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arXiv · 2112.03780

RSK tableaux and box-ball systems

Ben Drucker·Eli Garcia·Emily Gunawan·Aubrey Rumbolt·Rose Silver
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Abstract

A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation w is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of w, then the soliton decomposition of w and the RS insertion tableau of w are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.

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Ben Drucker, Eli Garcia, Emily Gunawan, Aubrey Rumbolt, Rose Silver. 2021-12-07. RSK tableaux and box-ball systems. https://doi.org/10.5070/c63261978

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