Diagram model for the Okada algebra and monoid

Abstract

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups expressing how irreducible representations restrict from $S_N$ to $S_{N-1}$. In 1988, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was realized as the Bratelli diagram of a family of algebras by Okada in 1994. In this paper, we realize the Okada algebra and its associated monoid using a labeled version of Temperley-Lieb arc-diagrams. We prove in full generality that the dimension of the Okada algebra is $n!$. In particular, we interpret a natural bijection between permutations and labeled arc-diagrams as an instance of Fomin's Robinson-Schensted correspondence for the Young-Fibonacci lattice. We prove that the Okada monoid is aperiodic and describe its Green relations. Lifting those results to the algebra allows us to construct a cellular basis of the Okada algebra. }