17 problems

Let $\mathfrak g$ be an affine Lie algebra, let $k\in I\setminus\{0\}$, and let $W(l\varpi_k)$ be the Kirillov–Reshetikhin module parametrized by $k$ and a positive integer $l$. Wr…

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $\mathcal H_{\mathbf n}=\prod_{\alpha,m}{{\rm KR}}_{\alpha,m}^{\otimes n_m^{(\alpha)}}$ be the Hilbert space of the g…

Let $B^{r,s}$ be a crystal of type $D_n^{(1)}$. Let $B(\Lambda)$ denote the classical $D_n$-crystal of highest weight $\Lambda$, and let the complement of a partition $\Lambda$ in…

Let $W^{r,s}$ be a Kirillov–Reshetikhin module for the relevant affine algebra, and let $B^{r,s}$ denote a crystal basis associated with it. Kirillov–Reshetikhin crystal-basis exis…

Type D affine crystal conjecture. If $B^{r,s}$ exists, then

Twisted Kirillov–Reshetikhin crystal conjecture. The perfect $U_q^{\prime}(\widehat{\mathfrak{g}})$-crystal $\widehat{\mathcal{B}}^{i,s}$ is isomorphic to the conjectural crystal b…

Right-splitting morphism conjecture. The stated right-splitting morphism and its extension after tensoring with any such $B$ exist with the asserted injectivity and preservation of…

Transfer-matrix q-character conjecture. The identification

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let $\hat{\mathfrak{g}}$ be its associated untwisted affine Lie algebra, and let $U_q'(\hat{\mathfrak{g}})$ be the co…

Let $\mathcal M_\mathbf C$ be the collection of Kirillov–Reshetikhin modules associated with a non-nested collection of supports as described in the paper. Simplicity conjecture. T…

Kirillov–Reshetikhin K-matrix conjecture. Symmetrizable invertible solutions of the generalized spectral reflection equation in tensor products of Kirillov–Reshetikhin modules, wit…

Let $I$ be the set of Dynkin labels, let $I_<\subset I$ be the labels of the short roots in types $B_N$ and $C_N$, and let $t_a=2$ for $a\in I_<$ and $t_a=1$ otherwise. Let…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra, let $i\in I$, and let $m\geq 0$. For each $\boldsymbol m\in\mathscr P_m$, let $T_i(\boldsymbol m)$ be the quotient of…

Let ${\mathfrak g}$ be an affine Kac–Moody Lie algebra, let $U_q'({\mathfrak g})$ be its quantum affine algebra without the degree operator, and let $W^{r,\ell}$ be a Kirillov–Resh…

Let $M^Y_{\lambda,\mathbf n}$ denote the multiplicity of the irreducible $\mathfrak g$-module $V(\lambda)$ in a generic tensor product of Kirillov–Reshetikhin modules. Let…

Let $\mathfrak{g}$ be a simple Lie algebra. For dominant integral $\lambda$ and multiplicity data $\mathbf n$, let $M_{\lambda,\mathbf n}^{(k)}$ and $N_{\lambda,\mathbf n}^{(k)}$ b…

Let $\mathfrak{g}$ be a simple Lie algebra, let $V(\lambda)$ be the irreducible $\mathfrak{g}$-module of dominant integral highest weight $\lambda$, and let…