44 problems

For a Coxeter system $(W,S)$, the extended weak order is the partial order introduced by Dyer that generalizes the weak order to possibly infinite Coxeter groups. Dyer's lattice co…

Ceballos–Pons geometric realization conjecture. The Hasse diagram of the ${\boldsymbol{s}}$-weak order can be realized as an orientation of the skeleton of a polyhedral subdivision…

Hart–Rowley ancestor property conjecture. Let $(W,S)$ be a finite Coxeter system, with $|W|<\infty$. If $w\in W$ and $w\neq 1$, then $w$ has the ancestor property.

Let $W$ be a finite Coxeter group, and let $u,v\in W$. For $w\in W$, write $T_L(w)$ for its left-reflection set. Given $A\subseteq T$, define its Bruhat preclosure by … Here an $A$…

Let $A,A'\in\mathrm{ASM}_n$ with $A'\leq_{\mathsf{L}} A$. Let $\Delta_{\mathsf{L}}(A',A)$ be the order complex of the open interval $[A',A]_{\mathsf{L}}\setminus\{A',A\}$. For a se…

Dyer's conjecture. The Bruhat preclosure of the union of the inversion sets of $u$ and $v$ is the inversion set of their join:

Let $(W,S)$ be a finite Coxeter system, let $T$ be its set of reflections, and let $\ell$ be the Coxeter length function. For $w\in W$, define its left-reflection set by … For…

Let $W$ be a rank-$3$ Coxeter group with positive root system $\Phi^+$. A subset $R\subseteq\Phi^+$ is weakly separable if there is a hyperplane $H$ such that $R\setminus H$ and…

Weak-order chain-length conjecture. The weak-order saturated chains from the identity to $A$ are all of the same length if and only if $X_B$ is Cohen--Macaulay for every…

Ultra-log-concavity conjecture. For every $\pi\in\mathfrak{S}_n$, $Q_W^\pi$ is ultra-log-concave.

Completeness conjecture. It is conjectured that $\mathsf{Dyer}(W)$ is a complete lattice.

Profinite semidistributivity conjecture. If $W$ is any Coxeter group, then $\mathsf{Dyer}(W)$ is a profinite semidistributive lattice.

Let $(W,S)$ be a Coxeter system with length function $\ell:W\to\mathbb N$. For $w\in W$, let $T(w)=\{t\in T\mid \ell(tw)<\ell(t)\}$ be its inversion set, where…

Let $\mathrm{Int}(n)$ be the set of nonempty left weak Bruhat intervals in $S_n$. For $I,J\in\mathrm{Int}(n)$, write $I\mathrel{\Deq}J$ if there exists a descent-preserving poset i…

Cellular regularity conjecture. The following conditions are equivalent: the quotient $W_{\boldsymbol{s}}/\equiv$ is cellularly regular; every arc minimal in subarc order among the…

Endpoint invariance conjecture for s-Cambrian congruences. For fixed ${\boldsymbol{s}}$, the following depend only on the endpoints of $\alpha$: the cardinality of the $\alpha$-Cam…

Maximality conjecture. The element $\widetilde{w}_{b}$ is maximal in the weak order on $\widetilde{W}/W$ among all dominant elements

Thiel's inversion-set conjecture. The inversion set of $\widetilde{w}_{b}$ contains the inversion sets of all other affine elements corresponding to elements of…

Let $s$ be a weak composition. For a pure interval $[T,T+A]$ of the $s$-weak order, consider the poset of pure intervals contained in it and the restriction of the $s$-weak order t…

Let $Q$ be a sorting and alternating word in a Coxeter group with longest element $\omega_\circ$, let $\mathsf{Brick}(Q,\omega_\circ)$ be the associated brick polytope, and let…

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 $w$ be a permutation in the symmetric group $S_n$. An Eeta-win descent conjecture asserted that if $w$ is an Eeta win in the weak order on $S_n$, then … This conjecture was dis…

Let $S_n$ be the symmetric group equipped with its weak order, and let $\mathcal E(S_n)$ denote the expected number of steps for the Ungarian Markov chain to reach the bottom eleme…

Let the weak order on the symmetric group be quotiented by a lattice congruence, and represent its intervals by semi-crossing arc bidiagrams. An interval is inclusion minimal if it…

Let an $s$-associahedron be a polytopal complex associated with the $s$-Tamari lattice, whose Hasse diagram is the $1$-skeleton of a polytopal subdivision of a polytope. Let an $s$…