10 problems

Dyer–Hohlweg conjecture. Each equivalence class under $\sim_m$ contains a unique minimal length element, and the set of these minimal length elements is the set of $m$-low elements…

Characteristic-polynomial equality conjecture. The arrangements $\mathcal{S}(G)$ and $\mathcal{I}(G)$ have the same characteristic polynomial for any digraph $G$. This conjecture e…

Shi-region low-element conjecture. One has $L_{\operatorname{Shi}}(W,S)=L(W,S)$. Moreover, every Shi region contains a unique low element, which is its unique element of minimal le…

Let $(W,S)$ be a Coxeter system and let $\mathscr{S}_n$ denote the $n$-Shi partition of $W$. A Shi-gating conjecture. For every $n\in\mathbb{N}$, the partition $\mathscr{S}_n$ is g…

Let a Coxeter group have a generalized Shi arrangement, and let the analogues of minimal permutations be the affine-group elements whose corresponding alcoves are minimal in their…

Let $W$ be an affine Weyl group whose Coxeter graph is simply laced, and consider the admissible sign types associated with its Shi arrangement. The supplied text notes that, when…

Shi-tableau uniqueness conjecture. Every dominant $p$-stable Weyl group element is uniquely determined by its Shi tableau.

The ideal-arrangement conjecture. All ideal-subarrangements ${\mathcal{A}}_I$ and ideal-Shi arrangements $\operatorname{Shi}^k_{\pm I}$ are divisionally free.

Let $W$ be a crystallographic reflection group, let $\mathsf{Shi}^{(m)}(W)$ be the extended Shi arrangement, and let $\operatorname{coh}(R)$ be the coheight of a positive region…

Let $W$ be a crystallographic reflection group, and let $\operatorname{Cat}^{(m)}(W,q)$ be the combinatorial $q$-Fuss-Catalan number obtained by summing $q^{\operatorname{coh}(R)}$…