36 problems

Crystal structure conjecture. The localization $\widetilde{\mathscr C}_w$ possesses a crystal $\mathbb B(\widetilde{\mathscr C}_w)$.

Let $e\geq2$, let $i\in\mathbb{Z}_e$, and let $R_\theta({\tt A}_{e-1}^{(1)})$ be the KLR algebra for a root-lattice element $\theta$. Let $\mathcal{MP}_{\mathrm K}^{(i,i)}(\theta)$…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathcal{NP}_{\mathrm{2res}}(\theta)$ be the set of skew diagrams…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathsf{D}\in\mathcal{UL}(\theta)$ be an upper ledge diagram. Let…

Let $\boldsymbol{\kappa}=(\kappa_1,\kappa_2)$ be a bicharge, let $\boldsymbol{\lambda}\in\mathcal{MP}^{\boldsymbol{\kappa}}$, and let $\boldsymbol{\lambda}_{\mathrm K}$ denote its…

Let ${\tt X} \in \{{\tt A}_n, {\tt D}_n, {\tt E}_{6,7,8}\}$, and let $\mathcal{A}$ be the data of the zigzag algebra $\bar{Z}({\tt X})$ as described in the paper. Write…

Let $\mathfrak{g}$ be the Lie-theoretic datum of the paper, let $\mathcal O_{\mathrm{sh}}$ be the shifted category O, and let $R=\bigoplus_{\nu\in Q_+}R_\nu$ be the associated KLR…

Suppose $\mathfrak{g}$ is simply-laced, let $\lambda\in P_+$, let $\mu\in \operatorname{wt}(V(\lambda))$, and choose an integral set of parameters $\mathbf{R}$ of level $\lambda$.…

Let $L$ be a finite-dimensional simple module over the relevant KLR algebra, let $d_L$ be the corresponding vector in the dual canonical basis of ${\mathbb C}[N]$, and let $b_Z$ be…

Proper stratification conjecture. The preorder of is properly stratifying for $A_0(\vecn)$ in arbitrary characteristic.

RoCK core block conjecture. If $\Lambda_+^{\boldsymbol{\kappa}}(\omega)$ is a $\theta$-RoCK core block with…

Let $K$ be an arbitrary field, and let $d4dc_{\alpha}^{\Lambda}$ be a cyclotomic KLR algebra over $K$ of arbitrary type. Indecomposability conjecture. The $K$-algebra…

Center conjecture. The center of $\mathscr{R}_{\alpha,K}^{\Lambda}$ consists precisely of the symmetric elements in $\mathscr{R}_{\alpha,K}^{\Lambda}$.

Let $\beta\in Q_n^+$ and let $\Lambda\in P^+$. Let $p_\beta^\Lambda:\mathscr{R}_\beta\twoheadrightarrow\mathscr{R}_\beta^\Lambda$ be the canonical surjection. The center…

Leclerc's conjecture. The product has the form

Rigidity-reality conjecture. If $C$ is rigid, then $\pi(C)$ is real.

Let KLR algebras be the braid-like diagrammatic algebras associated with a Cartan datum, with strands labeled by simple roots and decorated by dots. Their suitable quotients are th…

Let $R_{{\mathfrak{b}}}(\nu)$ be the odd KLR algebra associated with a dimension vector $\nu$, and let $Z(R_{{\mathfrak{b}}}(\nu))$ denote its supercenter. Let…

Let $R$ be the integral coefficient ring, let $\overline R$ be the corresponding coefficient ring for the construction with $e$ replaced by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$…

Cyclic-cellularity conjecture. If $A$ is cyclic cellular, then so is $\mathcal{H}^{\mathbf{C}}_n(A)$.

Graded-symmetry conjecture. The algebra $\mathcal{H}^{\mathbf{C}}_n(A)$ is graded symmetric.

Basis conjecture. The spanning set of Proposition $$ constitutes a $\mathbbm{k}$-basis for $\mathcal{H}^{\mathbf{C}}_n(A)$.

Let $\delta$ be the minimal positive imaginary root of the affine ADE root system, let $n\in\mathbb{Z}_{>0}$, and let $k$ be either $F$ or $K$. Write $C_{n\delta,\mathcal O}$ for t…

Cosocle construction conjecture. Under the same hypotheses as the existence theorem,

Fix $\alpha$ and let $H_{\alpha,F}$ be the corresponding finite type KLR algebra over $F$. For each $\lambda\in\operatorname{KP}(\alpha)$, let $P(\lambda)_F$ be the projective modu…