# Non-crystallographic cluster algebras

## Background

Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{x},B)$ consists of a cluster $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ of certain elements (which are called cluster variables) and a skew-symmetrisable integer $n\times n$ matrix $B$. By definition, the cluster algebra is generated by all cluster variables in all seeds that are obtained from a given initial seed by a sequence of mutations.

A skew-symmetrisable integer $2\times 2$ matrix has the form $$B=\pm\begin{pmatrix}0&a\\\\-b&0\end{pmatrix}$$ for some natural numbers $a,b\geq 1$. Note that the two possible choices yield isomorphic cluster algebras that we denote by $\mathcal{A}(a,b)$. We can parametrise the cluster variables in $\mathcal{A}(a,b)$ by the set of integers, so that we obtain cluster variables $x_i$, with $i\in\mathbb{Z}$, and clusters $(x_{i-1},x_i)$, with $i\in\mathbb{Z}$. The equation \begin{align*} x_{i-1}x_{i+1}=\begin{cases}x_i^a+1,& \textrm{if } i \textrm{ is even}, \\\\ x_i^b+1,&\textrm{if } i \textrm{ is odd,}\end{cases} \end{align*} describes the mutation from the cluster $(x_{i-1},x_i)$ to the cluster $(x_i,x_{i+1})$.

There are two kinds of cluster algebras: cluster algebras of finite type and cluster algebras of infinite type. We declare a cluster algebra to be of finite type if it admits only finitely many cluster variables. Fomin and Zelevinsky have furthermore classified cluster algebras of finite type by finite type root systems. We therefore see that (coefficient-free) cluster algebras (without frozen variables) are in bijection with Dynkin diagrams of type $A_n (n \geq 1)$, $B_n (n\geq 2)$, $C_n (n\geq 3)$, $D_n (n\geq 4)$, $E_n (n=6,7,8)$, $F_4$, and $G_2$. The classification theorem implies that the cluster algebra $\mathcal{A}(a,b)$ from above is of finite type if and only if $ab<4$. In this case, the sequence is $(x_i)_{i\in\mathbb{Z}}$ is periodic.

Dynkin diagrams satisfy a crystallographic condition. In the context of cluster algebras, the crystallographic condition yields integer entries in the $B$-matrix. On the other hand, finite Coxeter groups are in bijection with Coxeter-Dynkin diagrams. Coxeter-Dynkin diagrams do not necessarily satisfy a crystallographic condition. Examples of non-crystallographic Coxeter groups are dihedral groups (with Coxeter-Dynkin diagram $I_2(m)$ with $m=5$ or $m\geq 7$) and the symmetry group of the icosahedron (with Coxeter-Dynkin diagram $H_3$).

## Recurrences associated with non-crystallographic root systems

The Dynkin diagrams associated with the cluster algebras $\mathcal{A}(1,1)$, $\mathcal{A}(2,1)$ and $\mathcal{A}(3,1)$ are $A_2$, $B_2$ and $G_2$, respectively. Viewed as Coxeter-Dynkin diagrams, the corresponding Coxeter groups are the dihedral symmetry groups of the equilateral triangle, the square and the regular hexagon. More generally, the Coxeter-Dynkin diagram associated with the dihedral group of symmetries of the regular $m$-gon, for some $m\geq 3$, consists of two vertices that are joined by an edge of weight $a=4\cos^2(\frac{\pi}{m})$. Putting $b=1$, we can define a sequence $(x_i)_{i\in\mathbb{Z}}$ as above.

In the case $m=5$ (where we have $a=4\cos^2(\frac{\pi}{5})=\frac12(3+\sqrt{5})\approx2.618033988$) my computer algebra system has randomly choosen various starting values $x_1$ and $x_2$ from the interval $(0,1)$ and computed the first few terms numerically. It turns out that after 14 steps, we always get close to our starting values, as the following example illustrates:

0.9449133500
0.2109364289
1.076295668
9.843229446
370.8734823
37.77962145
36.31787856
0.9877779910
0.05419694189
1.067240769
40.32970628
38.72575663
356.3222202
9.226991316
0.9462529267
0.2109303955

The same phenomenon also holds for other values of $m$, and the number of steps seems to be either $m+2$ or $2(m+2)$.

Questions:

1. What's going on? I have computed the first terms in the sequence for $m=5$ in exact form, and it seems unlikely to me that the sequence is periodic on the nose. (Although I might have to try harder and deviations in the experiments come from computation errors.)
2. Fomin and Reading and other authors have generalised the Lie theoretic combinatorics of cluster algebras to general root systems. Have some authors also generalised cluster algebras to general root systems? For example, does a sophisticated version of the Laurent phenomenon hold in this case?
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This has been asked a long time ago (2005), see page 13 of aimath.org/WWN/braidgroups/braidgroups.pdf – F. C. May 25 '13 at 7:13
Many thanks for the reference! – Philipp Lampe May 27 '13 at 7:04
There is a "non-crystallographic folding" (A_4 to the dihedral group of order 10, D_6 to H_3, E_8 to H_4, and other A's to other odd dihedral groups, and D's to even dihedral groups). I have hoped that something like this should be involved, but I don't see how it would be. – Hugh Thomas Jun 1 '13 at 3:23