Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ n . For an integer t, a finite subset X of ℝ n given together with a weight function w is a Euclidean t-design if $\sum_{i=1}^p\frac{w(X_i)}{|S_i|} \int_{S_i}f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x\in X}w(\boldsymbol x) f(\boldsymbol x)$\sum_{i=1}^p\frac{w(X_i)}{|S_i|} \int_{S_i}f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x\in X}w(\boldsymbol x) f(\boldsymbol x) holds for any polynomial f(x) of deg(f)≤ t, where {S i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X i = X∩ S i , and w:X→ ℝ> 0. (The case of X⊂ S n−1 with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.

Etsuko Bannai
Kazuki Kawasaki
Yusuke Nitamizu
Teppei Sato

In this paper we show that if X is an s-distance set in Ropf m and X is on p concentric spheres then |X| les Sigma i=02p-1 ( s-im+s-i-1). Moreover if X is antipodal, then |X| les 2Sigma i=0p-1 ( m-1m+s-2i-2)

It is known that for each matrix W i and it's transpose t W i in any four-weight spin model (X, W 1, W 2, W 3, W 4; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W i ) and N( t W i ) = N′(W i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W 1) = N′(W 1) = N(W 3) = N′(W 3) holds. In this paper we show that all of them coincide, that is, N(W i ), N′(W i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra.

Duality maps of finite abelian groups are classified. As a corollary； spin models on finite abelian groups which arise from the solutions of the modular invariance equations are determined as tensor products of indecomposable spin models. We also classify finite abelian groups whose Bose-Mesner algebra can be generated by a spin model.

Eiichi Bannai
Jung Rae Cho
Mitsugu Hirasaka
Hyun Kwang Kim
Jack H. Koolen
Yoshihiro Mizoguchi
Akihiro Munemasa
Andreas Holmsen
Younjin Kim
Sang-il Oum
Etsuko Bannai

• Etsuko Bannai (at Shanghai) Title: Tight relative t-designs on Q-polynomial association schemes. Abstract: The concepts of t-design and relative t-design are defined on Qpolynomial association schemes by P. Delsarte. Euclidean t-design is defined as a generalization of spherical t-design. During the studies of Euclidean designs, we found similarities between relative t-designs on Q-polynomial association schemes and Euclidean t-designs. We give the definition of relative t-design on Q-polynomial scheme in the new style, which is similar as Euclidean t-design. The Fisher type lower bounds for the cardinalities of relative 2e-design are known in terms of the dimension of the column space of primitive idempotents of the given Q-polynomial schemes. However the explicit formulas are very difficult to obtain for general cases. The works by Z. Xiang, B-B-Suda-Tanaka shows that in the following cases we have the explicit formula for the Fisher type bounds: the Hamming schemes H(n, q), Johnson schemes J(n, d), and P-and Q-polynomial schemes with some additional conditions. In this talk we consider the tight relative 2-desings on 2 shells in H(n, 2) and J(n, d) with small d. These are joint work with Eiichi Bannai, Hideo Bannai and Y. Zhu. The concepts of t-design and relative t-design are defined on Qpolynomial association schemes by P. Delsarte. Euclidean t-design is defined as a generalization of spherical t-design. During the studies of Euclidean designs, we found similarities between relative t-designs on Q-polynomial association schemes and Euclidean t-designs. We give the definition of relative t-design on Q-polynomial scheme in the new style, which is similar as Euclidean t-design. The Fisher type lower bounds for the cardinalities of relative 2e-design are known in terms of the dimension of the column space of primitive idempotents of the given Q-polynomial schemes. However the explicit formulas are very difficult to obtain for general cases. The works by Z. Xiang, B-B-Suda-Tanaka shows that in the following cases we have the explicit formula for the Fisher type bounds: the Hamming schemes H(n, q), Johnson schemes J(n, d), and P-and Q-polynomial schemes with some additional conditions. In this talk we consider the tight relative 2-desings on 2 shells in H(n, 2) and J(n, d) with small d. These are joint work with Eiichi Bannai, Hideo Bannai and Y. Zhu. • Jeongok Choi (GIST) Title: Decomposition of Regular Hypergraphs. Abstract: An r-block is a 0, 1-matrix in which every row has sum r. Let Sn be the set of pairs (k, l) such that the columns of any (k ； l)-block with n rows split into a k-block and an l-block. We determine Sn for n ≤ 5. In particular, S3 = {(k, l) : 2|kl}, S4 = {(k, l) : (6|k or l) and k, l > 1}, and S5 = {(k, l) : 11 6= min{k, l} > 7 and each value in {3, 4, 5} divides k or l}. The problem arose from a list-coloring problem in digraphs and is a refinement of the notion of indecomposable hypergraphs. This is joint work with D. B. West. An r-block is a 0, 1-matrix in which every row has sum r. Let Sn be the set of pairs (k, l) such that the columns of any (k ； l)-block with n rows split into a k-block and an l-block. We determine Sn for n ≤ 5. In particular, S3 = {(k, l) : 2|kl}, S4 = {(k, l) : (6|k or l) and k, l > 1}, and S5 = {(k, l) : 11 6= min{k, l} > 7 and each value in {3, 4, 5} divides k or l}. The problem arose from a list-coloring problem in digraphs and is a refinement of the notion of indecomposable hypergraphs. This is joint work with D. B. West.

