Standard homological properties for quantum \(GL_ n\)
type: Item
instance of: scholarly article
title: Standard homological properties for quantum \(GL_ n\)
zbMATH Open document ID: 0858.17010
author: Stephen Donkin
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The author studies the standard homological properties for quantum \(GL_n\). The quantum version of \(GL_n\) with which he works throughout was introduced by \textit{R. Dipper} [Proc. Lond. Math. Soc., III. Ser. 63, No. 1, 165-211 (1991; Zbl 0734.20018)]. He establishes the standard homological properties for the induction functor in the quantum setting, including the representation theory of various subgroups of quantum \(GL_n\), Kempf vanishing and further representation theoretic properties. The results given in this paper are valid for all \(q\neq 0\) and (except for the Grothendieck vanishing) are obtained without invoking the corresponding results in the classical case. In particular, his proof of Kempf's vanishing theorem does not rely on properties of the cohomology of projective varieties, instead, it is based on the Koszul resolution defined by a maximal parabolic subgroup. Another feature of the proofs is that no use is made of Hecke algebras.
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 883758
zbMATH Keywords: \(q\)-Schur algebra
zbMATH Keywords: quantum group
zbMATH Keywords: quantum \(GL_ n\)
zbMATH Keywords: homological properties
zbMATH Keywords: Kempf's vanishing theorem
zbMATH Keywords: Koszul resolution
MaRDI profile type: Publication
full work available at URL: jabr.1996.0118
OpenAlex ID: W1975303984
DOI: 10.1006/JABR.1996.0118
description: scientific article; zbMATH DE number 883758