Cylindric Young Tableaux and their Properties

TL;DR

This paper explores the properties of cylindric Young tableaux, extending classical combinatorial correspondences, and introduces new interpretations and methods to relate them to skew tableaux and Knuth equivalence.

Contribution

It extends the Robinson-Schensted-Knuth correspondence to cylindric tableaux and develops new tools for their analysis and applications.

Findings

Extended RSK correspondence for cylindric tableaux

Interpreted cylindric tableaux via marble-passing game

Provided a method to transfer results to skew Young tableaux

Abstract

Cylindric Young tableaux are combinatorial objects that first appeared in the 1990s. A natural extension of the classical notion of a Young tableau, they have since been used several times, most notably by Gessel and Krattenthaler and by Alexander Postnikov. Despite this, relatively little is known about cylindric Young tableaux. This paper is an investigation of the properties of this object. In this paper, we extend the Robinson-Schensted-Knuth Correspondence, a well-known and very useful bijection concerning regular Young tableaux, to be a correspondence between pairs of cylindric tableaux. We use this correspondence to reach further results about cylindric tableaux. We then establish an interpretation of cylindric tableaux in terms of a game involving marble-passing. Next, we demonstrate a generic method to use results concerning cylindric tableaux in order to prove results about skew Young tableaux. We finish with a note on Knuth equivalence and its analog for cylindric tableaux.