This talk concerns a family of complex, finite dimensional, semi-simple algebras Ok(N) introduced by S. Okada in 1994. The chief feature of these algebras, as observed by Okada, is that their associated Bratteli diagram, which expresses how irreducible representations restrict from Ok(N) to Ok(N-1), is R. Stanley's Young-Fibonacci lattice (YF for short).
Like the lattice of integer partitions (i.e. the Young lattice Y), the YF-lattice has the structure of a 1-differential poset, and, as a consequence, it supports a variant of the classical Robinson-Schensted (RS) correspondence. In particular this tells us that the dimension of the Okada algebra Ok(N) is N!. This sets up an interesting parallel between the symmetric groups (or Hecke algebras) and the Okada algebras, whose Bratteli diagrams are, respectively, the Young and Young-Fibonacci lattices, and whose structure and representation theory can both be probed using an appropriate RS-correspondence which evolves on the associated Bratteli diagram.
In joint work with F. Hivert, we pursue this analogy with the symmetric groups and the Young lattice, and realize the Okada algebra as a diagram algebra, with a multiplicative basis consisting of certain arc-labelled, non-crossing perfect matchings (as appear in both the Temperley-Lieb and Blob algebras). In particular, this requires us to interpret the YF-variant of the RS-correspondence diagrammatically. Like most diagram algebras, this basis is cellular and affords us with a novel, diagrammatic presentation of the irreducible representations of the Okada algebra(i.e. "cell" modules). If time permits, I'll discuss some open problems addressing Okada versions of the Razumov-Stroganov conjecture/result and meander determinants.