35 problems

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\beta\in Q_n^+$, and let $\Lambda\in P^+$ be a dominant integral weight. Write $\mathscr{R}_{\beta}^{\Lambda}$ fo…

Let $D^\lambda$ be the simple module indexed by a multipartition $\lambda$, and call a standard tableau of shape $\lambda$ a Dipper–James–Murphy tableau if every standard tableau w…

Let $e$ denote the quantum characteristic, and let $\boldsymbol{a}^{F}_q$ be the adjustment matrix of a cyclotomic quiver Hecke algebra of type $A$ over a field $F$. Kleshchev–Ram…

Let $R_n(\Gamma,p)$ be the quiver Hecke algebra corresponding to a quiver $\Gamma$ over a field of characteristic $p$, and let $\mu,\lambda\vdash n$ be arbitrary partitions. Type A…

Assume $p\nmid(e+1)$, let $I=\\{0,1,\ldots,e\\}$, and let $R^{\Lambda}_n(I)$ denote the cyclotomic quiver Hecke algebra associated with a dominant integral weight $\Lambda\in P^+$.…

Let $\mathcal{C}$ be a monoidal category satisfying the condition … is embedded into a rigid monoidal category. Such an embedding would place the module categories of quiver Hecke…

Let $i\in I$ for a quiver Hecke algebra of finite ADE type. Let $\refl_i$ denote the reflection functor constructed in the paper, and let the functor constructed by S. Kato be the…

Reflection functor conjecture. The functor $\refl_i$ induces an equivalence of categories

Let $R^{\Lambda}(\beta)$ be a cyclotomic quiver Hecke algebra associated with an affine root system of type $A$, where $\Lambda$ is a dominant weight and $\beta\in Q_+$. Graded cel…

Let $\Lambda\in P^+$ and $\alpha\in Q^+$. Let $\PD$ be the set of piecewise dominant sequences with respect to $\Lambda$, and define an equivalence relation by $\mu\sim\nu$ if and…

Let $\Lambda\in P^+$ and $\alpha\in Q^+$. Let $\mathscr{R}_{\alpha}^{\Lambda}$ be the cyclotomic quiver Hecke algebra over a ground field $K$, let $d_{\Lambda,\alpha}$ denote the r…

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\Lambda\in P^+$ be dominant, and let $\beta\in Q_n^+$. Let $\mathscr{R}_{\beta}^{\Lambda}$ be the associated cycl…

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\Lambda\in P^+$ be dominant, and let $\beta\in Q_n^+$. Let $\mathscr{R}_{\beta}^{\Lambda}$ be the associated cycl…

Let $R^\Lambda_\theta$ be a cyclotomic quiver Hecke superalgebra, where $\theta$ is a positive-root combination and $\Lambda$ is a dominant weight. Symmetry conjecture. Every cyclo…

Let $\rho$ be a $\bar p$-core and $d\in\mathbb{Z}_{\geq0}$, and let $\mathcal{R}^{\Lambda_0}_{\operatorname{cont}(\rho)+d\delta}$ be the associated cyclotomic quiver Hecke superalg…

Let $\rho$ be a $\bar p$-core and $d\in\mathbb{Z}_{\geq0}$. Let $\mathcal{R}^{\Lambda_0}_{\operatorname{cont}(\rho)+d\delta}$ be the associated cyclotomic quiver Hecke superalgebra…

Let $\rho$ and $\rho'$ be $\bar p$-cores, let $d,d'\in\mathbb{Z}_{\geq0}$, let $\operatorname{cont}(\rho)$ denote the residue content, and let $\delta$ be the null-root parameter u…

Let $\mathfrak{g}$ be a simple Lie algebra of finite simply-laced type, let $W$ be its Weyl group, let $N$ be the associated maximal unipotent subgroup, and let…

Let $\mathfrak{g}$ be of type $A_n$, let $I$ be the indexing set for the simple roots, and let $\mathcal{M}in_{0}^{+}$ denote the strict dominant minuscule elements of the Weyl gro…

Let $R$ be a quiver Hecke algebra of arbitrary symmetrizable type, let $w$ be an element of its Weyl group, and let $\widetilde{\mathscr{C}}_{w}$ be the localization of the monoida…

Let $U_q(\mathfrak{n})$ be the negative part of the quantum group associated with a symmetric Kac–Moody algebra $\mathfrak{g}$, and let the corresponding quiver Hecke algebra categ…

Langlands equivalence conjecture. The functor $\mathcal{F}_{\mathscr{Q}}\colon \operatorname{Rep}(R)\to\mathscr{C}_{\mathscr{Q}}$ is an equivalence of categories. This is proposed…

Fix a multicharge $\kappa\in\mathbb{Z}^l$ and a multipartition $\lambda$ of $n$. Let $\mathcal{S}^{\lambda}_{\kappa}$ be the graded Specht module with standard-tableau vectors…

Let $\lambda\in \mathscr{P}^l_n$, and let $A\in [\lambda]$ be a Garnir node. Suppose $\mathbf{B}^A$ contains $a$ bricks in the first row and $b$ bricks in the second row. For each…

Let $A$ be a symmetric generalized Cartan matrix, and let $cGamma$ be its associated quiver with no loops. Let $cmathscr{R}_{cbeta}(cGamma)$ and…