Identifying Orbit Lengths for Promotion
Elise Catania, Jack Kendrick, Heather M. Russell, Julianna Tymoczko
TL;DR
This paper investigates the promotion operator on standard Young tableaux using $m$-diagrams, providing an algorithm to compute orbit lengths and extending results to semi-standard Young tableaux.
Contribution
It introduces a graphical approach to analyze promotion on SYT, revealing preserved structures and deriving a formula for orbit lengths of rectangular SSYT.
Findings
Preservation of internal structures under promotion
Algorithm for computing promotion orbit lengths
Formula for orbit lengths of rectangular SSYT
Abstract
In this work we study Sch\"utzenberger's promotion operator on standard Young tableaux via a corresponding graphical construction known as diagrams. In particular, we prove that certain internal structures of SYT are preserved under promotion and correspond to distinct components of diagrams. By treating these structures as atomic parts of the diagram, we provide a simple algorithm for computing the promotion orbit length of rectangular SYT. We conclude the paper by applying our results to (column) semi-standard Young tableaux and prove a formula for the promotion orbit lengths of rectangular (column) SSYT.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Markov Chains and Monte Carlo Methods · Algebraic structures and combinatorial models