Title: Box-ball systems, RSK, and Motzkin paths
| Speaker: | Emily Gunawan |
| Affiliation: | University of Oklahoma |
| Location: | MC 5479, please contact Olya Mandelshtam for Zoom link. |
Abstract: A box-ball system (BBS) is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on n objects gives a BBS 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 classical Greene’s theorem statistics.
The well-known Robinson—Schensted algorithm is a bijection from permutations w to pairs of standard tableaux P(w), Q(w) of the same shape. We will discuss a few new results which relates BBS to these P and Q tableaux:
(1) The soliton decomposition of a permutation w is a standard tableau if and only if it is equal to P(w).
(2) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system.
(3) We present a bijection between Motzkin paths and a class of involutions whose soliton decompositions are standard.
This talk is based on joint work with B. Drucker, E. Garcia, A. Rumbolt, R. L. Silver (arxiv.org/abs/2112.03780); M. Cofie, O. Fugikawa, M. Stewart, D. Zeng (SUMRY 2021); and S. Hong, M. Li, R. Okonogi-Neth, M. Sapronov, D. Stevanovich, and H. Weingord (SUMRY 2022).