81 problems

Let $(W,S)$ be a finite Coxeter system equipped with a canonical filtration … This filtration gives a reflection-coloring map $\alpha:T\to\{1,\ldots,r\}$ by assigning to each refle…

Let $(W,S)$ be a finite Coxeter system. For a subset $J\subseteq S$, write $W_J$ for the corresponding standard parabolic subgroup. For pairs in finite Coxeter systems, let…

Let $(W_1,S_1)$ and $(W_2,S_2)$ be Coxeter systems, and let $\sigma1,\omega1 easymp W1$ and $\sigma2,\omega2 easymp W2$. For each pair, let $G[sigmai,omegai]$ denote the directed Bruha…

The conjectured formula for $\mathfrak{D}_w$. For special $w$,

Let $W$ be a Coxeter group, let $u,v\in W$ satisfy $u\leq v$ in the Bruhat order, and let $\Phi_{[u,v]}$ denote the cd-index of the interval $[u,v]$. Bruhat-interval non-negativity…

Let $[u,v]$ be an interval in the Bruhat order on a Coxeter group, with $l(u)=k$ and $l(v)=d+k+1$. Let $\Phi_{[u,v]}$ denote its cd-index, and let a dual stacked polytope be a poly…

Subdivision conjecture. The moment polyhedron $\Phi_T(G/B)$ has a locally finite subdivision into Bruhat interval polytopes

Let $\mathfrak A_n$ denote the alternating group, let $f_n$ be the function defined by the paper's Algorithm 1, and let $\leq$ denote the Bruhat order on $\mathfrak A_n$. Bruhat-or…

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 $W$ be a Coxeter group of type $A$, let $u,v\in W$, and let $M$ be a special matching of the Bruhat interval $[u,v]$. For $x,y\in [u,v]$ with $x\le y$, write $R_{x,y}(q)$ for t…

Let $W$ be a Coxeter group of type $A$, and fix a Bruhat interval $[u,v]$. Consider the equivalence relation on the amazing hypercube decompositions of $[u,v]$ generated by … where…

Double-shortcut triviality conjecture. The above equivalence relation is trivial, meaning that it has only one equivalence class. This is a conjecture of Ellenberg and McNamara and…

Ranked-poset conjecture for tilted Bruhat order. The poset $(S_n,\leq_{\mathbf{a}})$ is ranked.

Simply laced Bruhat preclosure conjecture. The Bruhat preclosure is idempotent:

Billey–Fan–Losonczy's conjecture. The interval $[e,w]$ admits a Lehmer code.

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…

Graph Combinatorial Invariance conjecture. The $\widetilde{R}$-polynomial $\widetilde{R}_{u,v}(q)$ depends only on the isomorphism type of the graph $B([u,v])$.

Monical–Tokcan–Yong's conjecture. If $\beta\in\widehat{\mathcal{P}}_{\alpha}$ and $\beta\in\mathbb{Z}^n_{\geq 0}$, then $\beta\leq_{S}\alpha$.

Weak combinatorial invariance conjecture. If $x,x',y,y'\in W_+$ satisfy that $[x,y]$ and $[x',y']$ are full and poset-isomorphic, and there exists $g_0\in G$ with…

Stabilization conjecture. For any dominant coweight $\lambda\in\Lambda^\vee$ and dominant $x,y\in W$, there exists an integer $N_0=N_0(x,y,\lambda)$ such that for every $N\geq N_0$…

Combinatorial Invariance Conjecture. If two Bruhat intervals, potentially from different Coxeter groups, are isomorphic as posets, then

Type-$A_n$ polynomiality conjecture. The cardinality of the lower Bruhat interval is a polynomial in the coordinates $\lambda_1,\dots,\lambda_n$ of degree $n$:

Paper Boat zone conjecture. For every affine Weyl group there is a partition $\mathcal{Z}$ of $(\Lambda^\vee)^+$ with at most $3^n$ parts such that, for every $Z\in\mathcal{Z}$ and…

The Ancestor Conjecture. Every finite Coxeter group has the ancestor property.

Let $[u,v]\subset S_n$ be a Bruhat interval, let $I$ be a hypercube decomposition of $[u,v]$, and let $\ell(u,v)$ denote the length of the interval. The polynomials $N_{u,v,I}$ and…