Richard P. Stanley Seminar in Combinatorics: A diagrammatic realization of the Okada algebra

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 1975 Stanley discovered a similar lattice called the Young-Fibonacci lattice which was later realized as the Bratelli diagram of a family of algebras by Okada in 1994. In joint work with Florent Hivert (Université Paris-Sud) we realize the n-th Okada algebra as a diagram algebra with a multiplicative/monoid basis consisting of n-strand Temperley-Lieb diagrams, each equipped with a “height” labeling of its strands. The proof involves a diagrammatic version of Fomin’s Robinson-Schensted correspondence for the Young-Fibonacci lattice. This basis is cellular, which affords us with a novel, diagrammatic presentation of the irreducible representations of the Okada algebra (i.e. cell modules).

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