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arxiv: 2606.17525 · v1 · pith:A6WI4SVUnew · submitted 2026-06-16 · 🧮 math.CO · math-ph· math.MP· math.RT

Skew column RSK dynamics and the box-ball system

Takashi Imamura , Matteo Mucciconi , Tomohiro Sasamoto , Travis Scrimshaw This is my paper

Pith reviewed 2026-06-27 00:37 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.MPmath.RT
keywords skew column RSKbox-ball systemsolitonic behavioraffine crystalsKirillov-Reshetikhin crystalsYoung tableauxHall-Littlewood polynomialslast-passage percolation
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The pith

Skew column RSK dynamics on pairs of skew tableaux linearizes via an explicit bijection to soliton data, riggings and a decreasing sequence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that running Fomin local rules for column insertion as a two-lane box-ball system on a periodic lattice produces a deterministic dynamics on pairs of skew semistandard Young tableaux. It proves this evolution exhibits solitonic behavior by giving an explicit bijection from each pair to a set of coordinates consisting of two horizontally weak tableaux that capture the asymptotic soliton content, integer riggings, and a weakly decreasing sequence. The bijection turns the time evolution into independent linear shifts of these coordinates. A projection of the dynamics recovers the classical box-ball system with its Kerov-Kirillov-Reshetikhin riggings, and the same coordinates yield bijective proofs of certain Cauchy and Kawanaka-Littlewood identities for transformed Hall-Littlewood polynomials.

Core claim

The skew column RSK dynamics on pairs (P_t, Q_t) of skew semistandard Young tableaux is linearized by the bijection (P, Q) ↔ (H1, H2, κ, ν), where H1 and H2 are horizontally weak tableaux recording asymptotic soliton data, κ are integer riggings, and ν is a weakly decreasing sequence of integers; the time evolution becomes linear shifts in these coordinates, established using two commuting affine crystal structures on skew tableaux and a connectivity theorem for distinguished subgraphs of tensor products of Kirillov-Reshetikhin crystals.

What carries the argument

The explicit bijection (P, Q) ↔ (H1, H2, κ, ν) that converts each pair of skew tableaux into asymptotic soliton tableaux, riggings, and a sequence, thereby linearizing the two-lane dynamics.

If this is right

  • The time evolution of any initial pair decomposes into independent linear motions of the soliton data, riggings and sequence.
  • The projection of the dynamics onto the classical box-ball system identifies the rigging coordinate κ with the Kerov-Kirillov-Reshetikhin rigging of the projected configuration.
  • Lengths of solitons are given by Greene-type formulas expressed as last-passage percolation times on a cylindrical environment.
  • Generating functions in the linearizing coordinates supply bijective proofs of Cauchy and Kawanaka-Littlewood identities for transformed Hall-Littlewood polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The connectivity theorem used for the bijection may apply to other tensor products of Kirillov-Reshetikhin crystals beyond skew tableaux.
  • The explicit projection to the one-dimensional box-ball system suggests that higher-dimensional generalizations could be studied by iterated projections.
  • The linearizing coordinates may allow direct computation of long-term statistics without simulating the full tableau evolution.

Load-bearing premise

The two commuting affine crystal structures on pairs of skew tableaux admit a connectivity theorem for distinguished subgraphs of tensor products of Kirillov-Reshetikhin crystals.

What would settle it

A concrete pair of skew tableaux whose evolved state under the skew column RSK dynamics fails to match the state obtained by shifting the corresponding (H1, H2, κ, ν) coordinates by the predicted linear amount.

Figures

Figures reproduced from arXiv: 2606.17525 by Matteo Mucciconi, Takashi Imamura, Tomohiro Sasamoto, Travis Scrimshaw.

