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Tableau evacuation and webs

Rebecca Patrias, Oliver Pechenik

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.

Tableau evacuation and webs

Abstract

Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of and . 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.

Paper Structure

This paper contains 6 sections, 2 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Webs
  4. Tableaux and evacuation
  5. Bijections
  6. Main result

Key Result

Lemma 3.1

Let $T$ be a row-strict tableau of rectangular shape $\lambda = (k,k, \dots, k)$ with maximum label $n$. Then $\epsilon(T)$ is given by rotating $T$ by $180^\circ$ and reversing the alphabet so that $x \mapsto n+1-x$.

Figures (1)

  • Figure 1: An $\mathfrak{sl}_2$ web (left) and an $\mathfrak{sl}_3$ web (right).

Theorems & Definitions (12)

  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Example 2.6
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:rotation']}
  • Example 3.2
  • Theorem 3.3
  • ...and 2 more