104 problems

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…

Choose a generic line in $V^\ast$ and let $\rho:S\to{\mathbb R}[T]$ be the corresponding graded surjection, with $T$ of degree $2$. For an $S$-module $M$, set…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^{[y]}\hookrightarrow {\mathscr B}(x)^y$ be the inclusion of the costalk int…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^y$ be the stalk of the sheaf ${\mathscr B}(x)$ at $y\in{\mathcal W}$. For a…

Let $\mathcal W$ be the Coxeter group, let $\mathcal H$ be its Hecke algebra, and let $B(x)$ be the indecomposable projective object associated with $x\in\mathcal W$. For a graded…

Let $\mathcal W$ be the Coxeter group with Bruhat order $<$ and length function $l$. For $x,y\in\mathcal W$ with $y<x$, let $B(x)^{[y]}$ denote the corresponding graded $S$-module…

Let $W$ be a Coxeter group with weight function $L$, let $W_I\subseteq W$ be a parabolic subgroup, and let $Y_I$ denote the corresponding set of distinguished coset representatives…

Let $V$ be a reflection-faithful representation of ${\cal W}$ over an infinite field, and let $R$ be the ring of regular functions on $V$. For $x\in{\cal W}$, let $C'_x$ be the cor…

Let $W$ be of type $B_n$, with generators $t,s_1,\ldots,s_{n-1}$, where $t$ is not conjugate to the $s_i$, and let a positive weight function be specified by … … Write the weight f…

Let Springer fibers be associated with two-row Young shapes, and let their irreducible components be described by the corresponding tableau and cup-diagram data. An iterated…

Let $W_S$ be the Weyl subgroup associated to $\gf_S$, and let $W_S\setminus W$ be the corresponding coset with its Bruhat ordering. For $\sigma\leq\tau$ in this coset, let…

Consider a finite W-algebra associated to an ${\bf sl}_2$ embedding that is principal in $\gf_S\subseteq\gf$. Let $M(\Lambda)$ be a Verma module, let $\bar{\beta}$ be a position in…

45312-avoidance conjecture. If $w \in \mathfrak{S}_n$ does not avoid the pattern $45312$, then $\widetilde{C}_{w}(q)$ has no reversal factorization.

Generator conjectures. The element $X^{\mu}_N$ generates $I^{\mu}_N$; the element $Y^{\mu}_N$ generates $I^{\mu}_N$; and

Let $\mathrm{Inv}_n$ be the set of involutions in $\mathrm S_n$, let $L_z^{(n)}$ be the simple highest weight module indexed by $z$, let $\theta_x^{(n)}$ be the projective functor…