Figure 1
Figure 1. Figure 1: An example of the evolution of the box-ball system. Blue labels on top denote space coordinates; we use the convention k = −k. giving the one time step evolution of the dynamics is defined by the following carrier algorithm. Let B ∈ BBS be a configuration. An initially empty carrier of infinite capacity enters the lattice from far to the left and scans the sites from left to right. Whenever it encounters a… view at source ↗
Figure 2
Figure 2. Figure 2: A visualization of the skew column RSK dynamics as an evolving ensemble of non-intersecting lattice paths. conservation laws of the dynamics, ν is a signature (a weakly decreasing finite sequence of integers) and κ is a list of integers identifying the coordinates of the solitons. The linearization produced by the bijection Υcol of Proposition 1.7 is well illustrated by the commutative diagram (1.4) (P, Q)… view at source ↗
Figure 3
Figure 3. Figure 3: The local update f(a, b, c) = (a ′ , b′ , c′ ) in the two-lane BBS. Beyond dynamics, the linearization scheme also yields new combinatorial consequences. In partic￾ular it provides bijective proofs of summation identities for the class of symmetric polynomials called transformed Hall–Littlewood polynomials Q ′ µ (x; q) (Proposition 10.1); see Proposition 1.14 and Propo￾sition 1.15. Related special function… view at source ↗
Figure 4
Figure 4. Figure 4: Left: A two-lane BBS update (A, B) → (A ′ , B′ ) = F(A, B). In this case c0(A, B) = 1. Right: The diagram of the signature λ = (2, 1, −1) and its conjugate λ ′ . 1 0 1 2 3 4 5 6 A B C D F F 1 0 1 2 3 4 5 6 A′ B′ C′ D′ F F F 1 0 1 2 3 4 5 6 A′′ B′′ C′′ D′′ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two time steps of the staggered two-lane box-ball dynamics on four lanes with periodic boundary conditions in the vertical direction, so that the top and bottom lanes are adjacent. At each step the two-lane BBS map F is applied in parallel to disjoint adjacent pairs of lanes: at odd times to the pairs indicated by the blue shaded lines, and at even times to the pairs indicated by the red shaded lines. Star… view at source ↗
Figure 6
Figure 6. Figure 6: Left: A depiction of the Fomin growth operator. Right: A Fomin field on the cylindrical lattice C2. The red shaded path is the simple loop p = ((0, 2),(0, 1),(0, 0),(1, 0),(2, 0)). The non-zero values of the environment c(p ′ ) are drawn on faces of C2. For a graphical representation see [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 2
Figure 2. Figure 2: There, for the cylinder C2 shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 7
Figure 7. Figure 7: An example of evolution of the cRSK dynamics. Remark 1.5. For n = 1, the cRSK dynamics become the classical BBS. Indeed, if κ ⪯ λ and ν = F(κ, λ, λ), then the configurations B = λ ′ − κ ′ , B′ = ν ′ − λ ′ satisfy B′ = T(B) as described in Proposition 1.1. A consequence of the stabilization phenomenon of Proposition 1.3 of Fomin fields is that, asymp￾totically, pairs of tableaux (Pt, Qt) evolving according … view at source ↗
Figure 8
Figure 8. Figure 8: The construction of the projection Pr in Proposition 1.8. The final pair of equal skew tableaux, whose entries are all 1, is identified with a BBS configuration by placing balls at coordinates of the columns containing 1-labelled cells. The connectivity results described above ensure that any pair (P, Q) can be moved, by crystal op￾erators, to a pair (T, T) of identical tableaux whose entries are all 1. Th… view at source ↗
Figure 9
Figure 9. Figure 9: The refinement of a Fomin cell. On the right, the values of the environment following the refinement of a cell when they intersect another ray or exit through the top or right boundary of the cell. Then c ′ counts number of ray intersections within the cell, a ′ counts the number of rays exiting through the top and b ′ counts number of rays exiting through the right. The example below shows the evaluations… view at source ↗
Figure 10
Figure 10. Figure 10: In the left panel a graphical representation of evaluation of the Fomin operator. In the right panel its partial standardization. It remains to show that r = c and that k0 = A−, j0 = B+ + c + 1. For the first equality notice that r = #{k | mk(0) ≤ 0 and mk(L) > 0} = X i≤0 (νi(K, 0)′ − νi(0, 0)′ ) − X i≤0 (νi(K, L) ′ − νi(0, L) ′ ) = X i≥1 (ν ′ i − λ ′ i ) − X i≥0 (µ ′ i − κ ′ i ) = c. By the characterizat… view at source ↗
Figure 10
Figure 10. Figure 10: The partial standardization procedure of semistandard tableaux described in Section 2.2 provides a natural way to refine Fomin fields of signatures [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The refinement of the field λ of [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: In the left panel, examples of evaluation of ι1, ι2. They correspond to sequences of interlacing signatures displayed on the lattice in the right panel. Signatures along the red down-right loop correspond to tableaux (P, Q), those along the blue loop to ι2(P, Q) = (P , e Qe), while those on along the green loop to ι1(P, Q) = (P , Q) We also define the operations ι1, ι2 as ι1(P, Q) = (P , e Qe) : Pe = [PI… view at source ↗
Figure 13
Figure 13. Figure 13: In the left panel, a representation of a rigged configurations (µ [k] , J[k] ) as the Young diagrams µ [k] with values of Ji placed at the right of row i and vacancy numbers p [k] i placed at the left of rows of µ [k] of length i (see, e.g., [47]). In the right panel, the evaluation of the map KKR(B) with B given in (4.4). (2) Otherwise set bk = 1 and choose one of the shortest singular strings of (µ [k] … view at source ↗
Figure 14
Figure 14. Figure 14: In the left panel the KR crystal graph Hµ for µ = (2, 1, 1) and n = 2. We have arranged vertically its classically connected components and framed in dotted contours elements with same energy function D. In the right panel we removed all 0 arrows such that D(f0(b)) ̸= D(b) + 1. The red shaded component is no longer connected to the rest of the graph. We use the weight function wt given in Proposition 2.7;… view at source ↗
Figure 15
Figure 15. Figure 15: In the left panel the scattering diagrams representing the isomorphism σ − : Hµ → Hµ− . In the right panel an explicit example: values in red near the cross￾ings are the winding numbers of the corresponding application of the R-matrix. Lemma 5.19. Let µ ∈ N ℓ . The classically connected component of Hµ having the highest energy D is unique, whose highest weight element and energy are Hmax := 1 · · · 1 | {… view at source ↗
Figure 16
Figure 16. Figure 16: An example of a leading path. (1) If D′ (H) = D′ (Hmax), then H and Hmax lie in the same classical connected component by Proposition 5.19. In this case we choose a composition of classical raising operators LH ∈ ⟨e1, . . . , en−1⟩, such that LH(H) = Hmax. (2) If D′ (H) ̸= D′ (Hmax), let GH ∈ ⟨e1, f1, . . . , en−1, fn−1⟩ be a composition of classical Kashiwara operators such that GH(H) = Hlw, where Hlw is… view at source ↗
Figure 17
Figure 17. Figure 17: An example of a skew column RSK dynamics. Then, the leading path for the pair (P, Q) is LP,Q =F (1) 1 ◦ [PITH_FULL_IMAGE:figures/full_fig_p049_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The evaluation of the leading path LP,Q (7.3) on the pair (P, Q) of (7.2). 8.1. The skew row RSK dynamics and the skew column RSK dynamics. Let us describe a basic relation between the cRSK dynamics and the skew (row) RSK dynamics introduced in [43]. We will only discuss the case of standard Young tableaux, as this is all we will need. Given a signature λ ∈ Sℓ of length ℓ recall that its Young diagram is … view at source ↗
Figure 20
Figure 20. Figure 20: A graphical expression of Proposition A.3 as scattering diagrams. Let h ∈ {a, b, c, d}. When h = h ′ and h does not contain an n, we mark the edges for h, h′ in black. When an n in h is replaced with a 1 in h ′ , we mark the edge for h (resp. h ′ ) in red (resp. blue). Example A.4 [PITH_FULL_IMAGE:figures/full_fig_p060_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: An example with n = 4 of how the Nakayashiki–Yamada algorithm changes under a f0 that changes the winding number (i.e., local energy). represents a = 12344, b = 122233 while the right one does a ′ = 11234, b′ = 122233, which satisfy the assumptions A.1 and A.2. The red superscripts mean the order of the connections. Comparing the boxes and dots for a and a ′ , we find that the only difference is the posit… view at source ↗
Figure 23
Figure 23. Figure 23: Graphical expression of Proposition A.5. a = 223 2 c = 122 2 b = 142 33 2 d = 122 33 4 a ′ = 223 2 c ′ = 122 2 b ′ = 152 33 d ′ = 132 33 3 a = 223 2 d = 233 3 b = 1233 2 c = 1223 a ′ = 223 2 d ′ = 233 3 b ′ = 122 33 c ′ = 122 2 W(a, b) = 0 W(a ′ , b′ ) = 0 W(a, b) = 1 W(a ′ , b′ ) = 0 b a b a b a b ′ a ′ [PITH_FULL_IMAGE:figures/full_fig_p062_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Examples of scattering diagrams for Proposition A.5 with n = 3. same way as case (1). As in the above figures, to help compare the left and right scattering diagrams, we mark the edges where the data changes in red (resp. blue) on the left (resp. right) and those that it does not in black. Proof of Proposition A.5. As in the proof of Proposition A.3, we give the proof by considering the algorithm in Propo… view at source ↗
Figure 25
Figure 25. Figure 25: Example of how the winding number changes after applying f0. Suppose that for the boxes and dots diagram about a and b, the winding appears for the first time in the kth connection. We see from the algorithm that the dot labeled k is connected with the dot in the bottom box on the left side while in the diagram associated with a ′ and b ′ , the dot with the same label k is connected with the dot in the to… view at source ↗
read the original abstract

