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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

  1. Symmetric groups and symmetric functions
    1. Representation theory of finite groups and semisimple algebras
    2. Symmetric functions and the Frobenius­-Schur isomor­phism
    3. Combinatorics of partitions and tableaux
  2. Hecke algebras and their representations
    1. Hecke algebras and the Brauer­-Cartan theory
    2. Characters and dualities for Hecke algebras
    3. Representations of the Hecke algebras at q=0
  3. Observables of partitions
    1. The Ivanov­-Kerov algebra of observables
    2. The Jucys­-Murphy elements
    3. Symmetric groups and free probability
    4. The Stanley­-FĂ©ray formula and Kerov polynomials
  4. Models of random Young diagrams
    1. Representations of the infinite symmetric group
    2. Asymptotics of central measures
    3. Asymptotics of Plancherel measures