11 problems

Stembridge's conjecture. Every strongly connected admissible $W$-graph is isomorphic to a Kazhdan–Lusztig left cell.

Self-dual $W$-graph conjecture. For any irreducible self-dual representation, there exists a self-dual $W$-graph defined over $K$. If there exists a $W$-graph $\Gamma=(X,I,\mu)$ de…

Let $W$ be a finite Coxeter group with simple generators $S$, let $\mathcal{H}_K$ be the corresponding Hecke algebra over $K$, and let $E_v^\lambda$ be a simple $\mathcal{H}_K$-mod…

Let $W$ be a finite Coxeter group, let $\mathcal{H}$ be its Iwahori--Hecke algebra, and let a $W$-graph representation mean a representation afforded by a $W$-graph. A representati…

Let $(W,S)$ be a Coxeter system of type $A$, let $J\subseteq S$, and let $(D_J,J)$ be the associated W-graph ideal with its cells and Kazhdan--Lusztig partial order. Suppose that…

Decomposition conjecture. The family $(F^\lambda)_{\lambda\in\operatorname{Irr}(W)}$ satisfies all of the following properties:

Let $W$ be a finite Coxeter group and let $\rho_v^E$ be a matrix representation associated with an irreducible representation $E$ of $W$, obtained from a $W$-graph. The representat…

Exotic Robinson–Schensted conjecture. The basis $\{C_{w,\alpha}\}$ is a $W$-graph basis of the $\mathcal{H}_{2n}$-module $M_K(V\times G/B)$; the label of $C_{w,\alpha}$ in the $W$-…

Let $W_1$ be the first group in an iterated induction construction, and let $\Lambda$ be the $W_1$-graph on … where ${\mathscr{H}}_1$ is of type $A$. For vertices…

Let ${\mathscr{H}}$ be the Hecke algebra of $(W,S)=({\mathcal{S}}_n,\{s_1,\ldots,s_{n-1}\})$, and let $J\subseteq S$ satisfy … A restriction is weakly multiplicity-free when there…

Let $(W,S)$ be a finite Coxeter system with Hecke algebra ${\mathbf{H}}_K$, and let a representation of ${\mathbf{H}}_K$ be given by a $W$-graph. The representation is called balan…