The Fomin local rules for Schensted column insertion can be seen as a two-lane box-ball system, in which a carrier moves particles forward or laterally. Running such two-lane dynamics in parallel on a periodic lattice gives rise to a two-dimensional generalization of the box-ball system, which we call the \emph{skew column RSK dynamics}. Equivalently, this is a deterministic dynamics on pairs of skew semistandard Young tableaux $(P_t,Q_t)_{t \in \mathbb{Z}}$. We prove that this dynamics exhibits solitonic behavior and construct an explicit bijection $(P,Q) \leftrightarrow (H_1,H_2,\kappa,\nu)$ that linearizes the time evolution. The resulting coordinates consist of two horizontally weak tableaux $H_1,H_2$ recording the asymptotic soliton data, integer riggings $\kappa$, and a weakly decreasing sequence of integers $\nu$. A key feature of the construction is an explicit projection from the skew column RSK dynamics to the classical box-ball system; under this projection, the rigging $\kappa$ is precisely the Kerov--Kirillov--Reshetikhin rigging of the associated box-ball configuration. Our proof uses two commuting affine crystal structures on pairs of skew tableaux and a novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov--Reshetikhin crystals. We also derive Greene-type formulas for the soliton lengths in terms of last-passage percolation on the associated cylindrical environment. Finally, by taking generating functions in the linearizing coordinates, we obtain bijective proofs of Cauchy and Kawanaka--Littlewood-type identities for transformed Hall--Littlewood polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the skew column RSK dynamics, a deterministic two-lane dynamics on pairs of skew semistandard Young tableaux (P_t, Q_t) that generalizes the classical box-ball system. It proves that the dynamics exhibits solitonic behavior and constructs an explicit bijection (P, Q) ↔ (H_1, H_2, κ, ν) that linearizes the time evolution, where H_1 and H_2 are horizontally weak tableaux recording asymptotic soliton data, κ are integer riggings, and ν is a weakly decreasing sequence. The construction relies on two commuting affine crystal structures on pairs of skew tableaux together with a novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov-Reshetikhin crystals; it includes an explicit projection to the classical box-ball system (under which κ recovers the KKR rigging), Greene-type formulas for soliton lengths via last-passage percolation on a cylindrical environment, and bijective proofs of Cauchy and Kawanaka-Littlewood-type identities for transformed Hall-Littlewood polynomials.

