Some combinatorial interpretations of the Macdonald identities for affine root systems

2023/04/03 by David Wahiche, Wahiche, David
Mathematics · #05A17 #05E10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2304.01394

openalex publication_date 2023/04/03 · openalex created_date 2023/04/07 · openalex updated_date 2026/07/28

Abstract

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven infinite families in terms of symplectic and special orthogonal Schur functions. From these results, we are able to derive q-Nekrasov--Okounkov formulas associated to each family. Nevertheless we only give results for types C and C^\vee, and give a sketch of the proof for type C.

Citations

• Generalizations of Nekrasov-Okounkov Identity
• A Nekrasov–Okounkov formula for Macdonald polynomials

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