16 problems

Cyclicity conjecture. Given a configuration $m=(m^{(1)},\ldots,m^{(n)}),$ either every path in $P_h(m)$ is cyclic or no path in $P_h(m)$ is cyclic.

Dynamical-period conjecture. The dynamical period of $p$, namely the minimum positive integer $k$ such that $T_l^k(p)=p$, is equal to ${\mathcal P}_l$ generically, and is otherwise…

Let $\mu \in \mathbb{N}^{\ell}$, and let $\ell(\mu):=\lvert\{i=1,\dotsc,\ell:\mu_i>0\}\rvert$ be the number of nonzero entries of $\mu$. For the $\widehat{\mathfrak{sl}}_n$-crystal…

For a permutation $w$, let $\operatorname{SD}(w)$ denote its soliton decomposition, and call $w$ BBS good when $\operatorname{SD}(w)$ is a standard tableau. A consecutive pattern i…

Let $w\in S_n$ be a permutation, and let the steady-state time be the number of box-ball system moves required for the system containing $w$ to reach its soliton decomposition. The…

For permutations $v,w\in S_n$, let $\operatorname{Q}(v)$ and $\operatorname{Q}(w)$ denote their recording tableaux under the Robinson–Schensted correspondence. The steady-state tim…

For a permutation $w\in S_n$, let $\operatorname{Q}(w)$ be its recording tableau under the Robinson–Schensted correspondence, and let $\widehat{Q}$ be the tableau … The steady-stat…

Let a box-ball system have $n$ blocks, with $Q_k$ balls in the $k$-th block, and suppose the blocks are separated sufficiently so that the asymptotic soliton structure can be ident…

Boundary-condition conjecture. The boundary condition … , holds universally for type $D_n$ and types $E_6$, $E_7$, and $E_8$.

Let $\Phi(n,k)$ have initial configuration $(\mathbf{Y}_i^0)_i$ consisting only of positive solitons. Interchange space and time so that $\mathbf{Y}_i^t$ and $\mathbf{Z}_i^t$ are r…

Finite-carrier conjecture. When only positive solitons occur in $\Phi(n,k)$, the carriers $\mathbf{Z}_i^t$ take values in the finite set $M$. This is intended to express the observ…

Loop symmetric functions are polynomial invariants of the birational symmetric-group action induced by the one-row geometric $R$-matrix, and their tropicalizations can be considere…

Let the box-ball system corresponding to the vector representation of type $A^{(1)}_1$ be given, and let its states be parametrized by rigged configurations via the rigged configur…

Let $X\in\mathbb{R}^{M\times N}$ and let $\mathrm{UD}(\eta)$ denote the ultradiscretization of the mapping $\eta$ constructed from the divisor component and the ultradiscretized ma…

Ultradiscretization conjecture for the extended box-ball system. The same fact should hold for the extended box-ball system: every state of the extended system is obtained by ultra…

The loops around the cylinder are called whurls, and a whurl relation is the transformation $R$ on the parameters of adjacent whurls that preserves boundary measurements. In the mi…