Symmetric Functions and Macdonald Polynomials

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. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. 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. Declaration. The first part of this thesis is entirely the author's own work. The idea for the second part was suggested by Ole Warnaar who also carried out the derivation of equation 15 on pages 59–60 and verified proposition 2.1. The proof of lemma 2.1 was suggested by Paul Zinn-Justin. Steps two and three in the proof of the second part of the thesis are the author's own work. and the two anonymous referees for useful comments and suggestions on the manuscript as well as the University of Montreal for hosting an interesting lecture series and workshop on Macdonald Polynomials and Combinatorial Hopf Algebras.