Ribbon Tableaux and the Heisenberg Algebra

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions Gλ which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: • a Pieri and dual-Pieri formula for ribbon functions, • a ribbon Murnagham-Nakayama formula, • ribbon Cauchy and dual Cauchy identities, • and a C-algebra isomorphism ωn : Λ(q) → Λ(q) which sends each Gλ to Gλ′ . Our study of the functions Gλ will be connected to the Fock space representation F of Uq(ŝln) via a linear map Φ : F → Λ(q) which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [KMS] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of Uq(ŝln). Identifying the Fock Space of H with the ring of symmetric functions Λ(q) we will show that Φ is in fact a map of H-modules with remarkable properties. We give a combinatorial proof that the ribbon Murnagham-Nakayama and Pieri rules are formally equivalent thus allowing us to describe the action of the genrators of H on F in terms of ‘border ribbon strips’. We will also connect the ribbon Cauchy and Pieri formulae to the combinatorics of ribbon insertion as studied by Shimozono and White [SW2]. In particular we give complete combinatorial proofs for the domino n = 2 case.