  We introduce a general framework for analysing general probabilistic
theories, which emphasises the distinction between the dynamical and
probabilistic structures of a system. The dynamical structure is the set of
pure states together with the action of the reversible dynamics, whilst the
probabilistic structure determines the measurements and the outcome
probabilities. For transitive dynamical structures whose dynamical group and
stabiliser subgroup form a Gelfand pair we show that all probabilistic
structures are rigid (cannot be infinitesimally deformed) and are in one-to-one
correspondence with the spherical representations of the dynamical group. We
apply our methods to classify all probabilistic structures when the dynamical
structure is that of complex Grassmann manifolds acted on by the unitary group.
This is a generalisation of quantum theory where the pure states, instead of
being represented by one-dimensional subspaces of a complex vector space, are
represented by subspaces of a fixed dimension larger than one. We also show
that systems with compact two-point homogeneous dynamical structures (i.e.
every pair of pure states with a given distance can be reversibly transformed
to any other pair of pure states with the same distance), which include systems
corresponding to Euclidean Jordan Algebras, all have rigid probabilistic
structures.
