Linear and projective representations of symmetric groups.

*(English)*Zbl 1080.20011
Cambridge Tracts in Mathematics 163. Cambridge: Cambridge University Press (ISBN 0-521-83703-0/hbk). xiv, 277 p. (2005).

The volume under review is a much awaited contribution to the literature on representation theory of symmetric groups and associated groups and algebras. Related topics are presented in several recent surveys and monographs, such as [S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, Am. Math. Soc. (2002; Zbl 1003.17008); M. Cabanes and M. Enguehard, Representation theory of finite reductive groups, Cambridge University Press (2004; Zbl 1069.20032); M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford: Clarendon Press (2000; Zbl 0996.20004); A. Mathas, Iwahori-Hecke algebras and the Schur algebras of the symmetric group, Am. Math. Soc. (1999; Zbl 0940.20018)]. This book is a comprehensive presentation of the progress in the modular representation theory achieved especially by the work of A. Lascoux, B. Leclerc, J.-Y. Thibon, S. Ariki, J. Grojnowski, J. Brundan and of the author.

The book is divided into two parts. Part I is devoted to the linear representations, and consists of 11 chapters. The first two chapters are introductory and present the approach of Okounkov and Vershik to the representation theory in characteristic zero. Chapter 3 contains the definition of the degenerate affine Hecke algebras \({\mathcal H}_n\) and their basic properties, including a discussion of the induction and restriction functors. Chapter 4 discusses central characters and blocks, and the so-called Kato module. Chapter 5 continues the study of \({\mathcal H}_n\)-modules by introducing the functors \(e_a\), the crystal operators \(\widetilde e\) and \(\widetilde f\), and proving several branching rules. In Chapter 6 methods for calculating characters of certain irreducible \({\mathcal H}_n\)-modules are developed. Chapter 7 presents the basic results concerning integral representations and cyclotomic Hecke algebras, and in the next chapter cyclotomic analogues of the functors \(e_i\) are studied. Chapter 9 establishes a connection between the Grothendieck group \(K(\infty)\) of integral representations of all affine Hecke algebras \({\mathcal H}_n\) and the affine Kac-Moody algebras. An important result here constructs an explicit isomorphism of graded Hopf algebras between the graded dual \(K(\infty)^*\) and the Kostant-Tits \(\mathbb{Z}\)-form \(U^+_\mathbb{Z}\) of the Kac-Moody algebra of type \(A^{(1)}_{p-1}\). As a consequence, the blocks of the cyclotomic Hecke algebra are classified. Chapters 10 and 11 study the crystal graph of the irreducible highest weight module and the socle branching graph of the cyclotomic Hecke algebras.

The second part of the book is devoted to projective representations. Their study is done by regarding the twisted group algebra \({\mathcal T}_n\) of \(S_n\) as a superalgebra, and to work with the category of \({\mathcal T}_n\)-supermodules. It turns out that it is even more convenient to work with the Sergeev superalgebra, which is “almost Morita equivalent” to \({\mathcal T}_n\). This gives a parallelism between the linear representation theory and the spin representation theory which is exposed in the remaining Chapters 14-22.

Most of the material appears for the first time in book form, but at the same time, the theory is developed from scratch. Therefore, this volume will be useful not only to researchers, but also to the graduate students who want to understand this fascinating and dynamic subject, which is relevant not only for group theory, but it has rich connections with combinatorics, Lie theory and algebraic geometry.

The book is divided into two parts. Part I is devoted to the linear representations, and consists of 11 chapters. The first two chapters are introductory and present the approach of Okounkov and Vershik to the representation theory in characteristic zero. Chapter 3 contains the definition of the degenerate affine Hecke algebras \({\mathcal H}_n\) and their basic properties, including a discussion of the induction and restriction functors. Chapter 4 discusses central characters and blocks, and the so-called Kato module. Chapter 5 continues the study of \({\mathcal H}_n\)-modules by introducing the functors \(e_a\), the crystal operators \(\widetilde e\) and \(\widetilde f\), and proving several branching rules. In Chapter 6 methods for calculating characters of certain irreducible \({\mathcal H}_n\)-modules are developed. Chapter 7 presents the basic results concerning integral representations and cyclotomic Hecke algebras, and in the next chapter cyclotomic analogues of the functors \(e_i\) are studied. Chapter 9 establishes a connection between the Grothendieck group \(K(\infty)\) of integral representations of all affine Hecke algebras \({\mathcal H}_n\) and the affine Kac-Moody algebras. An important result here constructs an explicit isomorphism of graded Hopf algebras between the graded dual \(K(\infty)^*\) and the Kostant-Tits \(\mathbb{Z}\)-form \(U^+_\mathbb{Z}\) of the Kac-Moody algebra of type \(A^{(1)}_{p-1}\). As a consequence, the blocks of the cyclotomic Hecke algebra are classified. Chapters 10 and 11 study the crystal graph of the irreducible highest weight module and the socle branching graph of the cyclotomic Hecke algebras.

The second part of the book is devoted to projective representations. Their study is done by regarding the twisted group algebra \({\mathcal T}_n\) of \(S_n\) as a superalgebra, and to work with the category of \({\mathcal T}_n\)-supermodules. It turns out that it is even more convenient to work with the Sergeev superalgebra, which is “almost Morita equivalent” to \({\mathcal T}_n\). This gives a parallelism between the linear representation theory and the spin representation theory which is exposed in the remaining Chapters 14-22.

Most of the material appears for the first time in book form, but at the same time, the theory is developed from scratch. Therefore, this volume will be useful not only to researchers, but also to the graduate students who want to understand this fascinating and dynamic subject, which is relevant not only for group theory, but it has rich connections with combinatorics, Lie theory and algebraic geometry.

Reviewer: Andrei Marcus (Cluj-Napoca)

##### MSC:

20C30 | Representations of finite symmetric groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20C08 | Hecke algebras and their representations |

20C15 | Ordinary representations and characters |

20C20 | Modular representations and characters |

20C25 | Projective representations and multipliers |

05E10 | Combinatorial aspects of representation theory |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |