16 problems

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

Fix $B^0,t_0,t$, and let $F_{\ell;t}^{B^0;t_0}(y_1,\dots,y_n)$ be the associated $F$-polynomial. Unique maximal-monomial conjecture. Each such polynomial has a unique monomial of m…

Fix an initial exchange matrix $B^0$, an initial vertex $t_0$, and a vertex $t$. Let $F_{\ell;t}^{B^0;t_0}(y_1,\dots,y_n)$ be the associated $F$-polynomial of the cluster variable…

Recursive decomposition inequality. For every relevant index $i$, one has $h_i\leq g_i$.

Let ${\mathcal A}$ be a TSSS cluster algebra with principal coefficients. For a coefficient-free cluster monomial, let its Newton polytope be the convex polytope determined by its…

Let $F_{i;v}$ be the $F$-polynomial associated with the cluster variable $x_{i;v}$, and let $\mathbf{g}(x_{i;v})=(g_1,\dots,g_m)$ be its g-vector. Write $u_1,\dots,u_m$ for tropica…

Let $F_M$ denote the $F$-polynomial associated with a representation, and call its support saturated if every lattice point in the Newton polytope of $F_M$ has a nonzero coefficien…

Let $Q$ be a framed quiver and consider the periodic mutation sequence … Assume that performing the mutations $v_1,\ldots,v_p$ on the base quiver returns it to the original quiver.…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. Polynomiality conjecture for deformed $F$-polynomials. The deformed $F$-polynomial $S_n$ is a polynomial…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. For each mutation index $i$, let $r_i$ denote the associated term and let $M$ be the set of relevant mon…

Let $\Sigma$ be a cluster algebra with principal coefficients, and let $F(x)$ denote the $F$-polynomial of a cluster variable $x$, obtained by evaluating its Laurent expansion at t…

For each vertex $v\in\mathrm{Ex}_0(B)$, let $H^v$ be the associated matrix. A collection of entries is sign-coherent when, in each coordinate, the relevant entries are all nonnegat…

An $F$-polynomial is the polynomial in the initial coefficient variables obtained from a cluster variable by specializing the initial cluster variables to $1$. Maximal-monomial con…

Sign-coherence and constant-term conjectures. (i) Every column of $C'$ is a nonzero vector, and its nonzero components are either all positive or all negative. (ii) Every $F$-polyn…

Fix a positive integer $n$, an $n$-regular tree $\mathbb{T}_n$, a vertex $t_0\in\mathbb{T}_n$, and a skew-symmetrizable integer matrix $B=(b_{i,j})$. For each $t\in\mathbb{T}_n$ an…

Fix a positive integer $n$, an $n$-regular tree $\mathbb{T}_n$, a vertex $t_0\mathbin{\in}\mathbb{T}_n$, and a skew-symmetrizable integer matrix $B=(b_{i,j})$. For each…