Fomin–Zelevinsky's conjectures on cluster algebras

Let $\mathbb{T}_n$ be the $n$-regular tree indexing seeds, let $B$ be an exchange matrix at a base vertex $t_0$, and let $F_{j;t}^{B;t_0}$ and $\mathbf{g}_{j;t}^{B;t_0}$ denote the associated $F$-polynomials and $g$-vectors. For adjacent vertices $t_0$ and $t_1$ related by mutation in direction $k$, write $B'=\mu_k(B)$.

Fomin–Zelevinsky's conjectures. The following statements hold:

(i) Each polynomial $F_{j;t}^{B;t_0}$ has constant term $1$.

(ii) Each polynomial $F_{j;t}^{B;t_0}$ has a unique monomial of maximal degree; this monomial has coefficient $1$ and is divisible by all the other occurring monomials.

(iii) For every $t\in\mathbb{T}_n$, the vectors $\mathbf{g}_{1;t}^{B;t_0},\dots,\mathbf{g}_{n;t}^{B;t_0}$ are sign-coherent: for each $i=1,\dots,n$, their $i$th components are either all nonnegative or all nonpositive.

(iv) For every $t\in\mathbb{T}_n$, the vectors $\mathbf{g}_{1;t}^{B;t_0},\dots,\mathbf{g}_{n;t}^{B;t_0}$ form a $\mathbb{Z}$-basis of the lattice $\mathbb{Z}^n$.

(v) For any $t\in\mathbb{T}_n$ and $j=1,\dots,n$, if $\mathbf{g}_{j;t}^{B;t_0}=(g_1,\dots,g_n)$ and $\mathbf{g}_{j;t}^{B';t_1}=(g'_1,\dots,g'_n)$, then

$$g'_i=\begin{cases} -g_k & \text{if }i=k,\\ g_i+[b_{ik}]_+g_k-b_{ik}\min(g_k,0) & \text{if }i\ne k. \end{cases}$$

These are the conjectures reproduced from earlier work and concern the constant terms and maximal monomials of $F$-polynomials, sign-coherence and basis properties of $g$-vectors, and their mutation rule. In the source they are presented in a section explaining consequences of sign-coherence and matrix identities; the supplied text does not establish their resolution, so their status is left open.