10 problems

Let $B^{1,s}$ be the Kirillov–Reshetikhin crystal of type $E_7^{(1)}$, and let $\eta_i$ denote the fundamental weights of $A_7$. For $a,b,c,d\in\mathbb{Z}_{\geq 0}$ satisfying…

Fix $a\in I_0$, set $c=\min\{c_b\mid A_{ba}\neq 0\}$, and let $\ell\geq\lceil m/c\rceil$. Write $L(k)=\left\lfloor\frac{mA_{ba}-k}{A_{ab}}\right\rfloor$. Q-system Demazure decompos…

Let $B$ be a tensor product of Kirillov--Reshetikhin crystals of type $\mathfrak{g}$. For a classical weight $\lambda$, let $X(B,\lambda;q)$ be the one-dimensional sum over classic…

Let $B^{r,s}$ be a Kirillov--Reshetikhin crystal for an affine type, and let $\phi$ and $(\gamma_a)_{a\in I}$ be the folding and scaling data defining virtual crystals. Define … Vi…

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…

Let … and let … be organized into $r_i\times s_i$ rectangles as in $T^{r_i,s_i}$. Let $\operatorname{fill}\colon B\to T$ be the filling map. Filling-map factorization conjecture. T…

HKOTT's perfectness conjecture. The crystal $B^{r,s}$ is perfect if and only if $\frac{s}{c_r}$ is an integer. If $B^{r,s}$ is perfect, its level is $\frac{s}{c_r}$.

Let $(\nu,J)$ be a rigged configuration and let $b=\Phi(\nu,J)$ be its image under the bijection to the corresponding tensor product of type $D_n^{(1)}$ Kirillov--Reshetikhin cryst…

Let $(\nu,J)\in\operatorname{RC}(B)$, where $B=B_1\otimes B_2$ is a two-fold tensor product of Kirillov--Reshetikhin crystals, and let $B'=B_2\otimes B_1$ be the tensor product wit…

Let $B=B^{r_1,s_1}\otimes B^{r_2,s_2}\otimes\cdots\otimes B^{r_L,s_L}$ be a tensor product of Kirillov--Reshetikhin crystals, and let $\operatorname{RC}(B)$ be its set of rigged co…