2024/04/25 by Hivert, Florent, Scott, Jeanne
#05E05 (primary) #05E10 #16G99 #20C30 #20M99 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2404.16733
It is well known that the Young lattice is the Bratelli diagram of the symmetric groups expressing how irreducible representations restrict from SN to SN-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.