6 problems

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 $\Pi$ be the set of pairs of partitions, let $i\in\mathbb{Z}_2$, and let $\mathcal{X}_i:\Pi\rightarrow\mathcal{UL}$ be the associated map to upper ledge diagrams. If…

Let $\mathcal{P}$ be the set of partitions, let $\mathcal{P}_{\mathrm{2reg}}$ be the set of 2-regular partitions, and let…

Odd-shift inclusion conjecture. If $k$ is odd, then $X_1\subset X_2$.

Crystal-structure conjecture. There exists a crystal structure on $\mathcal{T}_{\lambda}$ such that

Let $(I,\cdot)$ be a Cartan datum of finite type, let $Q_+$ be its positive root lattice, let $\alpha\in Q_+$, and let $R_\alpha$ be the corresponding Khovanov–Lauda–Rouquier algeb…