175 problems

Lusztig's character formula conjecture. If $p \geq h$ and $w \in W_{\mathrm{aff}}^\varnothing$ satisfies

Kazhdan–Lusztig positivity conjecture. The coefficients of the Kazhdan–Lusztig polynomials are non-negative, namely $h_{y,x}\in v\mathbb{Z}_{\geq0}[v]$, and the structure constants…

Let $(W,S)$ be the Coxeter system underlying the Soergel bimodule category, let $B_x$ denote the indecomposable Soergel bimodule indexed by $x\in W$, and let $b_x$ be the correspon…

For $n\geq 0$, let $S_n$ be the symmetric group, let $\mathrm{Inv}_n$ be its set of involutions, and let $\mathbf{K}_n(w)=-$ mean that Kostant's problem for the simple highest weig…

Lusztig's boundedness conjecture. The number $N_L(W)$ is a bound for any positively weighted Coxeter group $(W,S,L)$ of finite rank. This is a weighted generalization of Xi's bound…

Let $W_n^{\mathsf{BC}}$ be the type $\mathrm{B}$ Coxeter group equipped with a weight function determined by a parameter $s$, and let its cells be the Kazhdan–Lusztig cells. The do…

Let $W$ be a finite Coxeter group, let $C$ be a conjugacy class of involutions in $W$, and let $V_C$ be the real vector space with basis $\{a_w\mid w\in C\}$ equipped with the $W$-…

Let $\widehat{W}^{\circ}$ be the finite subset of an affine Weyl group used in the moment-graph construction, let $\mathcal{B}(w)$ be the indecomposable object associated with…

Let $[x,y]$ and $[x',y']$ be intervals in Bruhat orders, and suppose that they are isomorphic as posets. The Kazhdan–Lusztig polynomial associated with an interval is denoted by…

Polynomial-positive decomposition conjecture. For each permutation $w\in S_n$, there exist codominant permutations $w_1,\ldots,w_k\in S_n$ such that…

Let $W$ be a finite real reflection group with parameter $c$. A Calogero–Moser two-sided $c$-cell is denoted by $\Gamma$, and let $\operatorname{Irr}_\Gamma^{c,\mathrm{CM}}(W)$ and…

Let $\mathbf{a}$ and $\mathbf{b}$ be multisegments on the Zelevinsky line of the trivial representation, and let $P_{\mathbf{a},\mathbf{b}}(q)$ be the Kazhdan–Lusztig polynomial ob…

Let $(W,S)$ be a Coxeter group with weight function $L$, Hecke algebra structure constants $h_{x,y,z}$, Kazhdan–Lusztig $\mathbf{a}$-function, and sets $\mathcal{D}_{\geq N}$ and…

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

Let $(W,S,L)$ be a weighted Coxeter group of rank $3$ with positive weight function. Let $a(w)$ denote the relevant $a$-function, let $N$ be the bound from Lusztig's conjecture, an…

Let $(W,S,*)$ be a triple consisting of a Coxeter system $(W,S)$ and an $S$-preserving involution $*\in\operatorname{Aut}(W)$, with corresponding twisted involutions $\mathbf I_*$.…

Twisted property B-prime. The polynomials $P^\pm_{y,w}$ are decreasing for fixed $w$: whenever $y,z,w\in\mathbf I_*$ and $y\leq z$, both differences $P^+_{y,w}-P^+_{z,w}$ and…

Let $A$ be a local deformation algebra, let $\mathcal{J}$ be an open and bounded subset of $\widehat{\mathfrak{h}}^*$, and let $\lambda,\mu\in\mathcal{J}$ be of critical level. Wri…

Let $\mu$ be integral dominant and $r$-fold atypical with respect to the roots $\gamma_1<\gamma_2<\cdots<\gamma_r$ ($\gamma_i\in\Delta_{1,+}$). Let $\lambda_\theta$, for…

The composition-factor multiplicity conjecture. For each of the $2^r$ integral dominant weights $\lambda_\theta$, $a_{\lambda_\theta,\mu}=1$, and $a_{\lambda,\mu}=0$ for every othe…

Conjecture B. For any $w,w'\in W_n$, $w\sim_{\mathcal L}w'$ if and only if $Q^{r-1}(w)=M(Q^{r-1}(w'),C)$ for a set $C$ of open cycles; the analogous statement holds for right cells…

Integral-ratio cell conjecture. Assume that $b=ra$ for some $r\geqslant 1$. Then the Kazhdan–Lusztig left, right, and two-sided cells of $W_n$ are the smallest subsets of $W_n$ tha…

Cellularity conjecture. The Kazhdan–Lusztig basis of ${\mathcal H}_n$ is a cellular basis in the sense of Graham–Lehrer.

Conjecture A+. Let $r\geqslant 0$ and assume that $ra<b<(r+1)a$. Then the Kazhdan–Lusztig left, right, and two-sided cells for these parameters coincide with those obtained for the…

Conjecture A. Assume that $\Gamma=\mathbb{Z}$, $a=2$ and $b=2r+1$. Then, for $w,w'\in W_n$, they lie in the same Kazhdan–Lusztig left cell if and only if $Q^r(w)=Q^r(w')$, in the s…