VI-modules in non-describing characteristic. II
type: Item
instance of: scholarly article
title: VI-modules in non-describing characteristic. II
zbMATH Open document ID: 1482.20008
author: Rohit Nagpal
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2021-12-01T00:00:00Z
review text: This paper establishes fundamental results about the category \(\mathrm{Mod}_{\mathrm{VI}}\) of \(\mathrm{VI}\)-modules and its localization away from torsion. Here, \(\mathrm{VI}\) is the category of finite-dimensional \(\mathbb{F}_q\)-vector spaces and injective linear maps; \(\mathrm{Mod}_{\mathrm{VI}}\) is the category of functors from \(\mathrm{VI}\) to \(\mathbf{k}\)-vector spaces, where \(\mathbf{k}\) is taken to be of non-describing characteristic for the main results. When \(\mathbf{k}\) is of characteristic zero, the results recover and strengthen those of [\textit{W. L. Gan} and \textit{J. Watterlond}, Algebr. Represent. Theory 21, No. 1, 47--60 (2018; Zbl 1485.20036)]. The category of generic \(\mathrm{VI}\)-modules is defined to be the quotient \[ \mathrm{Mod}_{\mathrm{VI}} / \mathrm{Mod}_{\mathrm{VI}}^{\mathrm{tors}} \] where \(\mathrm{Mod}_{\mathrm{VI}}^{\mathrm{tors}}\) is the full subcategory of torsion \(\mathrm{VI}\)-modules. The main results lead to the classification of the irreducible generic \(\mathrm{VI}\)-modules: their isomorphism classes are in bijection with \[ \amalg_{n\geq 0} \mathrm{Irr}(\mathrm{Mod}_{\mathbf{k}[\mathrm{GL}_n]}). \] One construction of the irreducibles is as follows. The induction functor \(\mathcal{J}\) is defined as the left adjoint to restriction to the maximal subgroupoid of \(\mathrm{VI}\) and a \(\mathrm{VI}\)-module is said to be semi-induced if it admits a finite filtration with associated graded that is induced. Thus, for \(\Theta\) an irreducible \(\mathrm{GL}_d\)-module, one has the induced \(\mathrm{VI}\)-module \(\mathcal{J}(\Theta)\). The associated irreducible generic \(\mathrm{VI}\)-module \(L(\Theta)\) is represented by the \(\mathrm{VI}\)-module \(\mathcal{L}(\Theta)\subset \mathcal{J}(\Theta)\) given as the intersection of the kernels of all maps from \(\mathcal{J}(\Theta)\) to semi-induced modules generated in degree \(<d\). The proofs rely on a different, more explicit approach, and are established…
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 7436430
zbMATH Keywords: VI-module: generic VI-module
zbMATH Keywords: irreducible
zbMATH Keywords: non-describing characteristic
zbMATH Keywords: FI-module
reviewed by: Geoffrey Powell
MaRDI profile type: Publication
full work available at URL: crelle-2021-0054
OpenAlex ID: W3207479301
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DOI: 10.1515/CRELLE-2021-0054
description: scientific article; zbMATH DE number 7436430