Tableau evacuation and webs

TL;DR

This paper explores the relationship between web symmetries and tableau operations, specifically connecting web reflection to tableau evacuation, extending known bijections between webs and tableaux.

Contribution

It introduces the correspondence between web reflection and tableau evacuation, expanding the understanding of web-tableau symmetries beyond rotation and promotion.

Findings

Web reflection corresponds to tableau evacuation.

Bijections relate web symmetries to tableau operations.

Extends previous work on web rotation and promotion.

Abstract

Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. There are explicit bijections between webs and certain rectangular tableaux. Work of Petersen-Pylyavskyy-Rhoades (2009) and Russell (2013) shows that these bijections relate web rotation to tableau promotion. We describe the analogous relation between web reflection and tableau evacuation.