  The vector space of m x n complex matrices (m >= n) admits a natural action
of the group GL = GL_m x GL_n via row and column operations. For positive
integers a,b, we consider the ideal I_{a x b} defined as the smallest
GL-equivariant ideal containing the b-th powers of the a x a minors of the
generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all
a,b, together with their GL-equivariant structure, generalizing earlier results
of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the
powers of the ideals of maximal minors (a=n). Our methods rely on a nice
connection between commutative algebra and the representation theory of the
superalgebra gl(m|n), as well as on our previous calculation of Ext modules
done in the context of describing local cohomology with determinantal support.
Our results constitute an important ingredient in the proof by
Nagpal-Sam-Snowden of the first non-trivial Noetherianity results for twisted
commutative algebras which are not generated in degree one.
