143 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 $G$ be a crystallographic complex reflection group, meaning a subgroup of $\operatorname{AGL}_n(\mathbb C)$ generated by reflections that fix a hyperplane pointwise and have co…

Freeness conjecture for $G(n,n,3)$. The arrangement $\mathcal{A}$ is free if and only if $n$ is not prime, or $n$ is prime and the characteristic of $k$ is not $n$.

Let $V$ be a finite-dimensional complex vector space, let $W\subset \operatorname{GL}_{\mathbb{C}}(V)$ be a finite reflection group, and let $c$ be a complex-valued function on the…

Mühlherr's conjecture. For any Coxeter generating set $S\subseteq R^W$, there exists an automorphism $\alpha$ of $W$ such that $\alpha(S)\subseteq R^W$ and $\alpha(S)$ is sharp-ang…

Let $F$ be an irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$ for a K3 surface $S$, and let the reflection group act on the positive cone of divisor…

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

Geck's chamber conjecture. The parameter space for $\mathcal{H}_{\bf q}(G)$ should decompose into a finite number of chambers, and the cell structure should depend only on the cham…

Let $M$ be an even hyperbolic reflective lattice. Consider scaling and the two operations described in the source: replacing $M$ by $p^{-1}(M\cap p^2M^*)+M$ for a prime $p$ dividin…

Let $\Gamma$ be a complex crystallographic group acting on $E_\mathbb{C}^n$, with linear part a finite complex reflection group. The orbit space $E_\mathbb{C}^n/\Gamma$ is consider…

Let $L$ be the lattice in $\mathbb{R}^{n}$ spanned by the canonical basis, and let a coincidence isometry be an isometry of $\mathbb{R}^{n}$ whose intersection with its image of…

Let $p\geq 3$ be prime. The regular polytope $\mathcal{C}_{p,p,3}$ has facets $\mathcal{M}_{p,p}$ and vertex-figures $\mathcal{M}_{p,3}$. Universality conjecture for…

Let $p\geq 3$ be prime. The regular polytope $\mathcal{C}_{p,3,3}$ has facets $\mathcal{M}_{p,3}$ and vertex-figures $\{3,3\}$. Universality conjecture for $\mathcal{C}_{p,3,3}.$ F…

Let $n>1$, and consider maximal arithmetic reflection groups in $\operatorname{Isom}(\mathbb{H}^{n})$. Finiteness conjecture. There are only finitely many maximal arithmetic reflec…

Twist-equivalence conjecture. If $R\subseteq S^W$ is a Coxeter generating set of $W$ that is sharp-angled with respect to $S$, then $R$ is twist-equivalent to $S$.

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…

Let $P$ be a finite-volume all-right-angled polytope in hyperbolic space $H^n$, meaning that every pair of intersecting sides of $P$ meets at dihedral angle $\frac{\pi}{2}$. Vinber…

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…

Let $L$ be a Lorentzian lattice, and let a modular form be reflective when the singularities of its singular theta transform occur along the reflection hyperplanes of $L$. Reflecti…

A Lorentzian reflection group with Weyl vector is a reflection group associated with a Lorentzian lattice and admitting a Weyl vector. The maximal-dimension conjecture. The maximal…

Let $P\subset \mathbb{X}$ be a finite-volume Coxeter polytope with $k$ facets, and let $G_P$ be the group generated by the reflections in the facets of $P$. Reflection-generation c…

Let $R$ be a reflection group with a non-negative multiplicity function $k$, let $A_r$ be the Banach algebra of homogeneous polynomial series on the closed ball $B_r$, and let…

The solutions to isomonodromic deformation equations for two-dimensional Fuchsian systems with four singularities on the Riemann sphere $\mathbb{P}^1$ can be expressed through solu…