12 problems

Let $W$ be a finite reflection group containing $-1$, let $G$ be its Dynkin diagram, and let $\bar G$ be an orientation. Form the Coxeter element $c$ from the oriented diagram, ord…

Let $C(\bar G)$ be a Cambrian lattice associated with an oriented Dynkin diagram $\bar G$ of a finite reflection group. A lattice is trim if it has a left modular maximal chain and…

Let $W$ be a finite crystallographic Coxeter group with root system $\Phi$, simple roots $\alpha_i$, fundamental weights $\omega_i$, and a bipartition $G=I_+\cup I_-$ of its Coxete…

Let $W$ be a finite Coxeter group and let $\stackrel{\to}{G}$ be an orientation of its Coxeter diagram. The Cambrian fan … is the fan whose maximal cones are the unions of reflecti…

Ordovician lattice structure conjecture. The following hold:

Finite Coxeter Cambrian mutation conjecture. The Cambrian lattice $\operatorname{Camb}(Q)$ is mutable.

The $m$-Tamari lattice and the linear type $A$ $m$-Cambrian lattice are lattices associated with the same parameter $m$. Stump's interval-count conjecture. The $m$-Tamari lattice a…

Extremal image-size conjecture. For every Coxeter element $c$ of $A_n$, we have

Sublattice conjecture for c-Cambrian order interval posets. For every Coxeter element $c$, the set $\mathsf{COIP}(c)$ induces a sublattice of the weak order on $\mathcal{P}(\Phi)$.

Sublattice conjecture for c-Cambrian order element posets. For every Coxeter element $c$, the set $\mathsf{COEP}(c)$ induces a sublattice of the weak order on $\mathcal{P}(\Phi)$.

Snake-decomposition characterization. A $\Phi$-poset $\mathrel{\mathop{\rel}}\in\mathcal{P}(\Phi)$ is in $\mathsf{COEP}(c)$ if and only if it is in $\mathsf{COIP}(c)$ and every roo…

Forbidden-subgraph conjecture. The list in Proposition 1.6 is exhaustive: if $\Gamma_{\gamma}(W)$ contains none of the listed induced subgraphs, then $\mathcal{B}_{\gamma}$ is distrib…