  We study the PI degree of various quantum algebras at roots of unity,
including quantum Grassmannians, quantum Schubert varieties, partition
subalgebras, and their associated quantum affine spaces. By a theorem of De
Concini and Procesi, the PI degree of partition subalgebras and their
associated quantum affine spaces is controlled by skew-symmetric integral
matrices associated to (Cauchon-Le) diagrams. We prove that the invariant
factors of these matrices are always powers of 2. This allows us to compute
explicitly the PI degree of partition subalgebras.
  Our results also apply to certain completely prime (homogeneous) quotients of
partition subalgebras. In particular, our results allow us to extend results of
Jakobsen and Jondrup regarding the PI degree of quantum determinantal rings at
roots of unity [JJ01] and we present a method to construct an irreducible
representation of maximal dimension for quantum determinantal ideals.
  Building on these results, we use the strong connection between partition
subalgebras and quantum Schubert varieties through noncommutative
dehomogenisation [LR08] to obtain expressions for the PI degree of quantum
Schubert varieties. In particular, we compute the PI degree of quantum
Grassmannians.
