8 problems

Let $Q$ be a word in the simple generators of a finite Coxeter group, let $\omega$ be a Coxeter-group element, and let $[e,\omega]$ be the corresponding interval in the weak order.…

Let $Q$ be a word and let $\omega$ be an element of a Coxeter group. For the non-empty subword complex $\mathcal{SC}(Q,\omega)$, let $[e,\omega]$ be the corresponding weak-order in…

Let $\mathcal{S}(Q,\pi)$ be a spherical subword complex of infinite type. Strong vertex-decomposability conjecture. The complex $\mathcal{S}(Q,\pi)$ is strongly vertex decomposable…

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 $f_{(i_1,\dots,i_d)}$ be one of the maps considered in the paper, let $p\in Y_w^o$, and let $\Delta((i_1,\dots,i_d),w)$ denote the corresponding subword complex. Consider the c…

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 $k+1$.