13 problems

Let … be a tensor product of type $D_4^{(3)}$ KR crystals, and let $\operatorname{RC}(B)$ be its rigged-configuration set equipped with the conjectural affine crystal operators. Af…

Let $\mathfrak{g}$ be of type $D_{n+1}^{(1)}$, and let $\widetilde{B}^{n,s}$ be the affine crystal constructed by the stated classical decomposition and affine operators. Spin-crys…

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

Let $D$ be a $\pm$ diagram of shape $\Lambda/\lambda$ whose ambient $r\times s$ rectangle has complement of $\Lambda$ tiled by vertical dominos. Let $m_{i}^{\circ}$, $m_{i}^{+}$,…

Let $G=SO_{2n+1}$, $Sp_{2n}$, or $SO_{2n}$, let $frak{V}( u)$ denote the restriction to $G$ of the irreducible finite-dimensional $GLN$-module of highest weight $u$, and let…

Let $G$ be one of the classical groups and let $\mu^{(1)},...,\mu^{(r)}$ and $\lambda$ index the corresponding $q$-analogue $\frak{D}_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,G}(q)$ of a…

Let $B_\mu=B_{\mu_1}\otimes\cdots\otimes B_{\mu_m}$ be an affine crystal, and let $X$ denote the associated one-dimensional sum, obtained by grading highest-weight vertices in the…

Coherence conjecture. The family $\{B^{n,s}\}$ has a limit in the sense of KNO; equivalently, $\{B^{n,s}\}$ is a coherent family.

Let $\mathfrak{g}$ be of affine type other than $A_n^{(1)}$, with parameters $N_0$, $\kappa$, $c_a$, and $t_a$ as defined in the source. For a classically lowest weight element…

Let $\mathfrak{g}$ be of affine type other than $A_n^{(1)}$. Let $N_0$ and $\kappa$ be as in the source, and let $(c_a)_{a\in I}$ and $(t_a)_{a\in I}$ be the Kac-label data and ass…

Let $\mathfrak{g}$ be an affine Kac–Moody algebra, let $i$ be a node of its classical Dynkin diagram, and let $m$ be a positive integer. Denote by $B^{i,m}$ the crystal of the Kiri…

Order-independence conjecture. The resulting map $\mathbb{S}_R$ is independent of the order in which the splitting steps and combinatorial $R$-matrices are applied.

Schilling–Shimozono conjecture. For $\diamondsuit\in\{(1),(2),(1,1)\}$,