Iterating the RSK Bijection

TL;DR

This paper explores the iterative behavior of the RSK bijection on permutations, revealing fixed points and cycles depending on the reading word used, with implications for understanding its dynamical properties.

Contribution

It characterizes the fixed points and cycles of the RSK bijection under iteration on different reading words, including conditions for convergence to fixed points.

Findings

Unique fixed point per shape in row reading word case

Two-step convergence to fixed points or cycles

Self-conjugate shapes lead to fixed points in reversed reading word case

Abstract

We investigate the dynamics of the well-known RSK bijection on permutations when iterated on various reading words of the recording tableau. In the setting of the ordinary (row) reading word, we show that there is exactly one fixed point per partition shape, and that it is always reached within two steps from any starting permutation. We also consider the modified dynamical systems formed by iterating RSK on the column reading word and the reversed reading word of the recording tableau. We show that the column reading word gives similar dynamics to the row reading word. On the other hand, for the reversed reading word, we always reach either a 2-cycle or fixed point after two steps. In fact, we reach a fixed point if and only if the shape of the initial tableau is self-conjugate.