ | $10$ |
| $10$ | $15$ |
| $11$ | $19$ |
| $12$ | $31$ |
| $13$ | $41$ |
| $14$ | $64$ |
| $15$ | $94$ |
| $16$ | $143$ |
| $17$ | $211$ |
| $18$ | $329$ |
| $19$ | $493$ |
| $20$ | $766$ |
| $21$ | $1170$ |
| $22$ | $1811$ |
| $23$ | $2787$ |
| $24$ | $4341$ |
| $25$ | $6713$ |
| $26$ | $10462$ |
| $27$ | $16274$ |
| $28$ | $25415$ |
[/proof]
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ and $j \to j+2\pmod n$.
[proof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows exponentially in cardinality in $n$. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $5$ | $6$ |
| $6$ | $6$ |
| $7$ | $4$ |
| $8$ | $4$ |
| $9$ | $5$ |
| $10$ | $6$ |
| $11$ | $7$ |
| $12$ | $11$ |
| $13$ | $12$ |
| $14$ | $17$ |
| $15$ | $23$ |
| $16$ | $31$ |
| $17$ | $40$ |
| $18$ | $58$ |
| $19$ | $76$ |
| $20$ | $109$ |
| $21$ | $149$ |
| $22$ | $208$ |
| $23$ | $287$ |
| $24$ | $410$ |
| $25$ | $567$ |
| $26$ | $803$ |
| $27$ | $1128$ |
| $28$ | $1599$ |
| $29$ | $2249$ |
| $30$ | $3201$ |
[/proof]
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ and $j \to j - 2 \pmod n$
[proof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows exponentially in cardinality in $n$. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $5$ | $6$ |
| $6$ | $2$ |
| $7$ | $28$ |
| $8$ | $42$ |
| $9$ | $48$ |
| $10$ | $440$ |
| $11$ | $4000$ |
| $12$ | $768$ |
| $13$ | $50585$ |
| $14$ | $62630$ |
| $15$ | $86369$ |
[/proof]
[proof=Growth rate for $n = 5$ (finite)] For $n = 5$, there are finitely many restricted cluster variables. Code used to find the restricted cluster variables (requires this version of sage):
[/proof]
[proof=Growth rate for $n = 6$ (exponential?)] Computed up to depth 15. Code used to find the restricted cluster variables (requires this version of sage):
Scatter plot of data:
[/proof]
[proof=Growth rate for $n = 7$ (finite)] For $n = 7$, there are finitely many restricted cluster variables. [/proof]
[proof=Growth rate for $n = 8$ (finite)] For $n = 8$, there are finitely many restricted cluster variables. [/proof]
[proof=Growth rate for $n = 9$ (cubic?)] For $n = 9$, we seem to have cubic growth rate.
[/proof]
[proof=Growth rate for $n = 10$ (finite)] For $n = 10$, there are finitely many restricted cluster variables. [/proof]
Triangular quivers
[proof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows rapidly in cardinality in the side length. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| Side length | $|[Q]_\mathrm{res}|$ |
|---|---|
| $0$ | $1$ |
| $1$ | $1$ |
| $2$ | $2$ |
| $3$ | $19$ |
| $4$ | $17340$ |
For the case of a side length of $5$, the restricted mutation class appears to be extremely large (already $854{,}236$ quivers at depth $11$) yet none of the resultant quivers have edge multiplicity larger than $2$. These types of quivers may be good to check as an extreme case.
[/proof]
[proof=Growth rate for side length 2 (finite)] There are finitely many cluster variables. [/proof]
[proof=Growth rate for side length 3 (exponential?)] Code used to find the restricted cluster variables (requires this version of sage):
Scatter plot of data:
[/proof]
[proof=Growth rate for side length 4 (quintic?)]
[/proof]
Problem C
Many balanced and connected quivers generate infinitely many cluster variables, even if we only allow for sequences of mutations at $4$-valent vertices.
