20 problems

Gordon–Martino conjecture. There is a natural identification of the $CM_{\bf c}$-partition and the $KL_{\bf L}$-partition, induced by attaching a $KL_{\bf L}$-cell to an irreducibl…

Sommers's cell-region decomposition conjecture.

Canonical cell conjecture. The element of minimal length is unique, and $J(\lambda)$ is a completely prime maximal ideal of ${\mathbf U}({^L \mathfrak{g}})$ whose associated variet…

Ungraded Hecke-cell invariance conjecture. Let $x,y,w,x',y'\in S_n$ satisfy

Hecke-cell invariance conjecture. Let $x,y,w,x',y'\in S_n$ satisfy

Graded combinatorial Kåhrström conjecture. Let $x,y,w,w'\in S_n$ satisfy

Let $c$ be a two-sided Kazhdan–Lusztig cell in a finite Weyl group, let $G_c$ be the associated finite group, and for each left cell $c^L\subset c$ let $H_{c^L}\leq G_c$ be the ass…

Dlab–Parshall–Scott conjecture. There exist a preorder $\leq$ on $\Omega$, strictly compatible with $\overline\Omega$, and a right $\mathcal H$-module $X$ such that $X$ has a finit…

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…

Special-character conjecture. For each left cell $C$ of $W$, there is a unique $\chi\in\mathcal S_\varphi(W)$ such that $\langle[C],\chi\rangle_W\ne 0$; moreover, for this characte…

Let $(W,S)$ be a Coxeter system with weight function $\varphi:S\rightarrow\Gamma_{>0}$. Let $\mathcal{G}_\varphi(W)$ be the graph whose vertices are the irreducible characters of…

Let $(W,S)$ be a Coxeter system with weight function $\varphi:S\rightarrow\Gamma_{>0}$, and let $C$ and $C'$ be left cells contained in the same two-sided cell. For any subset…

Let $W$ be an irreducible affine Weyl group with generating set $S$, let $\mathfrak{C}$ be the union of its proper finite standard parabolic subgroups, and let…

Let $\Phi^+$ be the set of positive roots, with short and long roots distinguished, and define … by $\nu(\alpha)=0$ if $\alpha$ is short and $\nu(\alpha)=1$ if $\alpha$ is long. Le…

Let $W$ be a Coxeter group with generating set $S$, and let $C\subset W$ be a cell. Write $\mathtt{Red}(C)$ for the language of reduced expressions of elements of $C$. Casselman's…

Let $(W_n,S_n)$ be a Weyl group of type $B_n$, with $n\geq2$, and write ${\bar S}=\{s,t\}$ as in the source. Type $B_n$ conjecture. Conjectures A and B hold for $(W_n,S_n)$, and th…

Let $(W,S)$ be an affine Weyl group of type $\widetilde{G}_2$, with the preceding notation $\bar S=\{s,t\}$. The hyperplanes ${\mathrsfs{H}}_s$, ${\mathrsfs{H}}_t$,…

Assume Conjecture A for $(W,S)$, and let ${\mathfrak A}$ be the set of essential hyperplanes of $(W,S)$. Let $X\in{\mathrsfs{P}}\!os({\mathbb Z}[{\bar S}])$, let $C$ be a left cell…

Let $(W,S)$ be a Coxeter system and let ${\mathrsfs{P}}\!os({\mathbb Z}[{\bar S}])$ be the parameter space of weight functions. A finite arrangement ${\mathfrak A}$ of rational hyp…

Let $(W,S)$ be a Coxeter group with $S$ finite, decomposed as $S=S_1\ropto S_2$ into nonempty subsets whose elements from different subsets are not conjugate in $W$. For positive i…