Representation Theory of Symmetric Groups
I have written a book (published in 2017) which covers many aspects of the representation theory of the symmetric groups, including: the classical theory of Frobenius and Schur, the extension of the classical results to the setting of Hecke algebras (in the generic case and at q=0), the theory of so-called polynomial functions on Young diagrams, and the applications of this theory to the study of large random integer partitions. You can buy it here (CRC Press) or there (Amazon).
I plan to write a second volume focused on Lie groups instead of symmetric groups, with a similar approach (i.e., recount some recent advances while being auto-contained). However, this will certainly take me a very long time...
Table of contents
- Symmetric groups and symmetric functions
- Representation theory of finite groups and semisimple algebras
- Symmetric functions and the Frobenius-Schur isomorphism
- Combinatorics of partitions and tableaux
- Hecke algebras and their representations
- Hecke algebras and the Brauer-Cartan theory
- Characters and dualities for Hecke algebras
- Representations of the Hecke algebras at q=0
- Observables of partitions
- The Ivanov-Kerov algebra of observables
- The Jucys-Murphy elements
- Symmetric groups and free probability
- The Stanley-Féray formula and Kerov polynomials
- Models of random Young diagrams
- Representations of the infinite symmetric group
- Asymptotics of central measures
- Asymptotics of Plancherel measures