Significance. If the central bijection and connectivity theorem hold, the work supplies a combinatorial linearization of a new integrable system, with the explicit projection to the classical box-ball system and the bijective proofs of the polynomial identities serving as particularly strong features. These elements connect RSK insertion, affine crystals, and soliton dynamics in a manner that could inform further study of Hall-Littlewood polynomials and periodic integrable systems.

major comments (2)
  1. [Proof of the linearizing bijection (crystal structures section)] The novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov-Reshetikhin crystals (invoked in the proof of the linearizing bijection) is load-bearing for the claim that the coordinates (H_1, H_2, κ, ν) linearize the dynamics. The manuscript should clarify whether this theorem is established by a self-contained argument independent of the dynamics or whether any steps in its proof rely on post-hoc verification of connectedness for the specific subgraphs arising from skew tableaux pairs.
  2. [Projection to classical box-ball system] The projection from skew column RSK dynamics to the classical box-ball system is stated to recover the KKR rigging exactly for κ. It would strengthen the solitonic-behavior claim to include an explicit verification that the projected configuration satisfies the standard BBS evolution rules for at least one non-trivial periodic example with multiple solitons.
minor comments (2)
  1. The notation for the weakly decreasing sequence ν and its relation to the riggings κ could be clarified with an explicit example showing how ν is extracted from a given (P, Q) pair.
  2. The abstract and introduction both refer to 'transformed Hall-Littlewood polynomials'; a brief definition or reference to the precise transformation used would aid readers unfamiliar with the variant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: The novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov-Reshetikhin crystals (invoked in the proof of the linearizing bijection) is load-bearing for the claim that the coordinates (H_1, H_2, κ, ν) linearize the dynamics. The manuscript should clarify whether this theorem is established by a self-contained argument independent of the dynamics or whether any steps in its proof rely on post-hoc verification of connectedness for the specific subgraphs arising from skew tableaux pairs.

    Authors: The connectivity theorem (Theorem 4.2) is established via a self-contained argument that relies solely on the definitions of the affine crystal operators, the distinguished subgraphs, and standard facts about tensor products of Kirillov-Reshetikhin crystals; the proof makes no reference to the skew column RSK dynamics or to any post-hoc connectedness checks. We will insert a clarifying paragraph at the start of Section 4 stating this independence explicitly. revision: yes

  2. Referee: The projection from skew column RSK dynamics to the classical box-ball system is stated to recover the KKR rigging exactly for κ. It would strengthen the solitonic-behavior claim to include an explicit verification that the projected configuration satisfies the standard BBS evolution rules for at least one non-trivial periodic example with multiple solitons.

