25 problems

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 $\mathfrak{g}$ 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…

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

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.

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,

Wide-equivalence conjecture. The functor $\Pi^{\lambda}$ is injective on objects and therefore a wide equivalence of categories.

Let ${\tt C}$ be a generalized Cartan matrix and let $\alpha$ belong to the non-negative part of the corresponding root lattice. The KLR algebra $R_\alpha({\tt C})$ is defined for…

Fix a Weyl group element $w\in W$, write $m=\ell(w)$ for its length, and let a stably irreducible family be a set of irreducible modules closed under irreducible convolution up to…

Let $W$ be the Weyl group of a symmetrizable Cartan datum, let $w\in W$, and let $\mathcal{A}_v[N^w]$ denote the corresponding quantum unipotent cell. Quantum cluster structure con…

Let $(Q,\mathbf{d})$ be a valued quiver, let $\Omega$ be the quantum shuffle character, and let $[V]^*$ denote the dual Hall--Ringel algebra element associated with a representatio…

Kleshchev–Ram conjecture. The decomposition numbers for KLR algebras associated to Dynkin quivers are trivial.

Let $\Lambda=\sum_{i\in I}\lambda_i\Lambda_i\in X^+$ and let $R^\Lambda(\alpha)$ be the summand of the cyclotomic KLR algebra corresponding to $\alpha\in I^m$. Assume that $\alpha$…