10 problems

For an acyclic exchange matrix $B$ and a Coxeter element $c$, the source constructs a $c$-Cambrian fan and a collection of cones of cluster $\mathbf g$-vectors. Cambrian fan and…

Let $W$ be a finite Coxeter group, let $c$ be a Coxeter element, and let a $c$-cluster $C$ have upper roots among its roots. Associate to $C$ the intersection of the hyperplanes or…

Let $W$ be a finite Coxeter group of rank $n$, let $c$ be a Coxeter element, and let $(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_{n-k})$ be a $c$-cluster, where the $\alpha_i$…

Let $W$ be a finite Coxeter group with simple roots $\alpha_s$, fundamental weights $\omega_s$, and let $c$ be a Coxeter element. For a cluster variable $x^c(\alpha)$ corresponding…

Let $R_u$ and $R_v$ be regions of the Coxeter arrangement, and let the transformed cluster fan be the fan whose maximal cones are under consideration. Write $u\equiv v$ when $u$ an…

Simpliciality conjecture. The fan $\operatorname{biCamb}(W,c)$ is simplicial. Equivalently, its dual polytope is simple.

Let $B$ be an exchange matrix, and let Theorem denote the source's theorem identifying the combinatorial Cambrian data with denominator vectors of cluster variables. Generalized Ca…

Let $W$ be a Coxeter group, let $c$ be a Coxeter element, and let $s$ be initial in $c$. For a collection ${\mathcal F}$ of cones, define ${\mathcal F}_{\ge s}$ by intersecting eve…

Let $W$ be a Coxeter group with simple generators $S$, let $c$ be a Coxeter element, let $v$ be a $c$-sortable element, let $\mathrm{Cone}_c(v)$ be its $c$-Cambrian cone, and let…

Reading's conjecture. Any $c$-Cambrian fan $\mathcal F_c$ is the normal fan of a simple convex polytope.