Properties of the Edelman-Greene bijection

TL;DR

This paper investigates the properties of the Edelman-Greene bijection, revealing how the shape of the insertion tableau relates to permutation diagrams and exploring its behavior on specific pattern-avoiding permutations and non-reduced words.

Contribution

It establishes a precise relationship between the tableau shape evolution and permutation diagrams, and analyzes the bijection's properties on pattern-avoiding and non-reduced words.

Findings

Shape evolution matches the upper-left component of the permutation's diagram.

Properties are characterized for 132- and 2143-avoiding permutations.

Analysis extended to non-reduced words.

Abstract

Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.