68 problems

A 13-dimensional complex ball quotient is obtained from the Allcock lattice construction, with $X$ denoting the corresponding ball quotient and $X^{\rm sg}$ its singular or special…

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…

Ginzburg's conjecture. The quotient variety $X$ can be naturally desingularized, and this desingularization is holomorphically symplectic and admits a hyperkähler structure. The co…

Let $W$ be a real reflection group with reflection representation $V$. A positive integer $r$ is $W$-good if, for every parameter $c$ with $|c|=r$, the Cherednik algebra $H_c(W)$ a…

Let $W$ be a real reflection group of rank $\ell$, with Coxeter number $h$ and basic invariant degrees $d_1,\dots,d_\ell$. Let $r$ satisfy $r\equiv d_i-1\pmod h$ for some…

Let $W$ be a real reflection group with reflection representation $V$, let $H(W)$ be the universal rational Cherednik algebra, and let $\mathcal{QH}\subset H(W)$ be the space of un…

In either of the two settings, let $B$ be the associated braid group, let $W$ be the corresponding Coxeter or complex reflection group, let $p:B\twoheadrightarrow W$ be the natural…

Proposed colored-face model. The ideal $SI'_{n,r}$ is stable under ${\mathbb{Z}}_r\wr{\mathfrak S}_n$, and there is a module isomorphism

Let $W \subset \mathrm{GL}(V)$ be a finite reflection group preserving a lattice $M \subset V$ and an $M$-rational complete simplicial fan $\Sigma \subset V$. Let $X_\Sigma$ be the…

Extreme-point density conjecture. The extreme points of $\overline{\Omega_P}$ are dense in $\partial\Omega_P$ unless $W_P$ is affine of type $\widetilde{A}$.

Proximal limit set characterization.

Strong twist conjecture. If $U\subseteq A$ is an Artin generating set with $R_S=R_U$, then $S$ and $U$ are twist equivalent.

Weak twist conjecture. If $U\subseteq A$ is an Artin generating set with $R_S=R_U$, then $\Gamma_S$ and $\Gamma_U$ are twist equivalent.

Let $\mathbb{G}=(W,\Lambda)$ be a $\mathbb{Z}_\ell$-reflection group with $(|W|,\ell)=1$, and let $T_a=\Lambda/\ell^a\Lambda$. For $b\ge a$, regard each…

Reflection deletion conjecture. If

Let $W$ be a finite reflection group of rank $n$, let $R^{(1\mid1)}_W$ be its superspace coinvariant-type quotient, and let $\mathcal{OP}_W$ be the Coxeter complex formed by the fa…

Kapranov–Schechtman's conjecture. The metamatrix $M(W)$ is totally positive.

Let $R_{\mathfrak{B}_n}^{(1,1)}$ be the type $B$ $(1,1)$ coinvariant ring, and let $B_{\mathfrak{B}_n}^{(1,1)}$ be the type $B$ super-Artin set. Type $B$ super-Artin conjecture. ……

Quaternionic Haiman conjecture. There is a $(g+1)^n$-dimensional quotient ring of the zero-fiber ring of $W$.

Let $[E/\Gamma]$ be the orbifold curve, let $\Xi$ denote the deformation data, and let $IW_R^{ell}$ act on $HH_0(E)$. Elliptic-reflection functor conjecture. There is a group $G$ a…

Neighborhood domination conjecture. If $H$ is a vertex-transitive reflection graph with diameter $d$, then $H(\ell)$ dominates $H(k)$ for all $1\leq k\leq \ell\leq d$.

Lewis's conjecture. Two reflection factorizations of $g$ belong to the same Hurwitz orbit if and only if they generate the same subgroup $H\leq W$ and have the same multiset of $H$…

Frobenius-manifold Lyashko–Looijenga degree conjecture. The degree of the map $LL(\mathcal{F}_g)$ equals

Strong inscribability conjecture in higher rank. Every strongly inscribable arrangement of rank $d\geq 3$ is combinatorially isomorphic to the restriction of a reflection arrangeme…

Character-sum refinement. Assume that $\tau$ is regular and that the fixed-component Calogero–Moser isomorphism conjecture holds. If…