52 problems

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…

Explicit adjoint formulas conjecture. The adjoints satisfy

Let $B_{(W,S)}$ be the generalized braid group associated with the Coxeter system $(W,S)$, and let $2$-${\mathcal B}r$ be the monoidal category generated by the Rouquier complexes…

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 $(W,S)$ be a Coxeter system of finite rank, and let $\mathcal{J}$ be a two-sided cell of $W$. A two-sided cell is called connection-finite when the relevant connection sets are…

Let $W$ be a Coxeter group of finite rank, let $\mathcal{H}$ be a diagonal $H$-cell, and let $d$ be a Duflo involution in $\mathcal{H}$. Assume the Klein–Elias–Hogancamp conjecture…

Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a two-sided cell, and let $d$ be a Duflo involution in $\mathcal{H}$ with $\mathtt{a}(d)=a$. Let $\widetilde{\mathrm{C}}_d$ be…

Let $W$ be a Coxeter group of finite rank, let $\mathcal{H}$ be a two-sided cell with a diagonal $H$-cell, and let $d$ be a Duflo involution in that cell. Let…

Let $Q$ be the Jordan quiver, let ${\mathcal X}_a$ be the associated semi-nilpotent commuting stack, and suppose that there is a functor … from the Karoubi-completed trace of the a…

Let $m$ and $n$ denote objects indexing Rouquier complexes, and let $m\boxtimes n\rightarrow n\boxtimes m$ be the corresponding Rouquier complexes. Let $\tau_{m,n}$ be the naturali…

Let $(W,S)$ be a Coxeter system of type $\operatorname{\textbf{B}}_n$, let $\operatorname{\mathfrak{h}}$ be a Cartan realisation of the same type, and let $I\subset S$ be a parabol…

Let $n\in\mathbb{N}$, let $(m_i,n_i)\in\mathbb{Z}\times\mathbb{N}$, and let $d_i\geq 1$. For the affine Soergel trace category $\operatorname{Tr}(\mathrm{ASBim}_n)$, write $\star$…

Let $\operatorname{ASBim}_n$ and $\operatorname{SBim}_n$ denote the affine and finite Soergel-bimodule categories, respectively, and suppose a suitable flattening functor…

Let $n\in\mathbb{N}$ and let $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$ be the idempotent completion of the trace category of affine Soergel bimodules. For integers…

Frayed-projector diagonalization conjecture. The infinite frayed projector $(P^{\boldsymbol{\lambda}}_{(1^n)})^{\vee}$ and Cautis's HOMFLY clasp should lie at opposite ends of a di…

Fray functors' 2-functoriality conjecture. For each composition $\boldsymbol{\lambda}\vdash n$,

Singular light leaves rex move conjecture. $\mathbb{ELL}([n,m])$ is a rex move.

Let $W=S_n$ be the symmetric group, let $k$ be a field of characteristic $p$, and let $b_x$ denote the Kazhdan–Lusztig basis element of the Hecke algebra $H(W)$. For $x\in W$, let…

Categorical half-twist conjecture. If $w$ lies in the cell $\lambda$, then

For ${\bm{b}}=(b_1,\ldots,b_m)$, let $E_{\bm{b}}$ be the colored polynomial algebra and let $I_{\bm{b}}\vartriangleleft E_{\bm{b}}$ be the colored full-twist ideal. Let…

Let $a_1,\ldots,a_m,b\geq0$ and set $\bm{a}=(a_1,\ldots,a_m)$. A one-sided twisted complex of threaded digons is a chain of complexes obtained by increasing the number of crossings…

Let $\mathsf{B}$ be the fully faithful functor from the category of Soergel bimodules $\mathrm{SBim}_n$ to the category of stable matrix factorizations $\textup{MF}_n^{st}$, and le…

Let $X$ be a Bott–Samelson bimodule, let $W$ be the relevant Weyl group, and write … where the $B_w$ are indecomposable objects. Let $F_1$ be the maximal finite-dimensional…

Let $\operatorname{Tr}$ be the universal dg monoidal trace, let $\operatorname{Tr}(\mathbbm{1})$ be the trace of the monoidal unit, and let $X$ be a twisted complex built out of su…

Let $\mathrm{SBim}_n$ be the category of Soergel bimodules for the symmetric group, let $\mathcal{K}^b(\mathrm{SBim}_n)$ be its bounded homotopy category, and let…