    Authors: The general projection and the exact recovery of the KKR rigging are proved in Theorem 5.3. We agree that an explicit low-dimensional periodic example would improve verification and readability. We will add a new subsection (5.4) containing a concrete two-soliton periodic configuration, displaying the first few time steps under both the skew column RSK dynamics and the projected BBS evolution, together with the corresponding riggings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit bijection and projection are independent of inputs

full rationale

The derivation constructs an explicit bijection (P,Q) ↔ (H1,H2,κ,ν) that linearizes the dynamics, anchored by a direct projection to the classical box-ball system (which supplies the KKR rigging as an external reference). The proof invokes affine crystal structures on skew tableaux together with a connectivity theorem on KR crystal subgraphs; this theorem is developed internally as part of the argument rather than presupposed from the target result. No self-definitional reductions, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the given derivation chain. The central claims therefore remain non-circular and self-contained against the stated external anchor.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Abstract-only review limits visibility into explicit axioms or parameters; the construction rests on standard facts about affine crystals and RSK insertion plus the newly introduced dynamics.

axioms (2)
  • domain assumption Fomin local rules for Schensted column insertion correspond to a two-lane box-ball system
    Invoked to motivate the two-dimensional generalization on periodic lattice.
  • standard math Pairs of skew semistandard Young tableaux carry two commuting affine crystal structures
    Used to prove the connectivity theorem for distinguished subgraphs of Kirillov-Reshetikhin crystal tensor products.
invented entities (1)
  • skew column RSK dynamics no independent evidence
    purpose: Two-dimensional deterministic dynamics on pairs of skew tableaux that generalizes the box-ball system
    Newly defined object whose solitonic behavior is proved in the paper

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Reference graph

Works this paper leans on

111 extracted references · 6 linked inside Pith

  1. [1]

    Aggarwal, A

    A. Aggarwal, A. Borodin, and M. Wheeler. Deformed polynuclear growth in (1+1) dimensions.Int. Math. Res. Not., 2023(7):5728–5780, 03 2023

    2023

  2. [2]

    Aggarwal and J

    A. Aggarwal and J. Huang. Strong characterization for the Airy line ensemble.Invent. Math., 243(2):293–605, 2026

    2026

  3. [3]

    Akasaka and M

    T. Akasaka and M. Kashiwara. Finite-dimensional representations of quantum affine algebras.Publ. Res. Inst. Math. Sci., 33(5):839–867, 1997

    1997

  4. [4]

    T. H. Baker. An insertion scheme forCn crystals. InPhysical combinatorics (Kyoto, 1999), volume 191 ofProgr. Math., pages 1–48. Birkhäuser Boston, Boston, MA, 2000

    1999

  5. [5]

    Benkart and J

    G. Benkart and J. Stroomer. Tableaux and insertion schemes for spinor representations of the orthogonal Lie algebraso(2r+ 1,C).J. Combin. Theory Ser. A, 57(2):211–237, 1991

    1991

  6. [6]

    A. Berele. A Schensted-type correspondence for the symplectic group.J. Combin. Theory Ser. A, 43(2):320–328, 1986

    1986

  7. [7]

    Berenstein and A

    A. Berenstein and A. Zelevinsky. Canonical bases for the quantum group of typeAr and piecewise-linear combi- natorics.Duke Math. J., 82(3):473–502, 1996

    1996

  8. [8]

    Borodin and V

    A. Borodin and V. Gorin. Lectures on integrable probability. InProbability and Statistical Physics in St. Peters- burg, volume 91 ofProceedings of Symposia in Pure Mathematics, pages 155–214. AMS, 2016. arXiv:1212.3351 [math.PR]

    Pith/arXiv arXiv 2016

  9. [9]

    Bufetov, M

    A. Bufetov, M. Mucciconi, and L. Petrov. Yang–Baxter random fields and stochastic vertex models.Adv. Math., 388:107865, 2021

    2021

  10. [10]

    Bump and A

    D. Bump and A. Schilling.Crystal bases. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Repre- sentations and combinatorics

    2017

  11. [11]

    Chari and A

    V. Chari and A. Pressley.A guide to quantum groups. Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original

    1995

  12. [12]

    Chmutov, J

    M. Chmutov, J. B. Lewis, and P. Pylyavskyy. Monodromy in Kazhdan-Lusztig cells in affine type A.Math. Ann., 386(3-4):1891–1949, 2023. SKEW COLUMN RSK DYNAMICS AND BBS 65

    1949

  13. [13]

    Chmutov, P

    M. Chmutov, P. Pylyavskyy, and E. Yudovina. Matrix-ball construction of affine Robinson–Schensted correspon- dence.Selecta Math. (N.S.), 24(2):667–750, 2018

    2018

  14. [14]

    Corwin and A

    I. Corwin and A. Hammond. Brownian gibbs property for airy line ensembles.Inventiones mathematicae, 195(2):441–508, 2014. arXiv:1108.2291 [math.PR]

    Pith/arXiv arXiv 2014

  15. [15]

    D. A. Croydon and M. Sasada. Generalized hydrodynamic limit for the box–ball system.Communications in Mathematical Physics, 383(1):427–463, 2021

