104 problems

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…

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…

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

Let $(W,S)$ be a Coxeter system, and let $P_{y,w}$ denote its Kazhdan–Lusztig polynomials for $y,w\in W$, with $\leq$ the Bruhat order. Property B. The polynomials $P_{y,w}$ are de…

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…

Bonnafé–Geck–Iancu–Lam conjecture. If $s\notin\mathbb{N}$, the Kazhdan–Lusztig left cells coincide with the irreducible combinatorial left cells of rank $\lfloor s\rfloor$. If…

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…

Haiman's conjecture. For any $w\in S_n$, the dual Frobenius character $\operatorname{ch}(q^{\frac{\ell(w)}{2}}C'_w)$ is $h$-positive.

Let $w\in S_n$ and define the generalized LLT polynomial by … where $C'_w$ is a Kazhdan–Lusztig element and $\omega$ is the standard involution on symmetric functions. A symmetric…

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 $\mathfrak g$ be a symmetrizable complex Kac–Moody algebra with Weyl group $\mathcal W$ and simple reflections $\mathcal S$. Fix a Weyl vector $\rho$, let…

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…

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

Let $\Gamma_{J,\mathsf{v}}$ be the cell recognised by the bounded representation $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$, let $\mathbf{a}_{J,\mathsf{v}}$…

Let $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$ be a bounded representation, let $\mathbf{a}_{J,\mathsf{v}}$ denote its bound, let $J$ be the relevant subset…

Lusztig's conjecture on the function $a$. For every $w\in W$, one has

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…

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…

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…

Let $R$ be a reduced, irreducible, finite root system, let $\widehat{\mathcal W}$ be its affine Weyl group, and let $k$ be a field. Form the associated moment graph…

Let $R$ be a reduced, irreducible, finite root system, $X$ its weight lattice, and $k$ a field whose characteristic is bigger than the Coxeter number of $R$. Let $\mathfrak g$ be t…

Choose a generic line and the associated graded algebra ${\mathbb R}[T]$, with $T$ of degree $2$, and let ${\mathscr B}(x)^{\delta y}_T$ be the corresponding module. The Hard Lefsc…