Markov quiver
The Markov quiver is the quiver on $[3]$ where with arrows $1 \to 2$, $2 \to 3$, and $3 \to 1$ all of multiplicity $2$. It is so named because it the cluster variables it generates can be used to construct the solutions to the Markov equation (see 3.4 of https://arxiv.org/abs/1608.05735).
For this quiver, mutating at any vertex simply reverses all the arrows so restricted mutation sequences are just the usual non-restricted mutation sequences. Since this quiver is not mutation-equivalent to a Dynkin ADE diagram, it generates infinitely-many cluster variables.
Multiplicity 2 square
The multiplicity two square is the quiver on $[4]$ with arrows $i \to i + 1$ modulo $4$ of multiplicity two. Here is code to plot these quivers:
One can produce infinitely many cluster variables by performing the sequence of mutations at the vertices
\[1 \to 3 \to 2 \to 4 \to 1 \to 3 \to 2 \to 4 \to \cdots.\]Here is code that implements this mutation sequence and displays the resultant cluster seeds (note: gets messy very fast).
[proof=Proof of infinitely many cluster variables] Let $Q$ be our quiver as described above and $(Q, \{x_1, \ldots, x_4\})$ be our initial seed. We will specialize by letting $x_1 = x_3 = x$ and $x_2 = x_4 = y$. By direct computation, one obtains
\[\mu_4 \mu_2 \mu_3 \mu_1(Q, \{x, y, x, y\}) = \left(Q, \left\{\frac{2y^2}{x}, \frac{8y^3}{x^2}, \frac{2y^2}{x}, \frac{8y^3}{x^2}\right\}\right).\]Under this specialization, we define
\[X_n = \frac{2^{n(2n - 1)} y^{2n}}{x^{2n - 1}} \quad\text{ and }\quad Y_n = \frac{2^{n(2n + 1)}y^{2n + 1}}{x^{2n}}.\]for $n \ge 1$ and $X_0 = x$ and $Y_0 = y$. By induction and the formula we computed for $\mu_4 \mu_2 \mu_3 \mu_1(Q, {x, y, x, y})$, one can show that
\[(\mu_4 \mu_2 \mu_3 \mu_1)^n(Q, \{x, y, x, y\}) = (Q, \{X_n, Y_n, X_n, Y_n\})\]for all $n \ge 0$. Thus, $Q$ cannot produce finitely many cluster variables. [/proof]
Square with multiplicity 2 diagonal
Code to plot the quiver in question:
Mutating this quiver at the vertex $0$ or vertex $1$ gives a quiver that is isomorphic to the original quiver. In fact, the resultant quiver is the original quiver with all its arrows reversed.
Since vertices $0$ and $1$ are the only vertices that can be mutated at, this construction is equivalent to an ice quiver of type $\widetilde{A}_1$. So this quiver produces infinitely many cluster variables.
Surfaces of type A
The quivers of surface type $A$ are constructed by taking a triangulation of a regular $(n+3)$-gon adding vertices to all edges of the polygon and adding arrows so that there is a clockwise-oriented triangle within each triangle of the triangulation. There is an example of this in Example 2.2, Figure 3 of Lauren William’s survey article. The $4$-valent vertices are precisely the vertices of the quiver corresponding to the diagonals of the triangulation. These quivers are known to be both restricted-mutation finite as well as produce finitely-many cluster variables.
Surfaces of type D
The quivers of surface type $D$ are constructed similarly to type $A$ but instead take place within a regular $n$-gon with one of the triangulation vertices sitting in the interior of the $n$-gon. See Figure 5.8 of Introduction to Cluster Algebras. Chapters 4-5. We can also add the arrows between the frozen vertices of the quiver without changing the cluster algebra to get a balanced quiver. The resultant quiver will be restricted mutation-finite and the cluster variables generated through sequences of restricted mutations give rise to the type $D$ cluster algebras (with coefficients).