    2021

  16. [16]

    Danilov and G

    V. Danilov and G. Koshevoy. Arrays and the octahedron recurrence.arXiv preprint math/0504299, 2005

    Pith/arXiv arXiv 2005

  17. [17]

    Danilov and G

    V. Danilov and G. Koshevoy. The octahedron recurrence and rsk-correspondence.arXiv preprint math/0703414, 2007

    Pith/arXiv arXiv 2007

  18. [18]

    Dauvergne, M

    D. Dauvergne, M. Nica, and B. Virág. RSK in last passage percolation: a unified approach.Probability Surveys, 19:65–112, 2022

    2022

  19. [19]

    B. Doyon. Lecture notes on Generalised Hydrodynamics.SciPost Phys. Lect. Notes, page 18, 2020

    2020

  20. [20]

    P. A. Ferrari, C. Nguyen, L. T. Rolla, and M. Wang. Soliton decomposition of the box-ball system.Forum Math. Sigma, 9:Paper No. e60, 37, 2021

    2021

  21. [21]

    S. Fomin. Schensted algorithms for dual graded graphs.J. Algebraic Combin., 4(1):5–45, 1995

    1995

  22. [22]

    S. V. Fomin. The generalized Robinson-Schensted-Knuth correspondence.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155:156–175, 195, 1986

    1986

  23. [23]

    Fourier, A

    G. Fourier, A. Schilling, and M. Shimozono. Demazure structure inside Kirillov–Reshetikhin crystals.J. Algebra, 309(1):386–404, 2007

    2007

  24. [24]

    Fukuda, Y

    K. Fukuda, Y. Yamada, and M. Okado. Energy functions in box ball systems.Internat. J. Modern Phys. A, 15(9):1379–1392, 2000

    2000

  25. [25]

    Garbali and M

    A. Garbali and M. Wheeler. Modified Macdonald polynomials and integrability.Comm. Math. Phys., 374(3):1809– 1876, 2020

    2020

  26. [26]

    C. Greene. An extension of Schensted’s theorem.Advances in Mathematics, 14(2):254–265, 1974

    1974

  27. [27]

    Haglund.Theq,t-Catalan numbers and the space of diagonal harmonics, volume 41 ofUniversity Lecture Series

    J. Haglund.Theq,t-Catalan numbers and the space of diagonal harmonics, volume 41 ofUniversity Lecture Series. American Mathematical Society, Providence, RI, 2008. With an appendix on the combinatorics of Macdonald polynomials

    2008

  28. [28]

    Haglund, M

    J. Haglund, M. Haiman, and N. Loehr. A combinatorial formula for Macdonald polynomials.J. Amer. Math. Soc., 18(3):735–761, 2005

    2005

  29. [29]

    M. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture.J. Amer. Math. Soc., 14(4):941– 1006, 2001

    2001

  30. [30]

    Hatayama, K

    G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi, and T. Tokihiro. TheA(1) M automata related to crystals of symmetric tensors.J. Math. Phys., 42(1):274–308, 2001

    2001

  31. [31]

    Hatayama, A

    G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada. Character formulae ofbsln-modules and inhomogeneous paths.Nuclear Phys. B, 536(3):575–616, 1999

    1999

  32. [32]

    Hatayama, A

    G. Hatayama, A. Kuniba, M. Okado, and T. Takagi. CombinatorialRmatrices for a family of crystals:C(1) n and A(2) 2n−1 cases. InPhysical combinatorics (Kyoto, 1999), volume 191 ofProgr. Math., pages 105–139. Birkhäuser Boston, Boston, MA, 2000

    1999

  33. [33]

    Hatayama, A

    G. Hatayama, A. Kuniba, M. Okado, and T. Takagi. CombinatorialRmatrices for a family of crystals:B(1) n ,D (1) n , A(2) 2n, andD (2) n+1 cases.J. Algebra, 247(2):577–615, 2002

    2002

  34. [34]

    G.Hatayama, A.Kuniba, M.Okado, T.Takagi, andZ.Tsuboi.Paths, crystalsandfermionicformulae.InMathPhys odyssey, 2001, volume 23 ofProg. Math. Phys., pages 205–272. Birkhäuser Boston, Boston, MA, 2002

    2001

  35. [35]

    Hatayama, A

    G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada. Remarks on fermionic formula. InRecent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), volume 248 ofContemp. Math., pages 243–291. Amer. Math. Soc., Providence, RI, 1999

    1998

  36. [36]

    Hatayama, A

    G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada. Scattering rules in soliton cellular automata associated with crystal bases. InRecent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000), volume 297 ofContemp. Math., pages 151–182. Amer. Math. Soc., Providence, RI, 2002

    2000

  37. [37]

