2013/01/18 by Eliezer Posner, Posner, Eliezer, Kris Hatch +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1301.4518
24 pages, completed as undergraduates in the Research Experience for Undergraduates program at UC Santa Barbara
arxiv created 2013/01/18 · openalex publication_date 2013/01/18 · arxiv updated 2013/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2010, Tom Halverson and Georgia Benkart introduced the Motzkin algebra, a generalization of the Temperley-Lieb algebra, whose elements are diagrams that can be multiplied by stacking one on top of the other. Halverson and Benkart gave a diagrammatic algorithm for decomposing any Motzkin diagram into diagrams of three subalgebras: the Right Planar Rook algebra, the Temperley-Lieb algebra, and the Left Planar Rook algebra. We first explore the Right and Left Planar Rook monoids, by finding presentations for these monoids by generators and relations, using a counting argument to prove that our relations suffice. We then turn to the newly-developed Motzkin monoid, where we describe Halverson's decomposition algorithm algebraically, find a presentation by generators and relations, and use a counting argument but with a much more sophisticated algorithm.