• Etsuko Bannai (at Shanghai) Title: Tight relative t-designs on Q-polynomial association schemes. Abstract: The concepts of t-design and relative t-design are defined on Qpolynomial association schemes by P. Delsarte. Euclidean t-design is defined as a generalization of spherical t-design. During the studies of Euclidean designs, we found similarities between relative t-designs on Q-polynomial association schemes and Euclidean t-designs. We give the definition of relative t-design on Q-polynomial scheme in the new style, which is similar as Euclidean t-design. The Fisher type lower bounds for the cardinalities of relative 2e-design are known in terms of the dimension of the column space of primitive idempotents of the given Q-polynomial schemes. However the explicit formulas are very difficult to obtain for general cases. The works by Z. Xiang, B-B-Suda-Tanaka shows that in the following cases we have the explicit formula for the Fisher type bounds: the Hamming schemes H(n, q), Johnson schemes J(n, d), and P-and Q-polynomial schemes with some additional conditions. In this talk we consider the tight relative 2-desings on 2 shells in H(n, 2) and J(n, d) with small d. These are joint work with Eiichi Bannai, Hideo Bannai and Y. Zhu. The concepts of t-design and relative t-design are defined on Qpolynomial association schemes by P. Delsarte. Euclidean t-design is defined as a generalization of spherical t-design. During the studies of Euclidean designs, we found similarities between relative t-designs on Q-polynomial association schemes and Euclidean t-designs. We give the definition of relative t-design on Q-polynomial scheme in the new style, which is similar as Euclidean t-design. The Fisher type lower bounds for the cardinalities of relative 2e-design are known in terms of the dimension of the column space of primitive idempotents of the given Q-polynomial schemes. However the explicit formulas are very difficult to obtain for general cases. The works by Z. Xiang, B-B-Suda-Tanaka shows that in the following cases we have the explicit formula for the Fisher type bounds: the Hamming schemes H(n, q), Johnson schemes J(n, d), and P-and Q-polynomial schemes with some additional conditions. In this talk we consider the tight relative 2-desings on 2 shells in H(n, 2) and J(n, d) with small d. These are joint work with Eiichi Bannai, Hideo Bannai and Y. Zhu. • Jeongok Choi (GIST) Title: Decomposition of Regular Hypergraphs. Abstract: An r-block is a 0, 1-matrix in which every row has sum r. Let Sn be the set of pairs (k, l) such that the columns of any (k ； l)-block with n rows split into a k-block and an l-block. We determine Sn for n ≤ 5. In particular, S3 = {(k, l) : 2|kl}, S4 = {(k, l) : (6|k or l) and k, l > 1}, and S5 = {(k, l) : 11 6= min{k, l} > 7 and each value in {3, 4, 5} divides k or l}. The problem arose from a list-coloring problem in digraphs and is a refinement of the notion of indecomposable hypergraphs. This is joint work with D. B. West. An r-block is a 0, 1-matrix in which every row has sum r. Let Sn be the set of pairs (k, l) such that the columns of any (k ； l)-block with n rows split into a k-block and an l-block. We determine Sn for n ≤ 5. In particular, S3 = {(k, l) : 2|kl}, S4 = {(k, l) : (6|k or l) and k, l > 1}, and S5 = {(k, l) : 11 6= min{k, l} > 7 and each value in {3, 4, 5} divides k or l}. The problem arose from a list-coloring problem in digraphs and is a refinement of the notion of indecomposable hypergraphs. This is joint work with D. B. West.

A spin model is a triple (X,W；,W−), where W； and W− are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from (X,W；,W−)). We show that these equations imply the existence of a certain isomorphism 9 between two algebras M and H associated with (X,W；,W−). When M = H = A, A is the Bose-Mesner algebra of some association scheme, and 9 is a duality of A. These results had already been obtained in [15] when W；,W− are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of “spin models at the algebraic level”. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.