16 problems

Let $N \in {\mathbb Z}_{\ge 2}$. Fix positive integers $k_1,\ldots,k_N$ with $k_1>\cdots>k_N$, and elements $b_i \in B_{k_i}$ for $i=1,\ldots,N$ in the…

Consider the two-soliton solutions of the $A_M^{(1)}$ automaton obtained from the ultradiscrete limit, where the phase factor $A(j)$ has several possible choices. The authors conje…

The $A_M^{(1)}$ automaton is described by the $N$-soliton solution constructed above, with parameters satisfying the stated constraints. For $N=1$ this solution is the general solu…

Rigged-configuration conjecture. A set of action-angle variables for this system is given by the rigged configuration.

Let $v=v_1\otimes v_2\otimes\cdots\otimes v_{\mathfrak{d}}\in(B^{r,1})^{\otimes\mathfrak{d}}$ be a soliton. For each factor $v_j$, let $k_j$ be the value of $k$ assigned by the pre…

Let $B^{r,1}$ be the $U_q(\mathfrak{gl}(m|n))$-crystal of rectangular semistandard Young tableaux of height $r$ and width $1$, with $r\leq m$. Consider the subset of…

Let $p$ be any state, and let $T_{\infty}$ be the time-evolution map. A soliton separation conjecture asserts that there exist a positive integer $\widetilde{t}$, solitons…

Let $v$ be a coupled soliton. A coupled-soliton decomposition conjecture asserts that there exist positive integers $d$ and $\widetilde{t}$ such that, whenever $w$ is an uncoupled…

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…

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…

Positive solitons are configurations of the cellular automaton $\Phi(n,k)$ in their minimal forms, and a multi-soliton solution is a configuration consisting of several such positi…

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

Uniform soliton parametrization conjecture. For all soliton cellular automata, solitons are parameterized by decoupled Kirillov–Reshetikhin crystals, the scattering rule is determi…

Rigged configurations are combinatorial objects associated with tensor products of Kirillov--Reshetikhin crystals, and soliton cellular automata are generalizations of box-ball sys…

Let $B^{r,s}$ be a perfect crystal, and let $g'$ and $g”$ be the simple Lie algebras associated with the two connected components obtained by removing the Dynkin node $r$ from the…