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

TL;DR

This work establishes a precise link between evacuation of rectangular standard Young tableaux and reflection of $\mathfrak{sl}_n$ web graphs. The authors construct a bridge via multicolored noncrossing matchings $\mathcal{M}^T$ derived from a tableau $T$, and define a reflection operation on both the matchings and the resulting webs. They prove that evacuation $E(T)$ corresponds to the reflected web $\varphi(w_T)$ up to a controlled edge-flip equivalence, generalizing earlier results for $n=2,3$ and aligning with promotion/reflection phenomena in related settings. The framework leverages Fontaine’s $\mathfrak{sl}_n$ web model to realize evacuation diagrammatically, and provides exact results in the standard $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$ conventions, thereby strengthening the connection between tableau combinatorics and $\mathfrak{sl}_n$ representation-theoretic webs with potential implications for categorification and invariant theory.

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.

Figures (14)

• Figure 1: Example of a tableau $T$ and its evacuation, plus the multicolored noncrossing matchings and web graphs associated to each. The second row has the reflection $\varphi(w_T)$ of $w_T$ over the line $x=5.5$ which differs from $w_{E(T)}$ only in a few edge directions and weights.
• Figure 2: Commutative diagram describing our argument, where $E$ denotes the evacuation map, $\varphi$ denotes reflection maps, and NCM abbreviates noncrossing matching
• Figure 3: Evacuation of a SYT of shape $3\times2$. The first row shows the original tableau $T$ (left) and its evacuation $E(T)$ (right). The second row shows the removal of the entry of the cell in the northwest corner — indicating the empty cell with $\bullet$ — followed by jeu de taquin slides. The third row shows the intermediate SYT after each iteration of jeu de taquin slides, indicating fixed, or negative, cells with green.
• Figure 4: Example of Lemma \ref{['PP']}, showing the original rectangular standard Young tableau $T$, the 180 degree rotation $\rho(T)$, and the evacuation $E(T)$.
• Figure 5: Example illustrating an NCM (left) and a crossing matching (right), on the set $\{1,2,3,4,5,6\}$. The matching on the left has arcs $\{(1,4),(2,3), (5,6)\}$ and the matching on the right has arcs $\{(1,3),(2,4),(5,6)\}$.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Definition 3
• Lemma 4: Patrias-Pechenik
• Definition 5
• Definition 6
• Definition 7
• Lemma 8
• Lemma 9
• proof