14 problems

Let $s$ be a positive integer, let $\Phi$ be a root system of rank $n$ with Coxeter number $h$ and exponents $e_1,\ldots,e_n$, and let $X$ be the set of facets of the generalized c…

Let $\Phi$ be a crystallographic root system of rank $\ell$, let $m\geq 1$, and let $\Delta^m_+(\Phi)$ be the positive part of the generalized cluster complex, with $N^+(\Phi,m-1)$…

Let $\Phi$ be an irreducible crystallographic root system of rank $\ell$, let $m\geq 1$, and let ${\mathcal A}^m_\Phi$ be its generalized Catalan hyperplane arrangement. Let…

Let $\Phi$ be a finite crystallographic root system of rank $n$, and let $m$ be a positive integer. The generalized cluster complex $\Delta^m(\Phi)$ is the simplicial complex assoc…

Cluster-complex characterization conjecture. The following statements are equivalent: (i) up to commutations of consecutive commuting letters, $Q=c\,\mathrm{w_\circ}(c)$ for some C…

Let $\mathcal{A}(B)$ be a cluster algebra, let $x,x'$ be cluster variables, and let $(x\parallel x')$ denote their compatibility degree. Compatibility-degree exchangeability conjec…

Let $(W,S)$ be a finite Coxeter system, let $J\subseteq S$, and let $c\in W$ be a Coxeter element. The $(W^J,c)$-cluster complex is the subword complex … and its facets carry the f…

Generalized cluster complex conjecture. Every generalized cluster complex $\mathcal C(m,\Phi)$ is pure-EKR.

Let ${\mathcal A}$ be a linear LP algebra of rank $r$. Simple-polytope conjecture. There exists a convex simple polytope $P({\mathcal A})$ of dimension $r$ such that its vertices a…

Multi-cluster polytopality conjecture. The multi-cluster complex $\Delta_c^k(W)$ is the boundary complex of a simplicial polytope.

Facet-maximality conjecture. The number of facets of $\Delta(Q,w_\circ)$ is at most the number of facets of $\Delta_c^k(W)$. Moreover, if the two numbers are equal, then $Q$ has th…

SIN-property characterization conjecture. The subword complex $\Delta(Q,\pi)$ is isomorphic to a multi-cluster complex if and only if $Q$ has the SIN-property and

Minimal non-face conjecture. Every minimal non-face of $\Delta_c^k(W)$ has cardinality $k+1$.

Let $\Phi$ be a finite root system of rank $n$, let $m$ be a positive integer, and let $F^m_\Phi(x,y)$ be the $F$-triangle of the generalised cluster complex. Let $NC^m(\Phi)$ be t…