Aims

The aim of the module is to introduce students to the modern representation theory by working with the classical example, the group of all permutations of a finite set.

Learning objectives

At the end of the module students should know:

• classical combinatorics related to the symmetric group;  
• general properties of representations and characters of finite groups;
• explicit construction of irreducible representations of the symmetric group.

Syllabus

• Conjugacy classes in the symmetric group and combinatorics of Young diagrams.  
• Robinson-Schensted-Knuth algorithm.
• General properties of representations of finite groups.
• General properties of characters of finite groups.
• Young symmetrizers and the regular representation of the symmetric group.
• Hook formula for the dimensions of irreducible representations of the symmetric group.

Recommended texts

• G D James, The representation theory of the symmetric group, Lecture Notes in Mathematics #682, Springer, 1978 (S 0.4 LEC).
• I G Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979 (S 2.86 MCD).
• B E Sagan, The symmetric roup: representations, combinatorial algorithms and symmetric functions, Wadsworth and Brooks/Cole, 1991 (S 2.86 SAG).

Teaching

• Autumn Term 
• 2 lectures per week
• problems classes by request

Assessment

• Examination in week 1 Spring Term (90%),
• Coursework (10%). Note that coursework submitted after the advertised deadlines will be given a mark of zero.

Elective information

This module is not available as an elective. 

Prerequisites

• Linear Algebra
• Groups Rings and Fields 
• Matrices