矩阵结合方案

Note from the Translator

Preface

Foreword to the Chinese Edition

List of Symbols

Chapter 1 Basic Theory of Association Schemes

1.1 Definition of Association Scheme

1.2 Examples

1.3 The Eigenvalues of Association Schemes

1.4 The Krein Parameters

1.5 S-Rings and Duality

1.6 Primitivity and Imprimitivity

1.7 Subschemes and Quotient Schemes

1.8 The Polynomial Property

1.9 The Automorphisms

Chapter 2 Association Schemes of Rectangular Matrices

2.1 Definition and Primitivity

2.2 The Polynomial Property of Association Schemes of Rectangular Matrices.

2.3 Recurrence Formulas for Intersection Numbers

2.4 The Duality of Association Schemes of Rectangular Matrices

2.5 The Automorphisms ofMat(m × n,q)

Chapter 3 Association Schemes of Alternate Matrices

3.1 Primitivity and P-polynomial Property

3.2 The Parameters of Г(1)

3.3 Recurrences for Intersection Numbers

3.4 Recurrences for Intersection Numbers: Continued

3.5 The Self-duality of Alt(n,q)

3.6 The Automorphisms of Alt(n,q)

Chapter 4 Association Schemes of Hermitian Matrices

4.1 Primitivity and P-polynomial Property

4.2 The Parameters of Graph Г(1)

4.3 Recurrences for Intersection Numbers

4.4 Recurrences for Intersection Numbers: Continued

4.5 The Self-duality of Her(n,q2)

4.6 The Automorphisms of Her(n,q2)

Chapter 5 Association Schemes of Symmetric Matrices in Odd Characteristic

5.1 The Normal Forms of Symmetric Matrices

5.2 The Association Schemes of Symmetric Matrices and Their Primitivity

5.3 Sym(n,q) for Small n

5.4 A Few Enumeration Formulas from Orthogonal Geometry

5.5 Calculation of Intersection Numbers

5.6 Calculation of Intersection Numbers: Continued

5.7 The Association Scheme Quad(n,q)

5.8 The Self-duality of Sym(n,q)

5.9 The Automorphisms of Sym(n,q)

Chapter 6 Association Schemes of Symmetric Matrices in Even Characteristic

6.1 The Normal Forms of Symmetric Matrices

6.2 The Imprimitivity of Sym(n,q)

6.3 The Association Scheme Sym(2,q)

6.4 Some Results of Pseudo-symplectic Geometry

6.5 Calculation of Intersection Numbers

6.6 Calculation of Intersection Numbers: Continued

6.7 A Fusion Scheme of Sym(n,q)

6.8 The Automorphisms of Sym(n,q)

Chapter 7 Association Schemes of Quadratic Forms in Even Characteristic--.

7.1 The Normal Forms of Quadratic Forms

7.2 Qua(2,q) and Qua(3,q)

7.3 Some Enumeration Formulas from Orthogonal Geometry

7.4 Calculation of Intersection Numbers

7.5 The Duality of Association Schemes of Quadratic Forms

7.6 The Imprimitivity of Association Schemes of Quadratic Forms

7.7 Two Fusion Schemes of Qua(n,q)

7.8 The Automorphisms of Association Schemes of Quadratic Forms

Chapter 8 The Eigenvalues of Association Schemes of Quadratic Forms

8.1 The Eigenvalues of Association Scheme Qua(2,q)

8.2 Some Lemmas on Additive Characters

8.3 The 1-extensions and fr(n)

8.4 Values of fr(n) on the Union Classes C2i(n)

8.5 The 2-extensions and f2k*(n)

8.6 Values of f2k*(n) on Classes C2i(n) and C2i(n)∪C2i-1(n)

8.7 Dual Schemes of Two Fusion Schemes of Qua(n,q)

8.8 Eigenvalues of Small Association Schemes of Quadratic Forms

References

Index

This title is the first in a series of mathematics texts and monographs co-published byScience Press， China and Jones & Bartlett Learning of Sudbury， Massachusetts. Theprimary goal of this partnership is to bring Chinese mathematical research findings to theglobal， English-speaking mathematics community. Each title in this series is put through acareful translation process to promote accuracy.
　　Association Schemes of Matrices is an 8-chapter monograph that has met with widesuccess throughout the Chinese mathematics community. Themain emphasis of this bookis on the association schemes of various types of matrices. The matrix method here iselementary and requires little mathematical background. An undergraduate student who hascompleted a course in linear algebra and abstract algebra will be able to take on a researchproject on matrix groups and their geometries with the matrix techniques that are presentedby the mathematicians that have contributed to this project. The matrix method has provento be effective when the underlying field has fewer elements or in lower dimensions.