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研究生课程《代数组合论》教学大纲

  发布日期:2016-12-23  浏览量:305


课程编号:Math2101

课程名称:代数组合论

英文名称:Algebraic Combinatorics

 

开课单位:数学科学学院

开课学期:春

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:考查

预修课程:高等代数,图论,近世代数,群表示论,代数组合引论

[Algebraic Combinatorics] 代数组合论

 

一.教学目标与要求

This course is aimed at giving a self contained proof to the Leonard Theorem about (P and Q)-polynomial schemes, by classifying Leonard pairs. We basically follow Terwilliger's original paper. Students are expected to have basic knowledge about group representation theory, association schemes and Terwilliger algebras.

 

二.课程内容与学时分配

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

 

Chapter 1 (P and Q)-polynomial schemes revisited (10课时)

1.1 association schemes: quick review

1.2 Distance-regular graphs (P-polynomial schemes) revisited

1.3 Q-polynomial schemes revisited in terms of the Terwilliger algebras

1.4 (P and Q)-polynomial schemes

1.5 orthogonal polynomials

 

Chapter 2 TD-pairs (tridiagonal pairs) (4 课时)

2.1 weight space decomposition

2.2 TD-relations (tridiagonal relations)

 

Chapter 3 L-pairs (Leonard pairs) (14课时)

3.1 the standard/ dual standard basis and a dual system of orthogonal polynomials

3.2 pre L-pairs

3.3 Terwilliger's  lemma

3.4 AW-relations (Askey-Wilson relations)

3.5 classification of L-pairs

3.6 dual systems of AW-polynomials

3.7 the classification of TD-pairs: outline

 

Chapter 4 the list of known (P and Q)-polynomial schemes (6课时)

4.1 the list of the cores

4.2 the list of the relatives

4.3 towards the classification of (P and Q)-polynomial schemes

 

四.教材

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

Benjamin/Cummings, Menlo Park, California, 1984.

P. Terwilliger, Two linear transformations each tridiagonal with respect

to an eigenbasis of the other, Linear Algebra Appl., 330 (2001), 149-203.

 

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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