  We derive symmetric and antisymmetric kernels by symmetrizing and
antisymmetrizing conventional kernels and analyze their properties. In
particular, we compute the feature space dimensions of the resulting polynomial
kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric
and antisymmetric Gaussian kernels are dense in the space of symmetric and
antisymmetric functions, and propose a Slater determinant representation of the
antisymmetric Gaussian kernel, which allows for an efficient evaluation even if
the state space is high-dimensional. Furthermore, we show that by exploiting
symmetries or antisymmetries the size of the training data set can be
significantly reduced. The results are illustrated with guiding examples and
simple quantum physics and chemistry applications.
