The AS–Cohen–Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS–Cohen–Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [T.H. Lenagan, L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006) 670–702]. Using Stanleyʹs Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS–Gorenstein.