26 problems

Let $R\trianglerighteq R'$ be a covering relation in the dominance order on sequences of rectangles. Let $\theta_R^{R'}:LRT(\lambda;R)\to LRT(\lambda;R')$ be the embedding obtained…

Let $R$ be a sequence of rectangles such that $R^t$ is dominant. Let $b_R$ be the map from rigged configurations to tableaux, and let $CCT(\lambda;R)$ denote the set of $R$-column-…

Let $R$ be a sequence of rectangles, let $\widetilde{R}$ and $\widetilde{\lambda}$ be the complementary dual data defined in the source, and let $\operatorname{dual}$ be the corres…

Let $\operatorname{ev}$ be evacuation on column-strict tableaux, restricting to a bijection $LRT(\lambda;R)\to LRT(\lambda;\operatorname{rev}(R))$. Let $\theta$ complement each rig…

Let $R$ be a sequence of rectangles and let $u$ be a permutation of the rectangle indices. The bijection $u_R:LRT(\lambda;R)\to LRT(\lambda;uR)$ is defined by composing the adjacen…

Let $R$ be a sequence of rectangles. Let $b_R:RC(\lambda^t;R^t)\to CST(\lambda,\gamma(R))$ be the restriction of the inverse rigged-configuration bijection to the indicated subset,…

Let $R$ be a dominant sequence of rectangles. Let $LRT(\lambda;R)$ be the corresponding Littlewood–Richardson tableaux, let $RC(\lambda^t;R^t)$ be the rigged configurations, and le…

Promotion conjecture. Theorem 4.11 remains true for any tensor product

Soliton characterization conjecture. The state $p$ corresponds to a soliton of length $\ell$ if and only if

Permutation-equivariance conjecture. The diagram relating $\operatorname{RC}(B)$ to $\operatorname{RC}(B')$ by the identity and $B$ to $B'$ by $R$, with vertical maps given by…

Crystal-bijection and cocharge conjecture. The map $\Phi$ is a classical crystal isomorphism and satisfies

Let $B$ be a tensor product of Kirillov--Reshetikhin crystals of affine type. Let $\operatorname{RC}(B)$ be the corresponding set of rigged configurations, let $\eta$ interchange r…

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 physical solutions of the Bethe ansatz equations be decomposed into strings, let $\nu$ be the partition formed by their string lengths, and let the riggings be assigned by the…

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 … 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 … be a tensor product of KR crystals, with virtualization map $v$ to the corresponding virtual crystal, and let $\Phi$ be the rigged-configuration bijection. Virtualization com…

Let … be a tensor product of type $D_4^{(3)}$ KR crystals. Let $\operatorname{RC}(B)$ be the corresponding rigged configurations, let $\Phi$ be the recursively defined map to $B$,…

X=M conjecture. One has

Let $B=B^{1,s_1}\otimes\cdots\otimes B^{1,s_m}$ be a product of one-row crystals, let $p=b_1\otimes\cdots\otimes b_m\in B$, and define $x_j^{(i+j-1)}$ to be the number of occurrenc…

Let $\mathrm{BA}(\ell)$ be the set of regular or physical singular Bethe solutions, and let $\iota:\mathrm{BA}(\ell)\to\mathrm{BA}(\ell)$ be defined by … For the generalized Heisen…

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 $\Phi$ be the map from rigged configurations of a tensor product of type $D_n^{(1)}$ Kirillov--Reshetikhin crystals to the corresponding crystal, and let $e_i$ and $f_i$ denote…