# Introduction

Chapter
Part of the Algebra and Applications book series (AA, volume 16)

## Abstract

Arad and Chen proved that a Normalized Integral Table Algebra (fusion ring) (A,B) generated by a non-real faithful element b 3B of degree three the non-identity elements of which have minimal degree 3 satisfies the condition $$b_{3} \bar{b}_{3} = 1 + b_{8}$$ where b 8B is an element of degree 8. They also showed that the general case naturally splits into four main sub-cases:
1. (1)

(A,B)≅ x (CH(PSL(2,7),Irr(PSL(2,7)));

2. (2)

$$b_{3}^{2} = b_{4} + b_{5}$$ where b 4,b 5B are elements of degrees 4 and 5;

3. (3)

$$b_{3}^{2} = \bar{b}_{3} + b_{6}$$ where b 6B is a non-real element of degree 6;

4. (4)

$$b_{3}^{2} = c_{3} + b_{6}$$ where c 3,b 6B are elements of degrees 3 and 6, $$c_{3}\neq b_{3},\bar{b}_{3}$$.

The cases (1), (3) and (4) are considered in Chap.  of the book. Chapter  deals with the case (2). Chapters  and analize the most complicated case—the third one. We developed new original methods for enumerating NITAs in the title. Using the developed technique we settled the above cases almost completely. The results obtained in the book and methods developed there will be of interest in the representation theory of finite groups, fusion rules algebras and association schemes.

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## Authors and Affiliations

• 1
• 2
Email author
• Xu Bangteng
• 3
• Guiyun Chen
• 4
• Effi Cohen
• 1
• Arisha Haj Ihia Hussam
• 5
• Mikhail Muzychuk
• 6
1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael