Relationships between two linearizations of the box-ball system : Kerov-Kirillov-Reshetikhin bijection and slot configuration

Technical Summary: Relationships Between Two Linearizations of the Box-Ball System

Problem Statement
The Box-Ball System (BBS), introduced by Takahashi and Satsuma, is a soliton cellular automaton whose dynamics can be linearized. Two primary linearization methods exist: the Kerov-Kirillov-Reshetikhin (KKR) bijection, which utilizes rigged Young diagrams and treats the system through the lens of quantum integrable systems (crystal bases), and the slot configuration method introduced by Ferrari, Nguyen, Rolla, and Wang (FNRW), which relies on soliton identification via the Takahashi-Satsuma (TS) algorithm and is useful for studying randomized BBS and generalized hydrodynamics. A significant open problem, noted in [FNRW], was the lack of a clarified relationship between the KKR bijection and the slot configuration. Furthermore, the slot configuration's definition via the TS algorithm is sequential only for finite configurations, limiting its applicability to general infinite configurations where soliton identification may fail.

Methodology
The authors introduce a novel intermediate framework called the seat number configuration to bridge these two linearization methods.

1. Carrier with Seat Numbers: They refine the standard carrier model (a carrier with capacity $\ell$) by indexing the carrier's seats with natural numbers $k \in \mathbb{N}$. The update rules are modified so that when a ball is picked up or put down, it occupies or vacates the seat with the smallest available number.
2. Seat Number Configuration: This process generates binary configurations $\eta^{\uparrow}_k$ and $\eta^{\downarrow}_k$, indicating whether a ball enters or exits the $k$-th seat at a specific site.
3. Sequential Construction: Unlike the slot configuration, which requires global information (or the TS algorithm) to define, the seat number configuration is defined sequentially based on the history of the ball configuration up to the current site.
4. Interlacing Young Diagrams: The authors construct a growing sequence of pairs of interlacing Young diagrams $(\mu^{\uparrow}(x), \mu^{\downarrow}(x))$ derived from the seat number configurations. They demonstrate that the KKR bijection corresponds to the projection of this pair onto the first component $(\mu^{\uparrow}, J^{\uparrow})$.

Key Contributions and Results

• Linearization via Seat Numbers: The authors prove that the seat number configuration linearizes the BBS dynamics. Specifically, they define "effective positions" $\xi_k(x)$ for solitons based on the seat numbers. They show that under time evolution $T$, these effective positions shift linearly by the soliton size $k$ (Theorem 2.1). This linearization holds for any configuration $\eta \in \Omega$, including those with infinite balls, without requiring the TS algorithm.
• Relation to KKR Bijection (Proposition 2.2): The paper establishes an explicit link between the seat number configuration and the KKR bijection. It proves that the partition $\lambda_k(x)$ (conjugate of the KKR partition) corresponds to the cumulative count of $(k, \uparrow)$-seats. Furthermore, the rigging values $J_{k,j}$ are shown to be determined by the "vacancy" functions derived from the seat numbers. This provides an intuitive, dynamical interpretation of the purely combinatorial KKR bijection.
• Relation to Slot Configuration (Proposition 2.3): The authors prove that for finite ball configurations, the seat number configuration is equivalent to the slot configuration. Specifically, a site $x$ is a $k$-slot in the FNRW sense if and only if it corresponds to a $(k, \uparrow)$-seat or a $(k, \downarrow)$-seat in the new framework. This allows the slot configuration to be constructed sequentially, resolving the limitation of the original definition for general configurations.
• Explicit Relation between KKR and Slots (Theorem 2.2): By combining the relations above, the paper derives an explicit formula connecting the KKR riggings $J$ to the slot decomposition $\tilde{\zeta}$. The elements of the rigging $J_{k,j}$ are identified as the effective positions of the $k$-solitons shifted by $k$.
• Linearization for Finite Capacity (Theorem 2.3): Using the established relations, the authors demonstrate that the slot configuration linearizes the BBS even when the carrier has finite capacity $\ell < \infty$. The time evolution of the slot decomposition follows the rule $(T_\ell \tilde{\zeta})_k(i) = \tilde{\zeta}_k(i - (k \wedge \ell))$.

Significance
The paper claims to resolve the open problem of clarifying the relationship between the KKR bijection and the slot configuration. Its significance lies in:

1. Unifying Perspectives: It connects the "classical" integrable system perspective (solitons and slot configurations) with the "quantum" integrable system perspective (KKR bijection and crystal bases) through the seat number configuration.
2. Generalization: It extends the definition of the slot configuration to general configurations (including those with infinite balls) where the TS algorithm fails, by showing the seat number configuration acts as a generalization.
3. Hydrodynamic Applications: The results provide a blueprint for generalizing slot-based linearization techniques to multi-color BBS models, which are currently difficult to analyze using existing KKR-based methods for rigorous hydrodynamic limits.
4. Finite Capacity Extension: It confirms that the linearization property of the slot decomposition holds for BBS with finite carrier capacity, a result necessary for deriving the generalized hydrodynamic limit for such systems.

The authors maintain a modest tone regarding future extensions, noting that while their results suggest a path for generalizing to multi-color models, such extensions are not yet available in the literature. They also clarify that their primary focus is the relationship between the two linearizations on the half-line, though they briefly note the adaptability of their methods to the whole line with appropriate offsets.