  Specht polynomials classically realize the irreducible representations of the
symmetric group. The ideals defined by these polynomials provide a strong
connection with the combinatorics of Young tableaux and have been intensively
studied by several authors. We initiate similar investigations for the ideals
defined by the Specht polynomials associated to the hyperoctahedral group
$B_n$. We introduce a bidominance order on bipartitions which describes the
poset of inclusions of these ideals and study algebraic consequences on general
$B_n$-invariant ideals and varieties, which can lead to computational
simplifications.
