116 problems

Let $W$ and $W'$ be Coxeter groups, with $u,v\in W$ and $u',v'\in W'$. Write $[u,v]$ and $[u',v']$ for the corresponding Bruhat intervals, and let $P_{u,v}(q)$ and…

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…

Panyushev's conjecture. Given orthogonal subsets $S,S'\subset\Psi(\mathfrak p^{\mathrm u})$, one has

Let $[u,v]$ be a Bruhat interval in a Coxeter group, and let $P_{u,v}(q)$ denote its Kazhdan–Lusztig polynomial. Combinatorial Invariance Conjecture. The polynomial $P_{u,v}(q)$ de…

Brownian-sheet persistence conjecture. The probability that $\pi$ is below $\tau$ in the strong Bruhat order satisfies

Completeness conjecture. The equivalences found in the study, including the zero and non-zero equivalences, form a complete collection of equivalences.

Let $[x,y]$ and $[x',y']$ be intervals in Bruhat orders, and suppose that they are isomorphic as posets. The Kazhdan–Lusztig polynomial associated with an interval is denoted by…

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

For permutations $a,b,v,w$, let $P_{a,b}(q)$ and $P_{v,w}(q)$ denote Kazhdan–Lusztig polynomials, and let $[a,b]$ and $[v,w]$ be the corresponding Bruhat order intervals. Lusztig's…

Let $G$ be a simply-laced group with Weyl group $W$, let $u,v\in W$ satisfy $u\leqslant v$ in the Bruhat order, and let $m_{u,v}=\mathcal{A}_v\psi_u(1)$. Let $Q_{u,v}$ be the inver…

Qualitative sharpness conjecture. For each of the ordinary and weak Bruhat orders, the upper bound on the number of comparable pairs is qualitatively close to the actual number of…

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

Goodearl–Lenagan's conjecture. There exists an explicit order-preserving bijection between

Let $W$ be a Coxeter group and let $v$ be a vertex of its Bruhat graph. A Bruhat path from $v_0=\operatorname{id}$ to $v$ is called $\Psi$-generating when the Woronowicz symmetrisa…

Let $(W,S)$ be a Coxeter system, and let … be the subposet of the Bruhat order induced by the involutions in $W$. For finite $W$, Incitti's results in types $A$, $B$, and $D$ show…

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…

Let the poset of configuration types for a line and two complete flags be ordered by the covering moves (i)--(v), with each cover labelled by its move type. Lexicographic shelling…

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

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…

A CSS code is obtained from a Coxeter group and its Bruhat order, with the code parameters determined by the associated Bruhat-poset construction. Family conjecture. Almost all suc…

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