Piecewise-linear promotion and RSK in rectangles and moon polyominoes

TL;DR

This paper develops piecewise-linear and birational versions of combinatorial maps related to promotion, evacuation, and RSK, demonstrating their effects on chain statistics and establishing Ehrhart equivalence for certain polytopes.

Contribution

It introduces new piecewise-linear and birational lifts of promotion, evacuation, and RSK, and constructs bijections between fillings of moon polyominoes that preserve chain statistics.

Findings

Chain statistics shift predictably under the maps.

Constructed bijections preserve chain statistics.

Established Ehrhart equivalence and period collapse for certain polytopes.

Abstract

We study piecewise-linear and birational lifts of Sch\"utzenberger promotion, evacuation, and the RSK correspondence defined in terms of toggles. Using this perspective, we prove that certain chain statistics in rectangles shift predictably under the action of these maps. We then use this to construct piecewise-linear and birational versions of Rubey's bijections between fillings of equivalent moon polyominoes that preserve these chain statistics, and we show that these maps form a commuting diagram. We also discuss how these results imply Ehrhart equivalence and Ehrhart quasi-polynomial period collapse of certain analogues of chain polytopes for moon polyominoes.