Introductions to Symmetric Polynomials and Symmetric Functions

Symmetric polynomials and symmetric functions are ubiquitous in mathematics and mathematical physics. For example, they appear in elementary algebra (e.g. Viete’s Theorem), representation theories of symmetric groups and general linear groups over C or finite fields. They are also important objects to study in algebraic combinatorics. Via their close relations with representation theory, the theory of symmetric functions has found many applications to mathematical physics. For example, they appear in the Boson-Fermion correspondence which is very important in both superstring theory and the theory of integrable system [2]. They also appear in Chern-Simons theory and the related link invariants and 3-manifold invariants [8]. By the duality between Chern-Simons theory and string theory [9] they emerge again in string theory [1], and in the study of moduli spaces of Riemann surfaces [6]. The following is a revised and expanded version of the informal lecture notes for a undergraduate topic course given in Tsinghua University in the spring semester of 2003. Part of the materials have also been used in a minicourse at the Center of Mathematical Sciences at Zhejiang University as part of the summer program on mathematical physics in 2003. I thank both the audiences for their participation. The purpose of this course is to present an introduction to this fascinating field with minimum prerequisite. I have kept the informal style of the original notes.