The Burge correspondence and crystal graphs

TL;DR

This paper explores the Burge correspondence linking graphs and Young tableaux, characterizes hook-shaped graphs via peak and valley conditions, and introduces a crystal structure on these graphs, advancing understanding of combinatorial bijections.

Contribution

It characterizes hook-shaped graphs through peak and valley conditions and establishes a crystal structure on these graphs, extending combinatorial bijection theory.

Findings

Characterization of hook-shaped graphs via peak and valley conditions

Introduction of a crystal structure on simple graphs of hook shape

Identification of extremal vectors with threshold and hook-shaped degree sequences

Abstract

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted's result for the RSK insertion which states that the length of the longest increasing subword of a word is the length of the largest row of the tableau under the RSK correspondence. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped.