22 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 $W$ 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 $(W,S)$ 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 $(W,S)$ be a Coxeter system with associated root system $\Phi$, positive roots $\Phi^+$, and reflection $s_\alpha$ corresponding to each $\alpha\in\Phi^+$. Let…

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.

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…

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 $C$ be a finite Coxeter group, equipped with its weak order. A ranked poset is strongly Sperner if, for every positive integer $r$, the maximum size of a union of $r$ antichain…

Let $\lambda$ be a rectangular staircase of the form … Let $w\in\mathfrak{S}_n$ be a vexillary permutation of shape $\lambda$, meaning that its Rothe diagram can be transformed to…

Homotopy-type conjecture. The complex $\Delta(T',T)$ is contractible unless

Let $(W,S)$ be a finite Coxeter system, let $J\subseteq S$, and let $c\in W$ be a Coxeter element. The $(W^J,c)$-cluster complex is the subword complex … and its facets carry the f…

Let $(W,S)$ be a finite Coxeter system, let $J\subseteq S$, and let $c\in W$ be a Coxeter element. An element of $W^J$ is $(W^J,c)$-aligned if, whenever…

Let $W_n$ be the weak order on the symmetric group $S_n$, ranked by inversion number, and let $U$ be the order-raising operator defined by … For $0\leq k<\frac12\binom n2$, let…

Let $\widetilde{W}$ be a simply-laced affine Weyl group with Coxeter number $h$, let $b$ be coprime to $h$, and let $\mathcal{S}_\Phi(b)$ be the relevant Sommers region. Let…

Let $(W,S)$ be a Coxeter system. Let $L_n(W)$ denote the set of $n$-low elements and let $\Lambda_n(W)$ denote the corresponding set of admissible collections of $n$-small roots; f…

Let $(W,S)$ be a Coxeter system. For $n\in\mathbb N$, define the dominance depth of a positive root as the number of positive roots it strictly dominates, let the $n$-small roots b…

Let $(W,S)$ be a Coxeter system with positive root system $\Phi^{{\scriptscriptstyle +}}$, and let $\mathcal B$ be the poset of biclosed subsets of $\Phi^{{\scriptscriptstyle +}}$,…

Inner tableau translation conjecture. If $S\lessdot_{weak}T$ is a covering relation in $SYT_n^R$ and $R'$ is obtained from $R$ by applying a single dual Knuth relation, then

Property of inner tableau translation. If