Evacuation of rectangular standard Young tableaux corresponds to reflection of $\mathfrak{sl}_n$ webs

TL;DR

This paper establishes a correspondence between tableau evacuation and web graph reflection, extending known results for specific cases and introducing multicolored noncrossing matchings as an intermediate tool.

Contribution

It proves that evacuation of standard Young tableaux corresponds to reflection of associated web graphs, generalizing previous results for $n=2,3$ and connecting to strandings.

Findings

Evacuation corresponds to web reflection up to edge-flip relations.

Extension of known results for $n=2,3$ to general $n$.

Introduction of multicolored noncrossing matchings as an analytical tool.

Abstract

Web graphs form a family of planar directed graphs with boundary that can be used to model quantum $\mathfrak{sl}_n$-invariant vectors. Standard Young tableaux on an $n \times k$ rectangle naturally index a basis for $\mathfrak{sl}_n$ web graphs. We prove that evacuation of the tableau $T$ corresponds to reflection of the associated web graph $w_T$ up to equivalence under a specific set of edge-flip relations. This extends a result of Patrias and Pechenik for the cases $n=2,3$ and mirrors analogous results about rotation of web graphs corresponding to promotion of tableau by Peterson-Pylyavskyy-Rhoades for $n=3$ and Gaetz-Pechenik-Pfannerer-Striker-Swanson for $n=4$. We use an intermediate object called a multicolored noncrossing matching, which is closely related to the notion of strandings recently introduced by Russell and the fourth author.