  This work is concerned with multi-party stabilizer states in the sense of
quantum information theory. We investigate the homological invariants for
states of which each party holds a large equal number N of quantum bits. We
show that in many cases there is a generic expected value of the invariants:
for large N it is approximated with arbitrarily high probability if a
stabilizer state is chosen at random. The result suggests that typical
entanglement of stabilizer states involves but the sets comprising just over
one half of the parties. Our main tool is the Bruhat decomposition from the
theory of finite Chevalley groups.
