18 problems

Let $B^{r,s}$ be a Kirillov–Reshetikhin crystal, with $r\in I_0$ and $s\in\mathbb{Z}_{>0}$, and let $c_r$ be the associated lacing parameter. Perfectness conjecture. The crystal…

Let $\Phi$ be the bijection between rigged configurations and tensor products of the Kirillov–Reshetikhin crystal $B^{1,1}$ in the setting of affine crystals. General KR-crystal ex…

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…

Global-basis existence conjecture. The module $W^{r,s}$ has a global crystal basis $B^{r,s}$.

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…

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

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 $B$ and $B'$ be tensor products of Kirillov–Reshetikhin crystals satisfying the hypotheses of Theorems demazure decomposition and red in the source, and let $D_\ell(B)$ and…

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 $B^{r,s}$ denote a Kirillov--Reshetikhin crystal in an exceptional affine type. The fermionic formula $M(B,\lambda;q)$ is the generating function obtained from rigged configura…

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…

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…

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…

Kirillov–Reshetikhin crystal conjecture. The module $W^{r,s}$ admits a crystal basis $B^{r,s}$, and this crystal is perfect whenever $s$ is a multiple of $c_r$.

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.