  Determining the relationship between composite systems and their subsystems
is a fundamental problem in quantum physics. In this paper we consider the
spectra of a bipartite quantum state and its two marginal states. To each
spectrum we can associate a representation of the symmetric group defined by a
Young diagram whose normalised row lengths approximate the spectrum. We show
that, for allowed spectra, the representation of the composite system is
contained in the tensor product of the representations of the two subsystems.
This gives a new physical meaning to representations of the symmetric group. It
also introduces a new way of using the machinery of group theory in quantum
informational problems, which we illustrate by two simple examples.
