  We study higher jumping numbers and generalized test ideals associated to
determinantal ideals over a field of positive characteristic. We work in
positive characteristic and give a complete characterization of both families
for ideals generated by maximal minors; and make a conjecture for the general
case. Although these invariants are understood asymptotically - as their
positive characteristic grows to infinity they essentially coincide with their
characteristic zero analogues - there are known examples where the behavior in
positive characteristic differs significantly from that of characteristic zero.
This new knowledge associated to determinantal ideals strengthens the case that
test ideals and multiplier exhibit essentially the same description even in
small characteristics. We conjecture that a similar result holds for
determinantal ideals of arbitrary size.
