Let G be a finite group not of prime power order. A gap G-module V is a finite-dimensional real G-representation space satisfying the following two conditions. The first is the condition dim VP > 2 dim VH for all P < H ≤ G such that P is of prime power order and the other is the condition that V has only one H-fixed point 0 for all large subgroups H : precisely to say, H ∈ L(G). If there exists a gap G-module, then G is called a gap group. We study G-modules induced from C-modules for subgroups C of G and obtain a sufficient condition for G to become a gap group. Consequently, we show that non-solvable general linear groups and the automorphism groups of sporadic groups are all gap groups.
Through the modular embedding of the complex n-dimensional ball BnC into the Siegel upper half-space Sn+1 of degree n + 1 with respect to the Eisenstein integers Z[ω], we pull back the theta constants on Sn+1. We find a condition on the characteristics of the theta constants so that the pullbacks are non-zero automorphic forms on BnC with respect to the congruence subgroup Γ(1−ω). These automorphic forms are real valued on the real ball naturally embedded in the complex ball.
In this paper we show that Hodge numbers hp,q of a compact Sasakian (2n + 1)-manifold M, 2n + 1 ≥ 5, are given by the dimension of certain linear subspaces of the Milnor algebras C[zi]/(∂f/∂zi), when M is a link of an isolated singularity associated with a weighted homogeneous polynomial f = f(z1,…,zn+2).
We study the extremal values of the double and the triple trigonometric functions defined via the infinite products. In particular, we show that the extremal values of the triple sine function are intimately related to the mysterious value ζ(3). The results also allow us to sketch their graphs.