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內容簡介

　　《有限群的綫性錶示》是一部非常經典的介紹有限群綫性錶示的教程，原版曾多次修訂重印，作者是當今法國最突齣的數學傢之一，他對理論數學有全麵的瞭解，尤以著述清晰、明瞭聞名。《有限群的綫性錶示》是他寫的為數不多的教科書之一，原文是法文(1971年版)，後齣瞭德譯本和英譯本。《有限群的綫性錶示》是英譯本的重印本。它篇幅不大，但深入淺齣的介紹瞭有限群的綫性錶示，並給齣瞭在量子化學等方麵的應用，便於廣大數學、物理、化學工作者初學時閱讀和參考。

內頁插圖

目錄

Part Ⅰ
Representations and Characters
1 Generalities on linear representations
1.1 Definitions
1.2 Basic examples
1.3 Submpmsentations
1.4 Irreducible representations
1.5 Tensor product of two representations
1.6 Symmetric square and alternating square

2 Character theory
2.1 The character of a representation
2.2 Schurs lemma; basic applications
2.3 0rthogonality relations 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
5.2 The group
5.3 The dihedral group
5.4 The group
5.5 The group
5.6 The group
5.7 The alternating group
5.8 The symmetric group
5.9 The group of the cube
Bibliography： Part Ⅰ

Part 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; Mackeys criterion
7.1 Induction
7.2 The character of an induced representation;
the reciprocity formula
7.3 Restriction to subgroups
7.4 Mackeys irreducibility criterion

8 Examples of induced representations
8. l Normal subgroups; applications to the degrees of the
ineducible representations
8.2 Semidirect products by an ahelian group
8.3 A review of some classes of finite groups
8.4 Syiows theorem
8.5 Linear representations of superselvable groups

9 Artins theorem
9.1 The ring R(G)
9.2 Statement of Artins 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 Brauers theorem

11 Applications of Brauers theorem
11.1 Characterization of characters
11.2 A theorem of Frobenius
11.3 A converse to Brauers theorem
11.4 The spectrum of A R(G)

12 Rationality questions
12.1 The rings RK(G) and RK(G)
12.2 Schur indices
12.3 Realizability over cyclotomic fields
12.4 The rank of RK(G)
12.5 Generalization of Artins theorem
12.6 Generalization of Brauers theorem
12.7 Proof of theorem 28

13 Rationality questions： examples
13. I The field Q
13.2 The field R
Bibliography： Part Ⅱ

Part Ⅲ
Introduction to Brauer Theory
14 The groups RK(G)， R(G)， and Pk(G)
14.1 The rings RK(G) and R，(G)
14.2 The groups Pk(G) and P^(G)
14.3 Structure of Pk(G)
14.4 Structure of PA(G)
14.5 Dualities
14.6 Scalar extensions

15 The cde triangle
15.1 Definition of c： Pk(G) ——Rk(G)
15.2 Definition of d： Rs(G) —— Rk(G)
15.3 Definition of e： Pk(G) —— RK(G)
15.4 Basic properties of the cde triangle
15.5 Example： p-gmups
15.6 Example： p-groups
15.7 Example： products ofp-groups and p-groups

16 Theorems
16.1 Properties of the cde triangle
16.2 Characterization of the image of e
16.3 Characterization of projective A [G ]-modules
by their characters
16.4 Examples of projective A [G ]-modules： irreducible
representations of defect zero

17 Proofs
17. I Change of groups
17.2 Brauers theorem in the modular case
17.3 Proof of theorem 33
17.4 Proof of theorem 35
17.5 Proof of theorem 37
17.6 Proof of theorem 38

18 Modular characters
18.1 The modular character of a representation
18.2 Independence of modular characters
18.3 Reformulations
18.4 A section ford
18.5 Example： Modular characters of the symmetric group
18.6 Example： Modular characters of the alternating group

19 Application to Artin representations
19.1 Artin and Swan representations
19.2 Rationality of the Artin and Swan representations
19.3 An invariant

Appendix
Bibliography： Part Ⅲ
Index of notation
Index of terminology

前言/序言

　　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.

有限群的綫性錶示 [Linear Representations of Finite Groups] 下載 mobi epub pdf txt 電子書

評分☆☆☆☆☆

經典圖書，細細品讀，很好的參考書

評分☆☆☆☆☆

要精確的解釋仿射和歐氏幾何之間的關係，我們要在仿射群中點齣歐氏幾何的群。歐氏群實際上是(采用前麵仿射群的錶述)正交(鏇轉和反射)群和平移群的準直積。

評分☆☆☆☆☆

好書 清晰易讀 值得推薦

評分☆☆☆☆☆

gtm42好看又好用，還是gtm52的親戚(●?∀?●)

評分☆☆☆☆☆

經常，兩個或者更多的不同的幾何有同構的自同構群。這就産生瞭從愛爾蘭根綱領的抽象群解讀齣具體的幾何的問題。

評分☆☆☆☆☆

在後來的工作上的影響

評分☆☆☆☆☆

例如n維射影幾何的群就是n維射影空間的對稱群(n+1階矩陣群,取和標量矩陣的商)。該仿射群是保持所選的無窮遠超平麵不變(映射集閤到自身，不是固定每一點)的子群。這個子群有一個已知的結構(n階矩陣群和平移子群的準直積)。這個錶述告訴我們什麼性質是'仿射的'。用歐氏平麵幾何術語，平行就是:仿射變換總是將一個平行四邊形變成另一個平行四邊形。而圓不是仿射地，因為仿射剪切可以把圓變成橢圓。

評分☆☆☆☆☆

當拓撲照例使用同胚下的不變量的術語來錶述時，我們可以看到操作背後的基礎思想。所涉及到的群在幾乎所有情況下 - 除瞭李群 - 都是無窮維的，但其方法是一樣。當然這隻是說剋萊因的影響啓發。諸如H.S.M. Coxeter所寫的書例行的采用愛爾蘭根綱領的方法來幫助'定位'幾何。用說教的術語，該綱領成瞭變換幾何，這是一個有一些不良影響的好事，它比歐幾裏得的風格建立在更強的直覺上，但是也更難轉換成為邏輯體係。

評分☆☆☆☆☆

一個例子: 可定嚮 (也就是說，反射是除外的)橢圓幾何 (也就是,把n球麵相對點等同的麯麵)和可定嚮 球麵幾何 (同樣的非歐幾何，但是相對的點沒有等同起來)有同構的自同構群，偶數n的SO(n+1)。這兩個看起來不同。但是事實上，這兩個幾何緊密相關，以一種可以精確描述的方式。

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