  We study quantum Schubert varieties from the point of view of regularity
conditions. More precisely, we show that these rings are domains which are
maximal orders and are AS-Cohen-Macaulay and we determine which of them are
AS-Gorenstein. One key fact that enables us to prove these results is that
quantum Schubert varieties are quantum graded algebras with a straightening law
that have a unique minimal element in the defining poset. We prove a general
result showing when such quantum graded algebras are maximal orders. Finally,
we exploit these results to show that quantum determinantal rings are maximal
orders.
