Event Details

Box-ball systems and RSK

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. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record permutation statistics similar to the well-known Greene’s theorem statistics.

We will discuss a few new results:

(1) If the soliton decomposition of a permutation w is a standard tableau or if the BBS partition of w coincides with the Robinson--Schensted (RS) partition of w, then the soliton decomposition of w and the RS insertion tableau of w are equal.

(2) The RS recording tableau of a permutation completely determines the dynamics of the corresponding box-ball system.

(3) The permutations whose BBS soliton partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and noncrossing involutions.

(4) Finally, we study the permutations whose RS insertion tableaux and soliton decompositions coincide and refer to them as "good". We prove that these "good" permutations are closed under consecutive pattern containment. Furthermore, we conjecture that the "good" RS recording tableaux are counted by the Motzkin numbers.

This talk is based on joint work with students Ben Drucker, Eli Garcia, Aubrey Rumbolt, Rose Silver (arxiv.org/abs/2112.03780) and Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng (SUMRY 2021).