In this note we shall determine which Mimura-type spherical 2-designs are rigid. Furthermore we show that spherical 2-designs of d + 2 points in S^d-1 are rigid by showing the spherical 2-design of d + 2 points is unique. We also establish that a 2-design of d + 3 points is non-rigid. This work is intended to shed some lights on conjectures of Bannai. Bannai conjectured [Ba2] that there exists a function f(d, t) such that if X is a spherical t-design in S^d-1 and |X| > f(d, t), than X is non-rigid. He has showed that f(2, t) = 2t + 1.

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The purpose of this paper is to provide a matrix representation of global transition functions of so-called Gauss-Seidel type finite celluar automata and their Jordan normal forms. The numbers of fixed points and the maximum length of cycles of these systems are calculated by using Jordan normal forms of the matrix representation of global transition functions

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Let X = X (3, 6) be the configuration space of six lines in general position in the projective plane; the space can be thought of the quotient space
X = GL (3) ? M*(3, 6; C)/(C*)⁶,
where M*(3, 6; C) is the set of 3 × 6 complex matrices such that any 3 by 3 minor does not vanish. In the previous paper [MSY], we constructed a projective compactification X of X and showed that it is the Satake compactification of the quotient of the 4-dimensional Hermitian symmetric domain H (2, 2) = {z∈M (2, 2; C)|(z − z*)/2i > 0} by an arithmetic group, say Λ acting on H (2, 2).
In this paper, we make a detailed study of combinatorial properties of the smooth affine variety X, the singular projective variety X and the parabolic parts of the arithmetic group Λ, and make, as an application of these, a toroidal compactification of X, which gives a non-singular model of X. We also treat several other arithmetic subgroups commensurable with Λ. As an appendix, we study the variety X as a toric variety.

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