A Combinatorial Generalization of the Boson-fermion Correspondence

This article is an attempt to understand the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from which Schur functions arise. Our work can be used to understand Hall-Littlewood polynomials, Macdonald polynomials and Lascoux, Leclerc and Thibon’s ribbon functions, together with other new families of symmetric functions.