**Course No.****：**21514Z

**Period****：**20

**Credits****：**1

**Course Category****：**Advanced Course

**Aims & Requirements:**

The course discusses some of the basic materials in symmetric functions and algebraic combinatorics. Students will learn the beautiful formula of Schur functions dated back to Jacobi’s time, which appeared in many branches of mathematics such as algebraic geometry and Lie theory. The simplest example of representation theory, the symmetric group will be covered thoroughly. This will naturally lead to some of the combinatorial concepts such as Young tableaux, Kostka numbers and hook formula. The last part will be devoted to generalizations, in particular, Macdonald polynomials. Macdonald polynomials are essential part of modern harmonic analysis and representation theory, and have interesting connections to many areas in mathematics.

**Prerequisites:**

Solid understanding of linear algebra is needed.

**Primary Coverage****：**

Ch.1. Group representations

Contents：matrix representations, Maschke’s theorem, Schur’s lemma, group characters, induced representations

Ch.. 2. Representations of the symmetric group

Contents: Young tabloids, dominance order, Specht modules, standard bases of S ,Young’s representations, Kostka numbers and Young’s rule, hook formulae

Ch. 3. Symmetric functions

Contents: Generating functions, Schur functions, Jacobi-Trudi theorem, LR rule, characteristic map, Frobenius formula

Ch. 4. Hall-Littlewood and Macdonald functions

Contents: Littlewood’s formula, flags of abelian groups, orthogonality, generalizations, Macdonald polynomials, Cherednick theorem

**References:**

(iv) B. Sagan, The symmetric groups, 2nd edition, Graduate Texts in Mathematics, Springer-Verlag, 2001.(Our main reference)

(v) I.G. Macdonal, Symmetric functions and Hall polynomials, Oxford, 1995

(vi) R. Stanley, Enumerative combinatorics, Cambridge, 1997-99.

**Author: **Naiheng Jing