  We study the problem of computing Gopakumar-Vafa invariants for
multiparameter families of symmetric Calabi-Yau threefolds admitting flops to
diffeomorphic manifolds. There are infinite Coxeter groups, generated by
permutations and flops, that act as symmetries on the GV-invariants of these
manifolds. We describe how these groups are related to symmetries in GLSMs and
the existence of multiple mirrors. Some representation theory of these Coxeter
groups is also discussed. The symmetries provide an infinite number of
relations between the GV-invariants for each fixed genus. This remarkable fact
is of assistance in obtaining higher-genus invariants via the BCOV recursion.
This proceedings article is based on joint work with Philip Candelas and Xenia
de la Ossa.
