47 problems

Rigged configuration virtualization conjecture. For all $\mathfrak{g}$ of symmetrizable type, there exists a simply-laced type $\widehat{\mathfrak{g}}$ and diagram folding $\phi$ w…

Let $\operatorname{BAE}(\ell)$ be the Bethe ansatz equations, and call a solution physical if it is either regular or a physical singular solution satisfying the Nepomechie–Wang co…

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 … 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 $R\trianglerighteq R'$ 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…

Let generalized Kostka polynomials be the polynomials studied in the paper, let ribbon tableaux carry their spin statistic, and let rigged configurations be the corresponding combi…

The crystal interpretation conjecture. The path $p$ is given by

Subtableau compatibility conjecture. The bijection $\overline{\Phi}$ is well-behaved with respect to fixing those subtableaux.

Promotion conjecture. Theorem 4.11 remains true for any tensor product

The $X=M$ conjecture. For all affine Kac–Moody algebras $\mathfrak{g}$,

Hatayama–Kirillov–Reshetikhin conjecture. The Kirillov–Reshetikhin modules $W^{r,s}$ have a crystal basis denoted by $B^{r,s}$.

Let $B_R$ be a crystal of type $A_{2n}^{(2)\dagger}$ and let $4\Lambda4$ be a dominant integral weight. With $X(B_R,\Lambda;q^{-1})$ the one-dimensional configuration sum, let…

$X=M$ conjecture.

Let $B$ be a tensor product of Kirillov–Reshetikhin crystals, let $X$ be the sum of the intrinsic energy statistic over paths in $B$, and let $L$ be the multiplicity matrix countin…

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…