群的表示与群的特征
2009-5
世界图书出版公司
詹姆斯
458
无
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.
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.
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
无
此书很适合初学者，很易读，包装也很好，印刷也很好，排版也很好。
这本表示内容非常好，我向大家推介这本书。