  We study a large family of products of Borel fixed ideals of maximal minors.
We compute their initial ideals and primary decompositions, and show that they
have linear free resolutions. The main tools are an extension of straightening
law and a very surprising primary decomposition formula. We study also the
homological properties of associated multi-Rees algebra which are shown to be
Cohen-Macaulay, Koszul and defined by a Gr\"obner basis of quadrics.
