This course serves as an introduction to representation theory. We will develop the basic ideas using the symmetric group as the main example. The course will include some new developments, including the Novelli-Pak-Stoyanovskii bijection of the hook formula, and recent applications to algebraic combinatorics.

COURSE OUTLINE

• Introduction to group representations (matrix representations, the group algebra, reducibility, Maschke's Theorem, Schur's Lemma, group characters...)
• Representations of the symmetric group (using Specht modules)
• Combinatorial algorithms in representation theory (Robinson-Schensted-Knuth algorithm, Novelli-Pak-Stoyanovskii hook formula, Frobenius-Young determinantal formula, Schützenberger's jeu de taquin)
• Introduction to Symmetric functions (Schur functions, Littlewood-Richardson and Murnaghan-Nakayama Rules)
• Applications (Stanley's theory of differential posets, Fomin's concept of growths, unimodality results, Stanley's symmetric function analogue of the chromatic polynomial of a graph)

Prerequisite: A graduate course in linear algebra or permission of instructor.