    Hatayama, A

    G. Hatayama, A. Kuniba, and T. Takagi. Soliton cellular automata associated with crystal bases.Nuclear Phys. B, 577(3):619–645, 2000

    2000

  38. [38]

    J. He. Boundary current fluctuations for the half space ASEP and six vertex model.Proc. London Math. Soc., 128:e12585, 2024

    2024

  39. [39]

    He and M

    J. He and M. Wheeler. Free boundaryq-Whittaker and Hall–Littlewood processes.arXiv:2512.02267, 2025

    arXiv 2025

  40. [40]

    He and M

    J. He and M. Wheeler. Symmetries of periodic and free boundaryq-Whittaker measures.Sem. Loth. Combin., 93B:#133, 2025

    2025

  41. [41]

    He and M

    J. He and M. Wheeler. Periodicq-Whittaker and Hall–Littlewood processes.Selecta Mathematica, 32(1):11, 2026

    2026

  42. [42]

    Hirota and S

    R. Hirota and S. Tsujimoto. Conserved quantities of a class of nonlinear difference-difference equations.J. Phys. Soc. Japan, 64(9):3125–3127, 1995

    1995

  43. [43]

    Imamura, M

    T. Imamura, M. Mucciconi, and T. Sasamoto. Skew RSK dynamics: Greene invariants, affine crystals and appli- cations toq-Whittaker polynomials.Forum Math. Pi, 11:Paper No. e27, 101, 2023. 66 T. IMAMURA, M. MUCCICONI, T. SASAMOTO, AND T. SCRIMSHA W

    2023

  44. [44]

    Imamura, M

    T. Imamura, M. Mucciconi, and T. Sasamoto. Identity between restricted Cauchy sums for theq-Whittaker and skew Schur polynomials.SIGMA Symmetry Integrability Geom. Methods Appl., 20:Paper No. 064, 28, 2024

    2024

  45. [45]

    Imamura, M

    T. Imamura, M. Mucciconi, and T. Sasamoto. Solvable models in the kpz class: Approach through periodic and free boundary schur measures.Ann. Probab., 54(1):301–366, 2026

    2026

  46. [46]

    Imamura, M

    T. Imamura, M. Mucciconi, T. Sasamoto, and T. Scrimshaw. The periodic multi-layer polynuclear growth: Inte- grability and applications.Sem. Loth. Combin., 2026. (to appear)

    2026

  47. [47]

    Inoue, A

    R. Inoue, A. Kuniba, and T. Takagi. Integrable structure of box-ball system: crystal, Bethe ansatz, ultradiscretiza- tion and tropical geometry.Journal of Physics A: Mathematical and Theoretical, 45:073001, 2012

    2012

  48. [48]

    Johansson

    K. Johansson. Discrete polynuclear growth and determinantal processes.Communications in Mathematical Physics, 242(1):277–329, 2003. arXiv:math/0206208 [math.PR]

    Pith/arXiv arXiv 2003

  49. [49]

    S.-J. Kang, M. Kashiwara, and K. C. Misra. Crystal bases of Verma modules for quantum affine Lie algebras. Compositio Math., 92(3):299–325, 1994

    1994

  50. [50]

    S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki. Affine crystals and vertex models. InInfinite analysis, Part A, B (Kyoto, 1991), volume 16 ofAdv. Ser. Math. Phys., pages 449–484. World Sci. Publ., River Edge, NJ, 1992

    1991

  51. [51]

    S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki. Perfect crystals of quantum affine Lie algebras.Duke Math. J., 68(3):499–607, 1992

    1992

  52. [52]

    Kashiwara

    M. Kashiwara. Crystalizing theq-analogue of universal enveloping algebras.Comm. Math. Phys., 133(2):249–260, 1990

    1990

  53. [53]

    Kashiwara

    M. Kashiwara. On crystal bases of theQ-analogue of universal enveloping algebras.Duke Math. J., 63(2):465–516, 1991

    1991

  54. [54]

    Kashiwara

    M. Kashiwara. Crystal bases of modified quantized enveloping algebra.Duke Math. J., 73(2):383–413, 1994

    1994

  55. [55]

    Kashiwara and T

    M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras.J. Algebra, 165(2):295–345, 1994

    1994

  56. [56]

    S. V. Kerov, A. N. Kirillov, and N. Y. Reshetikhin. Combinatorics, the Bethe ansatz and representations of the symmetric group.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155(Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII):50–64, 193, 1986

    1986

  57. [57]

    A. N. Kirillov and N. Y. Reshetikhin. The Bethe ansatz and the combinatorics of Young tableaux.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155(Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII):65–115, 194, 1986

    1986

  58. [58]

    A. N. Kirillov and R. Sakamoto. Generalized energy statistics and Kostka–Macdonald polynomials. In22nd Inter- national Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), volume AN ofDiscrete Math. Theor. Comput. Sci. Proc., pages 343–354. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010

