Approche Duale des représentations du groupe symétrique

Book in French — English title: A Dual Approach to the Representations of the Symmetric Group

This book is based on a series of four lectures given at the Collège de France in January and February 2013 under the auspices of the Claude-Antoine Peccot Foundation.

The aim of the course was to give a brief overview of some recent results on the representations of the symmetric group, in particular results of a combinatorial nature that fall within the framework of the dual approach initiated by S. Kerov and G. Olshanski in the 1990s.

After a brief review of the representations of finite groups in general, the first chapter gives a construction of the irreducible representations of the symmetric group.

This construction is then used in the second chapter to compute the irreducible characters, yielding a formula with a rich underlying combinatorics. The third chapter illustrates an application of this combinatorics to a problem posed by S. Kerov. In particular, a recently introduced combinatorial operation, cyclic inclusion–exclusion, arises naturally.

The fourth chapter presents S. Kerov’s approach to the study of large Young diagrams under the Plancherel measure. With the theory set out in the preceding chapters, it is then relatively straightforward to describe the limit shape of these diagrams. Finally, the fifth chapter sketches a framework in which most of the preceding results conjecturally generalise.

The first three chapters each correspond to a two-hour session, while chapters 4 and 5 correspond to the final session.