Abstract.
In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions 
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 Murnaghan-Nakayama formula,
• ribbon Cauchy and dual Cauchy identities,
• and a 
-algebra isomorphism ω
n
:Λ(q)→Λ(q) which sends each to 
.
Our study of the functions will be connected to the Fock space representation F of 
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 . 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. The study of this map will lead to our identities concerning ribbon tableaux generating functions. We will also give a combinatorial proof that the ribbon Murnaghan-Nakayama and Pieri rules are formally equivalent.
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Lam, T. Ribbon tableaux and the Heisenberg algebra. Math. Z. 250, 685–710 (2005). https://doi.org/10.1007/s00209-005-0771-3
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DOI: https://doi.org/10.1007/s00209-005-0771-3