Akihide Hanaki

Representations of Association Schemes and Their Factor Schemes

Abstract. In the present paper we investigate the relationship between the complex representations of an association scheme G and the complex representations of certain factor schemes of G. Our first result is that, similar to group representation theory, representations of factor schemes over normal closed subsets of G can be viewed as representations of G itself. We then give necessary and sufficient conditions for an irreducible character of G to be a character of a factor scheme of G. These characterizations involve the central primitive idempotents of the adjacency algebra of G and they are obtained with the help of the Frobenius reciprocity low which we prove for complex adjacency algebras.

Quantum Galois theory for finite groups

Dong and Mason [DM1] initiated a systematic research for a vertex operator algebra with a finite automorphism group, which is referred to as the “operator content of orbifold models” by physicists [DVVV]. The purpose of this paper is to extend one of their main results. We will assume that the reader is familiar with the vertex operator algebras (VOA), see [B],[FLM]. Throughout this paper, V denotes a simple vertex operator algebra, G is a finite automorphism group of V , C denotes the complex number field, and Z denotes rational integers. Let H be a subgroup of G and Irr(G) denote the set of all irreducible CG-characters. In their paper [DM1], they studied the sub VOA V H = {v ∈ V : h(v) = v for all h ∈ H} of H-invariants and the subspace V χ on which G acts according to χ ∈ Irr(G). Especially, they conjectured the following Galois correspondence between sub VOAs of V and subgroups of G and proved it for an Abelian or dihedral group G [DM1, Theorem 1] and later for nilpotent groups [DM2], which is an origin of their title of [DM1].

Characters of Association Schemes and Normal Closed Subsets

Abstract.We consider characters of association schemes. We define a kernel like subset for a character. It is a generalization of the kernel of a group character, but it need not be normal. We give a necessary and sufficient condition for it to be normal. Using this condition, we can read all normal closed subsets from the character table.

Nilpotent schemes and group-like schemes

We give a definition of nilpotent association schemes as a generalization of nilpotent groups and investigate their basic properties. Moreover, for a group-like scheme, we characterize the nilpotency by its character products.

Characters of association schemes containing a strongly normal closed subset of prime index

We consider a relation between characters of an association scheme and its strongly normal closed subsets with prime index. As an application of our result, we show that an association scheme of prime square order with a proper strongly normal closed subset is commutative.

Classification of association schemes of small order

This paper enumerates the isomorphism classes of association schemes of order 20-28 by using the computer. It also classifies all the association schemes of order 24-28 whether their automorphism groups are transitive or not, and whether they are group case or not. Some more properties of the obtained association schemes are computed.

Irreducible Representations of Wreath Products of Association Schemes

The wreath product of finite association schemes is a natural generalization of the notion of the wreath product of finite permutation groups. We determine all irreducible representations (the Jacobson radical) of a wreath product of two finite association schemes over an algebraically closed field in terms of the irreducible representations (Jacobson radicals) of the two factors involved.

Representations of finite association schemes

This article is a survey on representation theory of association schemes including recent developments and some applications. A lot of known formulas on complex characters are also obtained as corollaries to results on finite dimensional semisimple algebras. Clifford theory of association schemes and related results are explained. Also this article contains basic parts of modular representation theory. Modular representation theory is new and remarkable method in this area. Some open questions and related results are given.

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.

On Imprimitive NonCommutative Association Schemes of Order 6

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory; cf. [8] and [7]. One of these generalizations is the observation that, similar to groups, association schemes are commutative if they have at most five elements and not necessarily commutative if they have six elements; cf. [6, (4.1)]. (In this article, all association schemes are assumed to have finite valency). While any two noncommutative groups of order 6 are isomorphic to each other, there exist infinitely many isomorphism classes of noncommutative association schemes of order 6 (among them all finite projective planes). The present article is a first attempt to obtain insight into the structure of noncommutative schemes of order 6.