  We report a proof of the quantum Sanov Theorem by elementary application of
basic facts about representations of the symmetric group, together with a
complete characterization of the optimal error exponent in a situation where
the null hypothesis is given by an arbitrarily varying quantum source instead.
Our approach differs from previous ones in two points: First, it supports a
reasoning inspired by the method of types. Second, the measurement scheme we
propose to distinguish the two alternatives not only does that job
asymptotically perfect, but also yields additional information about the null
hypothesis. An example of that is given. The measurement is composed of
projections onto permutation-invariant subspaces, thus providing a direct link
between one of the most basic tasks in quantum information on the one hand side
and fundamental objects in representation theory on the other. We additionally
connect to representation theory by proving a relation between Kostka numbers
and quantum states, and to state estimation via a generalization of a
well-known spectral estimation theorem to non-i.i.d. sequences.
