Dugas’ Restricted Quiver Mutation Conjecture
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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 high concentrations of directed $3$-cycles (todo: elaborate) 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.
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ of multiplicity $2$
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.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $3$ | $1$ |
| $4$ | $2$ |
| $5$ | $3$ |
| $6$ | $5$ |
| $7$ | $5$ |
| $8$ | $8$ |
| $9$ | $10$ |
| $10$ | $15$ |
| $11$ | $19$ |
| $12$ | $31$ |
| $13$ | $41$ |
| $14$ | $64$ |
| $15$ | $94$ |
| $16$ | $143$ |
| $17$ | $211$ |
| $18$ | $329$ |
| $19$ | $493$ |
| $20$ | $766$ |
| $21$ | $1170$ |
| $22$ | $1811$ |
| $23$ | $2787$ |
| $24$ | $4341$ |
| $25$ | $6713$ |
| $26$ | $10462$ |
| $27$ | $16274$ |
| $28$ | $25415$ |
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ and $j \to j+2\pmod n$.
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.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $5$ | $6$ |
| $6$ | $6$ |
| $7$ | $4$ |
| $8$ | $4$ |
| $9$ | $5$ |
| $10$ | $6$ |
| $11$ | $7$ |
| $12$ | $11$ |
| $13$ | $12$ |
| $14$ | $17$ |
| $15$ | $23$ |
| $16$ | $31$ |
| $17$ | $40$ |
| $18$ | $58$ |
| $19$ | $76$ |
| $20$ | $109$ |
| $21$ | $149$ |
| $22$ | $208$ |
| $23$ | $287$ |
| $24$ | $410$ |
| $25$ | $567$ |
| $26$ | $803$ |
| $27$ | $1128$ |
| $28$ | $1599$ |
| $29$ | $2249$ |
| $30$ | $3201$ |
Vertices = $[n]$ and arrows $i \to i + 1 \pmod n$ and $j \to j - 2 \pmod n$ (todo)
Triangular quivers (todo)
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 the cluster variables can be used to construct every Markov triple (of which there are infinitely many), there must be infinitely many cluster variables. Todo: link felikson-shapiro-tumarkin exponential growth thing in section 4 of https://arxiv.org/pdf/1203.5558.