An introduction to the modern representation theory of big groups, exploring its connections to probability and algebraic combinatorics.

Alexei Borodin is a Professor of Mathematics at the Massachusetts Institute of Technology.

Introduction; Part I. Symmetric Functions and Thoma's Theorem: 1. Preliminary facts from representation theory of finite symmetric groups; 2. Theory of symmetric functions; 3. Coherent systems on the Young graph; 4. Extreme characters and Thoma's Theorem; 5. A toy model (the Pascal Graph) and de Finetti's Theorem; 6. Asymptotics of relative dimension in the Young graph; 7. Boundaries and Gibbs measures on paths; Part II. Unitary Representations: 8. Preliminaries and Gelfand pairs; 9. Classification of general spherical type representations; 10. Realization of irreducible spherical representations of (S(¿) × S(¿), diagS(¿)); 11. Generalized regular representations Tz; 12. Disjointness of representations Tz; References; Index.