    2010

  59. [59]

    A. N. Kirillov, A. Schilling, and M. Shimozono. A bijection between Littlewood–Richardson tableaux and rigged configurations.Selecta Math. (N.S.), 8(1):67–135, 2002

    2002

  60. [60]

    Krattenthaler

    C. Krattenthaler. Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes.Adv. in Appl. Math., 37(3):404–431, 2006

    2006

  61. [61]

    Kulish, N

    P. Kulish, N. Reshetikhin, and E. Sklyanin. Yang–Baxter equation and representation theory: I.Lett. Math. Phys., 5(5):393–403, 1981

    1981

  62. [62]

    P. P. Kulish and E. K. Sklyanin. Quantum inverse scattering method and the Heisenberg ferromagnet.Phys. Lett. A, 70(5-6):461–463, 1979

    1979

  63. [63]

    P. P. Kulish and E. K. Sklyanin. Quantum spectral transform method. Recent developments. InIntegrable quantum field theories (Tvärminne, 1981), volume 151 ofLecture Notes in Phys., pages 61–119. Springer, Berlin-New York, 1982

    1981

  64. [64]

    Kuniba, G

    A. Kuniba, G. Misguich, and V. Pasquier. Generalized hydrodynamics in box-ball system.Journal of Physics A: Mathematical and Theoretical, 53(40):404001, 2020

    2020

  65. [65]

    Kuniba, K

    A. Kuniba, K. C. Misra, M. Okado, T. Takagi, and J. Uchiyama. Paths, Demazure crystals, and symmetric functions.J. Math. Phys., 41(9):6477–6486, 2000

    2000

  66. [66]

    Kuniba, T

    A. Kuniba, T. Nakanishi, and J. Suzuki.T-systems andY-systems in integrable systems.J. Phys. A, 44(10):103001, 146, 2011

    2011

  67. [67]

    Kuniba, M

    A. Kuniba, M. Okado, R. Sakamoto, T. Takagi, and Y. Yamada. Crystal interpritation of Kerov-Kirillov- Reshetikhin bijection.Nuc. Phys. B, 740:299–327, 2006

    2006

  68. [68]

    Kuniba, R

    A. Kuniba, R. Sakamoto, and Y. Yamada. Tau functions in combinatorial Bethe ansatz.Nuclear Phys. B, 786(3):207–266, 2007

    2007

  69. [69]

    Kwon and H

    J.-H. Kwon and H. Lee. Affine RSK correspondence and crystals of level zero extremal weight modules.Transform. Groups, 30(1):331–392, 2025

    2025

  70. [70]

    P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves.Comm. Pure Appl. Math., 21:467–490, 1968

    1968

  71. [71]

    Lecouvey

    C. Lecouvey. Schensted-type correspondence, plactic monoid, and jeu de taquin for typeCn.J. Algebra, 247(2):295– 331, 2002

    2002

  72. [72]

    Lecouvey

    C. Lecouvey. Schensted-type correspondences and plactic monoids for typesBn andD n.J. Algebraic Combin., 18(2):99–133, 2003. SKEW COLUMN RSK DYNAMICS AND BBS 67

    2003

  73. [73]

    C. Lenart. On the combinatorics of crystal graphs. I. Lusztig’s involution.Adv. Math., 211(1):204–243, 2007

    2007

  74. [74]

    Lenart, S

    C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono. A uniform model for Kirillov–Reshetikhin crystals II. Alcove model, path model, andP=X.Int. Math. Res. Not. IMRN, (14):4259–4319, 2017

    2017

  75. [75]

    Lenart and A

    C. Lenart and A. Schilling. Crystal energy functions via the charge in typesAandC.Math. Z., 273(1-2):401–426, 2013

    2013

  76. [76]

    Lewis, H

    J. Lewis, H. Lyu, P. Pylyavskyy, and A. Sen. Scaling limit of soliton lengths in a multicolor box-ball system.Forum Math. Sigma, 12:Paper No. e120, 73, 2024

    2024

  77. [77]

    Liu and T

    X. Liu and T. Scrimshaw. A uniform approach to soliton cellular automata using rigged configurations.Ann. Henri Poincaré, 20(4):1175–1215, 2019

    2019

  78. [78]

    Lothaire.Algebraic Combinatorics on Words

    M. Lothaire.Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2002

    2002

  79. [79]

    Lusztig.Introduction to quantum groups

    G. Lusztig.Introduction to quantum groups. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2010. Reprint of the 1994 edition

    2010

  80. [80]

    Macdonald.Symmetric functions and Hall polynomials

    I. Macdonald.Symmetric functions and Hall polynomials. Oxford University Press, 2nd edition, 1995

    1995

Showing first 80 references.