Group Matrices, Group Determinants and Representation Theory

1. Front Matter

   Pages i-xxv

   PDF

2. Multiplicative Forms on Algebras and the Group Determinant

   • Kenneth W. Johnson

   Pages 1-53

3. Further Group Matrices and Group Determinants

   • Kenneth W. Johnson

   Pages 55-110

4. Norm Forms and Group Determinant Factors

   • Kenneth W. Johnson

   Pages 111-136

5. S-Rings, Gelfand Pairs and Association Schemes

   • Kenneth W. Johnson

   Pages 137-175

6. The 2-Characters of a Group and the Weak Cayley Table

   • Kenneth W. Johnson

   Pages 177-209

7. The Extended k-Characters

   • Kenneth W. Johnson

   Pages 211-229

8. Fourier Analysis on Groups, Random Walks and Markov Chains

   • Kenneth W. Johnson

   Pages 231-270

9. K-Characters and n-Homomorphisms

   • Kenneth W. Johnson

   Pages 271-286

10. Fusion and Supercharacters

    • Kenneth W. Johnson

    Pages 287-311

11. Other Situations Involving Group Matrices

    • Kenneth W. Johnson

    Pages 313-340

12. Back Matter

    Pages 341-384

    PDF

This book sets out an account of the tools which Frobenius used to discover representation theory for nonabelian groups and describes its modern applications. It provides a new viewpoint from which one can examine various aspects of representation theory and areas of application, such as probability theory and harmonic analysis. For example, the focal objects of this book, group matrices, can be thought of as a generalization of the circulant matrices which are behind many important algorithms in information science.

The book is designed to appeal to several audiences, primarily mathematicians working either in group representation theory or in areas of mathematics where representation theory is involved. Parts of it may be used to introduce undergraduates to representation theory by studying the appealing pattern structure of group matrices. It is also intended to attract readers who are curious about ideas close to the heart of group representation theory, which do not usually appear in modern accounts, but which offer new perspectives.

• Group Determinant
• Group Matrix
• k-Characters of Groups
• Norm forms on algebras
• Association schemes
• S-rings
• Fusion of characters
• Fission of characters
• Weak Cayley table
• Fourier transforms
• Random walks
• combinatorics

“The book is clearly written and presents many little known facts about group matrices. No other book deals so thoroughly with this topic.” (John D. Dixon, zbMATH 1458.20001, 2021)

• Department of Mathematics, Pennsylvania State University, Abington, USA

  Kenneth W. Johnson

The author was born in Manchester, England. He has an undergraduate degree from Trinity College, Oxford and a doctorate from Queen Mary College, London. He taught at the University of The West Indies in Jamaica from 1971-1984 and has been teaching at Penn State Abington from 1984 until the present, with visiting positions (a) Iowa State University (1988-90) , (b)  Queens College, CUNY, (first   Gorenstein Professor, Spring 1998) and Brigham Young University (Spring 2006).

Policies and ethics