MA 715, Representation Theory and the Symmetric Group.
This course serves as an introduction to
We will develop the basic ideas using
the symmetric group as the main example.
The course will include some
including the Novelli-Pak-Stoyanovskii bijection of the hook formula,
and recent applications to algebraic combinatorics.
Bruce E. Sagan,
The Symmetric Group:
Graduate Texts in Mathematics, Volume 203,
Introduction to group representations
(matrix representations, the group algebra,
reducibility, Maschke's Theorem, Schur's Lemma,
Representations of the symmetric group
(using Specht modules)
Combinatorial algorithms in representation theory
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)
(Stanley's theory of differential posets,
Fomin's concept of growths,
Stanley's symmetric function analogue of the chromatic polynomial of a graph)
A graduate course in linear algebra or permission of instructor.