268 problems

Let $(W,S)$ be a Coxeter system with positive roots $\Phi^+$. For biclosed subsets $A,B\subseteq\Phi^+$ whose join exists in the biclosed-set order, Dyer's conjecture asserts that…

For every finite Coxeter group $W$ of rank $r$, the $h$-polynomial of the order complex of its noncrossing-partition lattice admits a nonnegative real-rooted symmetric decompositio…

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…

Finite Coxeter Cambrian mutation conjecture. The Cambrian lattice $\operatorname{Camb}(Q)$ is mutable.

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…

Explicit adjoint formulas conjecture. The adjoints satisfy

Weighted Singer Conjecture. The weighted $L^2$-homology vanishes above the middle dimension:

Let $(W,S)$ be a Coxeter system, and let $P_{y,w}$ denote its Kazhdan–Lusztig polynomials for $y,w\in W$, with $\leq$ the Bruhat order. Property B. The polynomials $P_{y,w}$ are de…

Let $(W,S)$ be a Coxeter system, let $I\subseteq S$ be a subset of simple reflections, and let $W_I$ be the parabolic subgroup generated by $I$. Let $\mathfrak{h}$ be a realization…

Dual EL-shellability conjecture. The poset $\widehat{\operatorname{Cox}(\mu)^K}$ is dual EL-shellable.

Quasi-isometry classification conjecture. The groups $W_\Gamma$ and $W_{\Gamma'}$ are quasi-isometric if and only if they are isomorphic.

Let $(W,S)$ and $(W',S')$ be Coxeter systems, with $J\subseteq S$ and $J'\subseteq S'$. For $u,v\in W^J$ and $u',v'\in (W')^{J'}$ with $u\leqslant v$ and $u'\leqslant v'$, write ……

Perron-number conjecture. The growth rate of any Coxeter system of dimension at most $2$ is a Perron number.

Gordon–Martino conjecture. If $W$ is a Coxeter group, then

Let $\mathcal D_n$ be the average number of iterations of the ordinary stack-sorting map $\operatorname{\mathtt{s}}$ needed to send an element of $S_n$ to the identity permutation,…

Strong hull property theorem. The graph $\operatorname{Cay}(W)$ has the strong hull property.

Kazhdan–Lusztig's combinatorial invariance conjecture. The polynomial $P_{u,v}(q)$ depends only on the isomorphism class of the interval $[u,v]$ as a poset.

Let $(W,S)$ be a Coxeter group with weight function $L$, Hecke algebra structure constants $h_{x,y,z}$, Kazhdan–Lusztig $\mathbf{a}$-function, and sets $\mathcal{D}_{\geq N}$ and…

Involution word tableau enumeration conjecture. If $n\geq 3$, then

Let $S_n$ be the symmetric group, let $\mathcal{I}(S_n)$ denote its set of involutions, and for $z\in\mathcal{I}(S_n)$ let $\mathcal{A}(z)$ be the corresponding set of atoms, equip…

Let $W$ be a Coxeter group with reflection set $T$, and let ${\mathrm{Red}_T}(w)$ denote the set of minimal-length expressions of $w$ as a product of reflections. Suppose that … an…

Let $W$ be a finite Coxeter group, let $C$ be a conjugacy class of involutions in $W$, and let $V_C$ be the real vector space with basis $\{a_w\mid w\in C\}$ equipped with the $W$-…

Let $(W,S)$ be an irreducible finite Coxeter system with Coxeter number $h$, let $\vec{c}$ be a Coxeter word in $S$, let $p>0$ be an integer coprime to $h$, and let $J\subseteq S$.…

Let $(\mathbb W,S)$ be a Coxeter system with a geometric $R$-realisation over a fusion ring $R$, and let $(\check{\mathbb W},\check{S})$ be the corresponding unfolded Coxeter syste…

Let $R$ be a fusion ring and let $(r_{s,t})_{s,t\in S}$ be a matrix satisfying the geometric conditions for a Coxeter system $(\mathbb W,S)$, as in the fusion-realisation condition…