  To each discrete translationally periodic bar-joint framework $\C$ in $\bR^d$
we associate a matrix-valued function $\Phi_\C(z)$ defined on the d-torus. The
rigid unit mode spectrum $\Omega(\C)$ of $\C$ is defined in terms of the
multi-phases of phase-periodic infinitesimal flexes and is shown to correspond
to the singular points of the function $z \to \rank \Phi_\C(z)$ and also to the
set of wave vectors of harmonic excitations which have vanishing energy in the
long wavelength limit. To a crystal framework in Maxwell counting equilibrium,
which corresponds to $\Phi_\C(z)$ being square, the determinant of $\Phi_\C(z)$
gives rise to a unique multi-variable polynomial $p_\C(z_1,\dots,z_d)$. For
ideal zeolites the algebraic variety of zeros of $p_\C(z)$ on the d-torus
coincides with the RUM spectrum. The matrix function is related to other
aspects of idealised framework rigidity and flexibility and in particular leads
to an explicit formula for the number of supercell-periodic floppy modes. In
the case of certain zeolite frameworks in dimensions 2 and 3 direct proofs are
given to show the maximal floppy mode property (order $N$). In particular this
is the case for the cubic symmetry sodalite framework and some other idealised
zeolites.
