Iwahori-Hecke algebras and Schur algebras of the symmetric group.

*(English)*Zbl 0940.20018
University Lecture Series. 15. Providence, RI: American Mathematical Society (AMS). xiii, 188 p. (1999).

Although ordinary representations of the symmetric group are a well-understood subject, the modular representations of such groups still have many open problems. This book gives a self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups which includes as a special case the modular representations of \(S_n\). This (Iwahori-Hecke) algebra bridges the representations of the symmetric groups and the general linear groups. This connection brings to the stage an important algebra called the \(q\)-Schur algebra (the classical Schur algebra is a \(q\)-Schur algebra with \(q=1\)) introduced by R. Dipper and G. James [Proc. Lond. Math. Soc., III. Ser. 59, No. 1, 23-50 (1989; Zbl 0711.20007)].

The book has six chapters and three appendices. Chapter 1 establishes Hecke algebra of the symmetric group and its basic properties. The second chapter explains the theory of J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] on cellular algebras which is a unifying concept for the representation theory of Iwahori-Hecke algebras and \(q\)-Schur algebras. Chapter 3 embarks upon the study of the representation theory of the Iwahori-Hecke algebras in the light of G. E. Murphy’s results [J. Algebra 173, No. 1, 97-121 (1995; Zbl 0829.20022)]. In particular, a construction is given for the \(q\)-Specht modules for this algebra. Chapter 4 deals with the consequences of the fact that the \(q\)-Schur algebra and Iwahori-Hecke algebra are cellular algebras. Chapter 5 is devoted to classifying the blocks of the \(q\)-Schur algebras and Iwahori-Hecke algebras. The final chapter is a survey of some recent and important results and conjectures in the field. In addition the book contains three appendices treating basic representation theory, crystalized decomposition matrices and the elementary divisors of the Gram matrices of the integral Specht modules for \(n\leq 12\).

The book has six chapters and three appendices. Chapter 1 establishes Hecke algebra of the symmetric group and its basic properties. The second chapter explains the theory of J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] on cellular algebras which is a unifying concept for the representation theory of Iwahori-Hecke algebras and \(q\)-Schur algebras. Chapter 3 embarks upon the study of the representation theory of the Iwahori-Hecke algebras in the light of G. E. Murphy’s results [J. Algebra 173, No. 1, 97-121 (1995; Zbl 0829.20022)]. In particular, a construction is given for the \(q\)-Specht modules for this algebra. Chapter 4 deals with the consequences of the fact that the \(q\)-Schur algebra and Iwahori-Hecke algebra are cellular algebras. Chapter 5 is devoted to classifying the blocks of the \(q\)-Schur algebras and Iwahori-Hecke algebras. The final chapter is a survey of some recent and important results and conjectures in the field. In addition the book contains three appendices treating basic representation theory, crystalized decomposition matrices and the elementary divisors of the Gram matrices of the integral Specht modules for \(n\leq 12\).

Reviewer: A.Khammash (Makkah)

##### MSC:

20C30 | Representations of finite symmetric groups |

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

20C08 | Hecke algebras and their representations |