6 problems

Let $\Phi$ be a finite root system of rank $n$. Define … where $h_{k,\ell}$ counts antichains of cardinality $k$ containing $\ell$ simple roots. Let…

F–M triangle transformation conjecture. The two generating functions are related by the invertible transformation

Cluster complex face-lattice conjecture. This poset is the face lattice of a simple $n$-dimensional convex polytope $P(\Phi)$.

Cluster fan polytopality conjecture. The simplicial fan generated by the clusters is the normal fan of a simple $n$-dimensional convex polytope $P(\Phi)$.

Let $\mathcal{A}$ be a doubly extended cluster algebra of rank $n+2$. Its exchange complex is the complex associated with the exchange structure, and its doubly extended associahed…

An affine associahedron is the dual of the quotient cluster complex associated with an affine cluster algebra, and its dimension is determined by the rank of that algebra. The affi…