2009/07/22 by Robin Langer, Langer, Robin
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.0907.3950
openalex publication_date 2009/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood--Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. In the first part of this thesis we show that by using a generalization of the classical umbral calculus of Gian-Carlo Rota, one may deform the basis of Schur functions to find many other bases for which the Littlewood--Richardson numbers as coproduct structure constants. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.