In Solid State physics the reciprocal space is of utmost importance to predict the band structure and thus most of the electrical transport parameters like effective mass, etc.
The First Brillouin Zone of a certain crystal lattice with its symmetries specified by its space and point groups, is defined as the volume whose points are closer to a lattice point in the reciprocal lattice than to all other reciprocal lattice points (thus the equivalent to the Wigner Seitz cell in real space).
In the International Tables for Crystallography and also on the excellent Bilbao Crystallographic Server (listed in the row "GP"), the asymmetric unit (aka irreducible wedge) is specified as the polyhedron which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice. The geometry of this zone can be calculated from the given equations by doing a vertex enumeration and can be described by the inequality relation
$$\textbf{m}.\vec{r}-\vec{b}\leq0$$
with $\textbf{m}$ being the row matrix of the normal vectors of the surface planes and $\vec{b}$ being the distance of the plane to the coordinate center.
A similiar set of inequality relations can be defined for the full first Brillouin zone.
Now for my question: One can easily show that by applying all point group operations of the related point group of the lattice to the asymmetric unit we can get back to the first Brillouin Zone of the lattice. However is there an algebraic way how to deduce the asymmetric unit through the space group operations applied to the first Brillouin zone (thus the inverse operation applied on the inequality relation describing the first Brillouin zone)?