[Paper Review] A simple bijection between standard (n,n,n) tableaux and irreducible webs for sl_3

Julianna Tymoczko|arXiv (Cornell University)|May 26, 2010

Advanced Combinatorial Mathematics7 references4 citations

TL;DR

This paper presents a simple, explicit bijection between standard Young tableaux of shape $3\times n$ and irreducible webs for $\mathfrak{sl}_3$ with all boundary vertices as sources, using an intermediary object called an $\mathsf{m}$-diagram. The construction matches Khovanov-Kuperberg's recursive map and provides a geometric proof of the correspondence between web rotation and jeu-de-taquin promotion, while also introducing a shuffle operation on tableaux that corresponds to web joins.

ABSTRACT

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and sl_2. Building on a result of Kuperberg's, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape (n,n,n) and irreducible webs for sl_3 whose boundary vertices are all sources. In this paper, we give a simple and explicit map from standard Young tableaux of shape (n,n,n) to irreducible webs for sl_3 whose boundary vertices are all sources, and show that it is the same as Khovanov-Kuperberg's map. Our construction generalizes to some webs with both sources and sinks on the boundary. Moreover, it allows us to extend the correspondence between webs and tableaux in two ways. First, we provide a short, geometric proof of Petersen-Pylyavskyy-Rhoades's recent result that rotation of webs corresponds to jeu-de-taquin promotion on (n,n,n) tableaux. Second, we define another natural operation on tableaux called a shuffle, and show that it corresponds to the join of two webs. Our main tool is an intermediary object between tableaux and webs that we call an m-diagram. The construction of m-diagrams, like many of our results, applies to shapes of tableaux other than (n,n,n).

Motivation & Objective

• To provide a direct, non-recursive bijection between standard $3\times n$ Young tableaux and irreducible $\mathfrak{sl}_3$ webs with all boundary vertices as sources.
• To establish a geometric connection between web rotation and jeu-de-taquin promotion on $3\times n$ tableaux.
• To define a new tableau operation called 'shuffle' and show its correspondence to the join of two webs.
• To generalize the correspondence beyond $3\times n$ tableaux using $\mathsf{m}$-diagrams as an intermediary structure.

Proposed method

• Constructing an $\mathsf{m}$-diagram from a standard Young tableau by reading entries row by row from bottom to top and connecting each entry to the largest unconnected entry below it in the next row.
• Defining a geometric map from $\mathsf{m}$-diagrams to irreducible $\mathfrak{sl}_3$ webs via arc resolution and trivalent vertex placement.
• Proving that the resulting map from tableaux to webs is equivalent to Khovanov-Kuperberg’s recursive construction.
• Using the $\mathsf{m}$-diagram framework to show that rotation of webs corresponds to promotion on tableaux via a geometric argument.
• Introducing the 'shuffle' operation on tableaux as a composition of two tableaux with aligned heights, and showing it corresponds to the join of their associated webs.
• Establishing that the $\mathsf{m}$-diagram of a shuffled tableau is the join of the $\mathsf{m}$-diagrams of the component tableaux.

Experimental results

Research questions

• RQ1 Does a simple, explicit bijection exist between standard $3\times n$ Young tableaux and irreducible $\mathfrak{sl}_3$ webs with all boundary vertices as sources?
• RQ2 How does web rotation relate to the classical jeu-de-taquin promotion on $3\times n$ tableaux?
• RQ3 Can a natural operation on tableaux, such as 'shuffle', be defined and shown to correspond to a natural graph operation on webs?
• RQ4 To what extent can the $\mathsf{m}$-diagram construction be generalized beyond $3\times n$ tableaux?

Key findings

• The proposed bijection between standard $3\times n$ Young tableaux and irreducible $\mathfrak{sl}_3$ webs with all boundary vertices as sources is equivalent to Khovanov-Kuperberg’s recursive construction.
• A geometric proof is provided showing that web rotation corresponds to jeu-de-taquin promotion on $3\times n$ tableaux, using the $\mathsf{m}$-diagram framework.
• The shuffle operation on tableaux—defined as the composition of two rectangular tableaux of equal height—corresponds exactly to the join of their associated webs.
• The $\mathsf{m}$-diagram construction generalizes to tableaux of arbitrary shape, not just $3\times n$, and preserves structural properties under composition.
• The $\mathsf{m}$-diagram of a shuffled tableau is the join of the $\mathsf{m}$-diagrams of the individual tableaux, establishing a consistent correspondence between tableau operations and web operations.