qme-ng: Quiver Mutation Explorer

Welcome to qme-ng !

QME is a fast quiver manipulation program intended to:

• find the cardinality of the mutation class of a given quiver
• find the maximum length of a green suite starting from a given quiver

In order to do so, it uses various optimizations that are specific to the problem at hand, and which result from a careful study by Grégoire Dupont and Matthieu Pérotin.

QuickJump: Quickstart, Overview, Installation, Documentation, Dependencies, License, Contact.

Quickstart

• Find the mutation class of D₇
  qme-ng --type D --size 7
• Find the length of all the maximum green sequences of the quiver in quiver.qmu
  qme-ng --file quiver.qmu --green
• Try to find one green sequence of length at least 30 for the quiver in quiver.qmu
  qme-ng --file quiver.qmu --green --one 50 --min_depth 30 --max_depth 50

Overview

Quiver mutations were introduced by Fomin and Zelevinsky in the context of cluster algebras. Initially, the mutation was defined for skew-symmetric (actually skew-symmetrizable) matrices over the integers. Using incidence matrices, there is a 1-1 correspondence between skew-symmetric matrices and oriented graphs without 1-cycles or 2-cycles. These graphs are referred to as quivers in our context.

I will not explain precisely what the quiver (or matrix) mutation is. I think the first way to understand it is to try it on examples using Bernhard Keller’s java applet. Then, if you want to know the precise definition, you can consult the historical article by Fomin and Zelevinsky on the subject.

Roughly speaking, given a quiver Q, for any vertex i of Q, we can define a new quiver Q’, called mutation of Q at i, by changing locally around i the quiver Q. Iterating this process, we create a list of quivers which can be finite or infinite. The quiver we started with is called mutation-finite if you only create a finite number of new quivers this way, it is called mutation-infinite otherwise.

The aim of QME is to decide, for a given quiver Q, if Q is mutation-finite or not. Also, when Q is mutation-finite, QME gives the number of non-isomorphic quivers created during the process.

Maximal green sequences are particular sequences of quiver mutations which were introduced by Keller in his article On cluster theory and quantum dilogarithm identities and which appear independently in various areas of mathematics and theoretical physics.

For a purely combinatorial description, we refer to the to the first section of our preprint On Maximal Green Sequences (joint with T. Brüstle).

Regarding the maximal green sequences problem, the aim of QME is to answer the two following questions, given a quiver Q:

1. Does there exist a maximal green sequence for Q ?
2. If yes, how many maximal green sequences of each length are there in the set of maximal green sequences for Q ?

Download & Installation

Ubuntu packages

Packages are available for the Ubuntu distribution. You will not get the best performances from this install mode, yet you may consider it to be more convenient.

• qme-ng_1.1.1-0ubuntu1_i386.deb
• qme-ng_1.1.1-0ubuntu1_amd64.deb

The installation procedure is as follows:

1. Download the Ubuntu package corresponding to your architecture
   
2. Install the required libraries:
   sudo apt-get install libgmpxx4ldbl libboost-program-options1.46.1
3. Install the package
   sudo dpkg -i package.deb

Compilation from the sources

This installation mode guarantees the best performances, as you will obtained tailored binaries for your computer architecture.

1. Get the sources, by clicking on one of the icons on the top of this page or by executing git
   git clone https://github.com/mp-bull/qme-ng.git
2. Install the building dependencies
   sudo apt-get install libgmp-dev libboost-program-options-dev build-essential
3. Go into the source directory and compile the project
   make

You now have a qme-ng executable ready to be used !

Documentation

The main documentation of qme-ng is on the Wiki.

Dependencies/Requirements

• gmplib >= 5.0.0
• boost::program::options >= 1.46

Licence

Mainly BSD 2-Clause License. Incorporates nauty which has its own licence.

Contact & Support

Matthieu Pérotin matthieu.perotin(a)bull.net