## "Association schemes and fusion algebras (an introduction)", Eiichi Bannai, 1993

#### Reviewed January 9, 2024

Citation: Bannai, Eiichi. "Association schemes and fusion algebras (an introduction)." Journal of algebraic Combinatorics 2 (1993): 327-344

Tags: 0-categorical, Foundational, Conformal-field-theory

This paper ties together a long story in the literature. Way back in 1942, Yoshido Kawada introduced the notion of "character algebra" in the context of the non-abelian Fourier transform:

> Kawada, Yukiyosi. "Über den Dualitätssatz der Charaktere nichtkommutativer Gruppen." Proceedings of the Physico-Mathematical Society of Japan. 3rd Series 24 (1942): 97-109.

This notion of character algebra became very influential, spreading into the burgeoning field of association schemes. Association schemes and character algebras have since found a varied array of uses. The most important of which is in the statistics, with experimental design and the analysis of variance:

> Bailey, Rosemary A. Association schemes: Designed experiments, algebra and combinatorics. Vol. 84. Cambridge University Press, 2004.

Another application is in the theory of (classical) error correcting codes:

> Sloane, Neil James Alexander. "An introduction to association schemes and coding theory." Theory and application of special functions. Academic Press, 1975. 225-260.

Of course, the irreps of a non-abelian group are the most basic example of a fusion ring. In the 1970s, these fusion rings were becoming very important in conformal field theory. The similarity between association schemes/character algebras and fusion rings was noticed by Eiichi Bannai, who wrote this article to unify the two perspectives.

The main result is Theorem 3.1: There exist a canonical bijection between fusion algebras and character algebras. In a very real sense, people have been studying fusion algebras since the 1940s. Moreover, this bijection is nice enough that large amounts of information can be carried between both sides. For example, requiring that the fusion rules in the fusion algebras be integers corresponds to choosing the character algebra to be of "integral type".

Amazingly, some of the key results about conformal field theory are algebra well-known in the association scheme literature. For instance, there is a version of the Verlinde formula dating all the way back to the original 1942 paper! Hiding under the surface of association schemes there are also modular representations, just as one would expect. In particular, the modular representations exist on the nose for the so-called "Hamming association schemes":

> Bannai, Eiichi. "Modular invariance of the character table of the Hamming association scheme H (d, q)." Journal of Number Theory 47.1 (1994): 79-92.

In the years surrounding the publication of this paper, fusion rings attracted a lot of attention in the rational conformal field theory community. See e.g. the following:

> Gepner, Doron. "Fusion rings and geometry." Communications in Mathematical Physics 141.2 (1991): 381-411.

However, the now-undisputed reference for fusion rings is Chapter 3 of Etingof's book:

> Etingof, Pavel, et al. Tensor categories. Vol. 205. American Mathematical Soc., 2016.

Here, he collects a large amount of literature, works up the correct definition, gives ample examples, proves key theorems, and places fusion rings in the larger context of tensor categories.