Quiver Mutation Explorer

Please consult the new web page for updates.

• General overview
• Download
• How to?
  • Install
  • Compute mutation sizes
  • Detect mutation-equivalent quivers
• References
• Some results

Quiver Mutation Explorer (QME) is a C++ program designed to compute efficiently the number of isoclasses in the the mutation class of a quiver. The aim of this program is to give an efficient complement to Bernhard Keller's Java applet for computing mutation classes. The program is already stable but should be improved regularly, keep an eye on this page.

Let Q be a (finite) quiver (without loops and 2-cycles), we denote by Mut(Q) the set of all isoclasses of quivers mutation-equivalent to Q in the sense of Fomin and Zelevinsky [FZ]. If Mut(Q) is a finite set, we denote by m(Q) its cardinality. The aim of this QME is to decide whether Mut(Q) is finite and if it is, to compute the number m(Q).

It already turned out that these for quivers Q of Dynkin type A the number m(Q) was related to the Brown sequence (see [T]). A list of results is avalaible in [D] and shall be presented soon on this webpage.

The main idea of this program was to use Mc Kay's algorithm for quiver isomorphisms [M]. Also, a parallel version is currently under development.

The program also allows to decide whether two (mutation-finite) quivers are mutation-equivalent and if they are, it returns one of the shortest mutation sequences from the first to the second.

• Sequential version (QME lite)
• Version 1.2.The sequential version of the program is available here (tar.gz archive).
• Parallel version
• available soon...

• Download the file here.
• Unpack the file to a local installation directory.
• Go to the installation directory.
• Run make to compile the program.

The easiest way is to use Bernhard Keller's Java applet in order to create your quiver:

• Export a quiver from Bernhard Keller's Java applet as a *.qmu file (e.g. quiver.qmu).
• Go to the directory of QME.
• Run ./main quiver.qmu.
• You will get a message if the mutation class is infinite, otherwise the program computes until it returns the size of the mutation class up to isomorphism.

An alternative way is to use a text file. The text file shall contain the incidence (antisymmetric) matrix of the quiver in the standard Maple format (e.g. [[a,b],[c,d]] for a 2x2 matrix).

• Create your text file (e.g. quiver.txt).
• Go to the directory of QME.
• Run ./main quiver.txt.
• You will get a message if the mutation class is infinite, otherwise the program computes until it returns the size of the mutation class up to isomorphism.

For Dynkin quivers, you have an inner way.

• Choose your Dynkin type (A_n, D_n, E_n) (e.g. A_7)
• Go to the directory of QME.
• Run ./main A 7. (replace A by D for type D and E for type E)
• You will get a message if the mutation class is infinite, otherwise the program computes until it returns the size of the mutation class up to isomorphism.

For affine quivers of types \tilde D and \tilde E, you also have an inner way.

• Choose your affine type (\tilde D_n, \tilde E_n) (e.g. \tilde E_7)
• Go to the directory of QME.
• Run ./main DTILDE 7. (replace DTILDE by ETILDE for type \tilde E).
• You will get a message if the mutation class is infinite, otherwise the program computes until it returns the size of the mutation class up to isomorphism.

For affine quivers of type \tilde A, you also have an inner way.

• The affine type of \tilde A is determined by two integers r,s>0 corresponding to the number of arrows going clockwise and anticlockwise respectively. Choose thus an affine type \tilde A_{r,s} (e.g. \tilde A_{2,3})
• Go to the directory of QME.
• Run ./main ATILDE 2 3.
• You will get a message if the mutation class is infinite, otherwise the program computes until it returns the size of the mutation class up to isomorphism.

[D] G. Dupont, Algèbres amassées affines, Ph.D. thesis in preparation.

[FZ] S. Fomin and A. Zelevinsky, Cluster Algebras : Foundations, Journal of the AMS, 15:497--529, 2002.

[K] B. Keller, Quiver mutation in java, available here.

[M] B.D. Mc Kay, Practical Graph Isomorphism, Congressum Numerantium, 30:45--87, 1981.

[T] H.A. Torkildsen, Counting cluster-tilted algebras of type $A_n$, Arxiv preprint.

See also:

The cluster algebras portal by S. Fomin.

Quiver Mutation in Java by B. Keller