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We describe quantum symmetries associated with the $F_4$ Dynkin diagram. Our study stems from an analysis of the (Ocneanu) modular splitting equation applied to a partition function which is invariant under a particular congruence subgroup of the modular group.
We provide a complete classification of asymptotic quasinormal fre- quencies for static, spherically symmetric black hole spacetimes in $d$ dimensions. This includes all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi–Kodama master equations. The frequencies for Schwarzschild are dimension independent, while for Reissner–Nordström are dimension dependent (the extremal Reissner–Nordström case must be considered separately from the non–extremal case). For Schwarzschild de Sitter, there is a dimension independent formula for the frequencies, except in dimension $d = 5$ where the formula is different. For Reissner–Nordström de Sitter there is a dimension dependent formula for the frequencies, except in dimension $d = 5$ where the formula is different. Schwarzschild and Reissner–Nordström Anti–de Sitter black hole spacetimes are simpler: the formulae for the frequencies will depend upon a parameter related to the tortoise coordinate at spatial infinity, and scalar type perturbations in dimension $d = 5$ lead to a continuous spectrum for the quasinormal frequencies. We also address non–black hole spacetimes, such as pure de Sitter spacetime— where there are quasinormal modes only in odd dimensions—and pure Anti–de Sitter spacetime—where again scalar type perturbations in dimension $d = 5$ lead to a continuous spectrum for the normal frequencies. Our results match previous numerical calculations with great accuracy. Asymptotic quasinormal frequencies have also been applied in the framework of quantum gravity for black holes. Our results show that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity. In an effort to keep this paper self–contained we also review earlier results in the literature.