5 problems

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 $R_u$ 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…