11 problems

Let $W^L=\bigotimes (W_s^{(r)})^{\otimes L_s^{(r)}}$ be a finite tensor product of Kirillov–Reshetikhin modules, with $L=(L_s^{(r)})$ a collection of nonnegative integers. Let…

Let $\mu$ be as in the paper, let $H_{\mathrm{hw}}$ be the highest-weight element, and write $\ell(\mu)$ for the number of nonzero parts of $\mu$. Assume $n=\max(H_{\mathrm{hw}})$,…

Let $\mu \in \mathbb{N}^{\ell}$, and let $\ell(\mu):=\lvert\{i=1,\dotsc,\ell:\mu_i>0\}\rvert$ be the number of nonzero entries of $\mu$. For the $\widehat{\mathfrak{sl}}_n$-crystal…

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

Key-map conjecture. Our key map agrees with the representation-theoretic description coming from the Kashiwara crystal structure, the key map on Kashiwara–Nakashima tableaux, and t…

Let $\mathcal{LR}(\lambda/\mu,\nu)$ and $\mathcal{LR}(\lambda/\nu,\mu)$ be the sets of Littlewood–Richardson tableaux, and let $\rho_1$, $\rho_2$, and $\rho_2'$ be the bijections d…

Let $\mathfrak{g}$ be an affine Lie algebra of nonexceptional type, and suppose its rank is sufficiently large. The one-dimensional sum depends only on the attachment of the affine…