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

TL;DR

This paper proves that quantum coordinate rings of certain flag manifolds and related varieties are AS-Cohen-Macaulay and AS-Gorenstein, using quantum graded algebras with straightening laws and Stanley's Theorem.

Contribution

It establishes the AS-Cohen-Macaulay and AS-Gorenstein properties for quantum flag manifolds of minuscule weight, extending algebraic geometric properties to the quantum setting.

Findings

Quantum homogeneous coordinate rings are AS-Cohen-Macaulay.

Quantum flag manifolds of minuscule weight are AS-Gorenstein.

Results apply to Schubert varieties, big cells, and determinantal varieties.

Abstract

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 [J. Algebra 301 (2006), 670-702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein.