有限群的線性表示

前言

This book consists of three parts, rather different in level and purpose:The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra.The examples (Chapter 5) have been chosen from those useful to chemists.

內(nèi)容概要

本書是一部非常經(jīng)典的介紹有限群線性表示的教程，原版曾多次修訂重印，作者是當(dāng)今法國最突出的數(shù)學(xué)家之一，他對理論數(shù)學(xué)有全面的了解，尤以著述清晰、明了聞名。本書是他寫的為數(shù)不多的教科書之一，原文是法文（1971年版），后出了德譯本和英譯本。本書是英譯本的重印本。它篇幅不大，但深入淺出的介紹了有限群的線性表示，并給出了在量子化學(xué)等方面的應(yīng)用，便于廣大數(shù)學(xué)、物理、化學(xué)工作者初學(xué)時閱讀和參考。

作者簡介

作者：(法國)賽爾 (Serre.J.P)

書籍目錄

Part ⅠRepresentations and Characters  1  Generalities on linear representations　　1.1  Definitions　　1.2  Basic examples　　1.3  Subrepresentations　　1.4  Irreducible representations　　1.5  Tensor product of two representations　　1.6  Symmetiic square and alternating square  2  Character theory　　2.1  The character of a representation　　2.2  Schur's lemma; basic applications　　2.3  Orthogonality'reiations for characters　　2.4 Decomposition of the regular representation　　2.5  Number of irreducible representations　　2.6 Canonical decomposition of a representation　　2.7 Explicit decomposition of a representation  3  Subgroups, products, induced representations　　3.1  Abelian subgroups　　3.2  Product of two groups　　3.3  Induced representations  4  Compact groups　　4.1  Compact groups　　4.2  lnvariant measure on a compact group　　4.3  Linear representations of compact groups  5  Examples　　5.1  The cyclic Group Cn　　5.2  The group C　　5.3  The dihedral group D　　5.4  The group Dn　　5.5  The group D　　5.6  The group D　　5.7  The alternating group 　　5.8  The symmetric group 　　5.9  The group of the cube　Bibliography: Part IPart Ⅱ　Representations in Characteristic Zero  6  The group algebra　　6,1  Representations and modules　　6.2  Decomposition of C[G]　　6.3  The center of C[G]　　6.4  Basic properties of integers　　6.5  lntegrality properties of characters. Applications  7  Induced representations; Mackey's criterion　　7.1  Induction　　7.2  The character of an induced representation; the reciprocity formula　　7.3  Restriction to subgroups　　7.4  Mackey's irreducibility criterion  8  Examples of induced representations　　8.1  Normal subgroups; applications to the degrees of the irreducible representations　　8.2  Semidirect products by an abelian group　　8.3  A review of some classes of finite groups　　8.4  Sylow's theorem　　8.5  Linear representations of supersolvable groups  9  Artin's theorem　　9.1  The ring R(G)　　9,2  Statement of Artin's theorem　　9.3  First proof　　9.4  Second proof of (i)  (ii)  10  A theorem of Brauer　　10.1 p-regular elements;p-elementary subgroups　　10.2 Induced characters arising from p-elementary subgroups　　10.3 Construction of characters　　10.4 Proof of theorems 18 and 18'　　10,5 Brauer's theorem　……part Ⅲ Introduction to Brauer TheoryAppendixBibliography:Part ⅢIndex of notationIndex of terminology

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