群的表示与群的特征

出版时间：2009-5  出版社：世界图书出版公司  作者：詹姆斯  页数：458  
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前言

　　We have attempted in this book to provide a leisurely introduction tothe representation theory of groups. But why should this subjectinterest you？　　　Representation theory is concerned with the ways of writing a groupas a group of matrices. Not only is the theory beautiful in its own right，but it also provides one of the keys to a proper understanding of finitegroups. For example， it is often vital to have a concrete description of aparticular group； this is achieved by finding a representation of thegroup as a group of matrices. Moreover， by studying the differentrepresentations of the group， it is possible to prove results which lieoutside the framework of representation theory. One simple example: allgroups of order p2 （where p is a prime number） are abelian； this can beshown quickly using only group theory， but it is also a consequence ofbasic results about representations. More generally， all groups of order （p and q primes） are soluble； this again is a statement purely aboutgroups， but the best proof， due to Burnside， is an outstanding exampleof the use of representation theory. In fact， the range of applications ofthe theory extends far beyond the boundaries of pure mathematics， andincludes theoretical physics and chemistry - we describe one suchapplication in the last chapter.　The book is suitable for students who have taken first undergraduatecourses involving group theory and linear algebra. We have included twopreliminary chapters which cover the necessary background material.The basic theory of representations is developed in Chapters 3-23， andour methods concentrate upon the use of modules； although this accordswith the more modem style of algebra， in several instances our proofsdiffer from those found in other textbooks. The main results are elegantand surprising.

内容概要

　　Representation theory is concerned with the ways of writing a groupas a group of matrices. Not only is the theory beautiful in its own right，but it also provides one of the keys to a proper understanding of finitegroups. For example， it is often vital to have a concrete description of aparticular group； this is achieved by finding a representation of thegroup as a group of matrices. Moreover， by studying the differentrepresentations of the group， it is possible to prove results which lieoutside the framework of representation theory. One simple example： allgroups of order p2 （where p is a prime number） are abelian； this can beshown quickly using only group theory， but it is also a consequence ofbasic results about representations.

书籍目录

Preface1  Groups and homomorphisms2  Vector spaces and linear transformations3  Group representations4  FG-modules5  FG-submodules and reducibility6  Group algebras7  FG-homomorphisms8  Maschke's Theorem9  Schur's Lemma10  Irreducible modules and the group algebra11  More on the group algebra12  Conjugacy classes13  Characters14  Inner products of characters15  The number of irreducible characters16  Character tables and orthogonality relations17  Normal subgroups and lifted characters18  Some elementary character tables19  Tensor products20  Restriction to a subgroup21  Induced modules and characters22  Algebraic integers23  Real representations24  Summary of properties of character tables25  Characters of groups of order pq26  Characters of some p-groups27  Character table of the simple group of order 16828  Character table of GL(2, q)29  Permutations and characters30  Applications to group theory31  Burnside's Theorem32  An application of representation theory to molecular vibrationSolutions to exercisesBibliographyIndex

用户评论 (总计1条)

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