18 problems

Interior dual block complex conjecture. The interior dual block complexes of subword complexes should be homeomorphic to the fibers of the Chevalley exponentiation maps to totally…

A spherical subword complex is a simplicial complex associated with a word and a Coxeter group whose relevant subword complex is spherical; its dual graph has one vertex for each f…

Acyclic-facet lattice quotient conjecture. Acyclic triangular pipe dreams define quotients of the weak order for a large family of subword complexes on all Coxeter groups.

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 $A,B$ be spherical subword complexes of infinite type, and let $FR(A)$ and $FR(B)$ denote their facet-ridge graphs. Spherical subword-complex reconstruction conjecture. Every i…

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…

Let $\mathcal{S}(Q,\pi)$ be a spherical subword complex of infinite type, and let $\{i\}\in\mathcal{S}(Q,\pi)$ be a vertex. The infinite-type deletion conjecture. The deletion … is…

Pilaud–Stump's conjecture. Important properties of brick polytopes for spherical root-independent subword complexes should hold for all spherical subword complexes.

Consider the simplicial complexes corresponding to arbitrary elements in the poset-like structures formed by Demazure weaves, as in the comparison with subword complexes. Demazure-…

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…

Let $U$) be the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over $\mathbb{R}$. Let $f_{(i_1,\dots,i_d)}$ be one of the…

Let $W$ be a finite Coxeter group with longest element $w_\circ$, let $c$ be a Coxeter element, let $N:=\ell(w_\circ)$, and let $m$ be a positive integer. Consider the $m$-eralized…

Let $W$ be a finite Coxeter group, let ${\rm Q}$ be a realizing word, and let $\mathcal{S}({\rm Q})$ be its subword complex. Define the increasing flip order on the facets of…

Let $W$ be a finite Coxeter group, let ${\rm Q}$ be a word whose subword complex is the spherical subword complex $\mathcal{S}({\rm Q})$, and let the increasing flip order be the t…

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 $k+1$.