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研究生课程《代数组合引论》教学大纲

  发布日期:2016-12-23  浏览量:219


课程编号:Math2081

课程名称:代数组合引论

英文名称:Introduction to Algebraic Combinatorics

 

开课单位:数学科学学院

开课学期:秋

课内学时:36

教学方式:英文讲授

适用专业及层次:数学科学学院基础数学专业硕士

考核方式:考试

预修课程:高等代数,图论,近世代数

[Introduction to Algebraic Combinatorics] 代数组合引论

 

一.教学目标与要求

This course is aimed at giving basic knowledge about (i) representations of finite groups and (ii) association schemes. Students are required to have basic knowledge about linear algebra and about groups, rings, fields.

 

二.课程内容与学时分配

Preface: introduction: an overview on algebraic combinatorics (2课时)

Chapter 1 group representations (8 课时)

1.1 complete reducibility

1.2 orthogonality relations of characters

1.3 induced representations

1.4 applications: permutation representations

 

Chapter 2 semi-simple algebras (6课时)

2.1 Wedderburn's theorem

2.2 group rings: group representations revisited

2.3 Hecke algebras of transitive permutation groups

 

Chapter 3 association schemes (8课时)

3.1 Bose-Mesner algebra: the 1st/2nd eigenmatrix

3.2 parameter relations: the intersection matrices

3.3 Terwilliger algebra

3.4 primitivity: sub-schemes and quotient schemes

 

Chapter 4 Hamming schemes and Johnson schemes (4课时)

4.1 the Hamming scheme H(n,q)

4.2 the Johnson scheme J(v, k)

 

Chapter 5 P/Q-polynomial schemes (8课时)

5.1 Distance-regular graphs (P-polynomial schemes) and their duals

(Q-polynomial schemes): orthogonal polynomials

5.2 Moore graphs

5.3 Leonard pairs and Askey-Wilson polynomials

5.4 Towards the classification of (P and Q)-polynomial schemes: the list

of known examples

 

四.教材

E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes,

Benjamin/Cummings, Menlo Park, California, 1984.

 

四、 主要参考书

1A. E. Brouwer, A.M. Cohen, A.Neumaier, Distance-Regular Graphs

2N. Biggs, Algebraic graph theory, Cambridge University Press, 1974.

 

大纲撰写负责人:Tatsuro Ito, 徐静

 

授课教师:Tatsuro Ito

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