In 1994 S. Okada introduced a family of non-commutative polynomials satisfying a Pieri-type identity which recapitulates the branching rule of R. Stanley’s Young-Fibonacci lattice. More generally, products of these so called “clone” Schur functions were shown to obey a non-commutative version of the Littlewood-Richardson identity with structure constants determined combinatorially from the Young-Fibonacci lattice structure. In this talk I’ll survey Okada’s clone theory with the aim of drawing parallels with the (classical) theory of symmetric functions, the representation theory of the symmetric group, and the combinatorics of the Young lattice. I’d also like to use the opportunity to report on some speculative work based on discussions with Leonid Petrov: Specifically a new concept of total positivity related to clone Schur functions together with a corresponding “Stieltjes” moment problem. If there’s time, I’ll pose an open problem to the audience of whether a matrix model can be meaningfully associated to certain clone “tau” functions.

Brandeis University