  We propose a framework to describe and simulate a class of many-body quantum
states. We do so by considering joint eigenspaces of sets of monomial unitary
matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one
entry per row and column is nonzero. We show that M-spaces encompass various
important state families, such as all Pauli stabilizer states and codes, the
AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset
states, W states and the locally maximally entanglable states. We furthermore
show how basic properties of M-spaces can transparently be understood by
manipulating their monomial stabilizer groups. In particular we derive a
unified procedure to construct an eigenbasis of any M-space, yielding an
explicit formula for each of the eigenstates. We also discuss the computational
complexity of M-spaces and show that basic problems, such as estimating local
expectation values, are NP-hard. Finally we prove that a large subclass of
M-spaces---containing in particular most of the aforementioned examples---can
be simulated efficiently classically with a unified method.
