Plant a seed and grow a cluster

As the title says start with a seed.

It is a n tuple of variables.

Also give an n by n skew symmetric matrix that is called the exchange matrix.

For each i between 1 and n, there is a mutation.

The action on the exchange matrix is given as follows.

View the exchange matrix as the adjacency matrix for a quiver. Now flip the arrows that touch vertex i. Also create shortcuts whenever you flip two arrows that are lined up. Cancel any loops you might have created and then give back the adjacency matrix.

The action on the seed is as follows.

Leave everything except the ith in the tuple alone. Call that one y and replace it with w. The other elements in the tuple are t's.

Now we have an n valence tree where each vertex has a seed and an exchange matrix and traversing an edge tells you to mutate in that direction.

A cluster algebra is called finite type if you end up only getting a finite number of seeds. It has been shown that the classification of finite type cluster algebras is the same as the classification of semisimple Lie algebras and finite root systems.