  Cubic interactions between the simplest mixed-symmetry gauge field and
gravity are constructed in anti-de Sitter (AdS) and flat backgrounds.
Nonabelian cubic interactions are obtained in AdS following various
perturbative methods including the Fradkin-Vasiliev construction, with and
without Stueckelberg fields. The action that features the maximal number of
Stueckelberg fields can be considered in the flat limit without loss of
physical degrees of freedom. The resulting interactions in flat space are
compared with a classification of vertices obtained via the antifield
cohomological perturbative method. It is shown that the gauge algebra becomes
abelian in the flat limit, in contrast to what happens for totally symmetric
gauge fields in AdS.
