27 problems

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

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…

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 $\operatorname{Tr}$ 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 $W$ be a Coxeter group with simple reflections $S\subset W$, a realization $V$ over $\mathbb{C}$, and let $\mathrm{SBim}(W)$ be the associated monoidal category of Soergel bimo…

Let $\operatorname{vTr}(\operatorname{SBim})$ and $\operatorname{hTr}(\operatorname{SBim})$ denote the vertical and horizontal traces of the category of Soergel bimodules, and let…

Let $W$ be a finite Coxeter group with simple reflections $S$, let $C_\epsilon$ be the horizontal trace of the Rouquier complex associated with the Coxeter braid determined by…

Let $\operatorname{Hilb}^n(\mathbb{C}^2)$ be the Hilbert scheme of $n$ points, let $\mathcal{I}$ be its tautological ideal sheaf, and let $X,Y$ be its two commuting endomorphisms.…

Let $\mathsf{FM}_2^{\mathrm{Ext}}$ be the Ext-enhanced category of free-monodromic complexes, and let $\mathrm{Tilt}_2^{\mathrm{Ext}}$ be the full subcategory consisting of…

Let $(W,S)$ be a finite Coxeter system with longest element $w_0$, and choose a realization $\mathfrak{h}$ of $W$. Let…

Let $\operatorname{FT}_n$ be the full-twist complex and let $\alpha:\1\rightarrow\operatorname{FT}_n$ be an eigenmap. Form the directed system … with transition maps induced by ten…

Let $(W,S)$ be a Coxeter system, let $\operatorname{FT}(W)$ denote the Rouquier complex of its full-twist braid, and let $\lambda$ range over the two-sided cells of $W$. An idempot…

Let $(W,S)$ be a finite Coxeter system, let $\operatorname{FT}(W)$ be the Rouquier complex of the full-twist braid, and let $\lambda$ range over the two-sided cells of $W$. Full-tw…

Let $P_T$ be the categorified Jones--Wenzl projector indexed by a standard Young tableau $T$ of size $n$, and let $\mathring{\mathrm{FHilb}}_T(\mathbb{C})$ be the corresponding aff…

Let $W$ be the Weyl group under consideration, let $x_1,\dots,x_m\in W$, and let $B_{x_1}\dots B_{x_m}$ be equipped with its intersection form. For…

Let $({\mathcal{W}},{\mathcal{S}})$ be a Coxeter system with length function $\ell$, let $A\in\mathscr{B}(\Phi^+)$, and let $w_A$ be the associated Mikado braid. Set $\beta:=w_A$,…

Let $W$ be the Coxeter group, let $s\in S$, and let $\underline{w}$ and $\underline{w'}$ be reduced expressions, each ending in $s$. A rex move is a morphism between reduced expres…

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…