Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum grassmannians

We study quantum analogues of quotient varieties, namely quantum grassmannians and quantum determinantal rings, from the point of view of regularity conditions. More precisely, we show that these rings are AS–Cohen–Macaulay and determine which of them are AS–Gorenstein. Our method is inspired by the one developed by De Concini, Eisenbud and Procesi in the commutative case. Thus, we introduce and study the notion of a quantum graded algebra with a straightening law on a partially ordered set, showing in particular that, among such algebras, those whose underlying poset is wonderful are AS–Cohen–Macaulay. Then, we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a straightening law on a wonderful poset, hence showing that they are AS–Cohen–Macaulay. In this last step, we are led to introduce and study (to some extent) natural quantum analogues of Schubert varieties.