11 problems

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 $\Lambda\in P^+$ 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 $\mathfrak{g}$ 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…

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…

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…

Let $A$ be a simple $R(\nu)$-module in $\operatorname{Rep}^{\Lambda_i}$ with $|\nu|\geq 1$. Let $r(A)$ and $\mathcal{R}(A)$ be as in the preceding theorem for trivial modules, and…

Equivalence conjecture. For $t=1,2$, the functor