These java applets implement quiver mutation (and cluster
mutation) as invented in joint work by S. Fomin
Zelevinsky in 2000. Quiver mutation is related to a large number
of subjects in mathematics and to Seiberg duality
in physics, cf. for example section 6, page 21 of this article.
A quiver is an oriented graph: it has vertices (nodes) and
arrows between the vertices. To mutate with respect to a vertex, click
To adjust the picture after mutation, drag the vertices.
Note that edges may lie one over the other.
This example cannot be transformed to
a quiver without oriented cycles. This can be proved either
using representation theory (cf. section 2.3 of this preprint)
or using brute force: Indeed, it turns out that the mutation
class of this quiver is finite. It contains 5739 quivers up
to isomorphism and can be conveniently computed using the
mutation applet below.
One then checks that each of these quivers
contains at least one oriented cycle.
The quiver can be transformed into
a quiver containing a double arrow in eight mutations.
It cannot be transformed to a quiver containing arrows
of multiplicity 3 or greater (indeed, it is not hard
to check that this would imply that its mutation class
is infinite). This quiver is associated with the
elliptic root system of doubly extended type E8.
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Win the prize
A prize of 20 euros is yours if you are the first to transform the
following quiver into one without oriented cycles in strictly less
than 8 mutations.
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This applet opens a window on your
computer. You can resize the window
and experiment with quivers of your own invention. Make them by
adding/deleting nodes and arrows in the given examples (delete arrows
by adding arrows in the opposite direction!). In the cluster menu you
can activate the cluster variables and observe how they change under
mutations. This part of the applet is based on the Java ring library
from ring.perisic.com. Here
are variants of the same application:
In contrast to the applet, the
standalone application allows you to save, open and print quivers.
Moreover, as of October 2006, the standalone application and the
applet can compute the complete mutation class of a given quiver (if
it is finite).
To use the standalone version, you need to have a Java runtime
environment (JRE) installed. You run it
(without web access) by double-clicking it or via the command line:
java -jar MutationApp.jar
If you wish to use more than the 64 Mb of default memory, you can
java -Xmx200m -jar MutationApp.jar
to allocate 200 Mb to the application (for example for computing large