Semilinear representations of symmetric groups and of automorphism groups of universal domains
type: Item
instance of: scholarly article
title: Semilinear representations of symmetric groups and of automorphism groups of universal domains
zbMATH Open document ID: 1430.20013
published in: Selecta Mathematica. New Series
publication date: 2018-07-16T00:00:00Z
review text: Let \(K\) be a field and let \(G\) be a group of automorphisms of \(K\), endowed with the compact-open topology. The goal of this paper is to study the smooth semilinear representations of the group \(\mathcal{S}_\Psi\) of all permutations of an infinite set \(\Psi\). Suppose \(k\) is a field and \(k(\Psi)\) is the field freely generated by \(\Psi\) over \(k\). The author describes the Gabriel spectrum of the category of smooth \(k(\Psi)\)-semilinear representations of \(\mathcal{S}_\Psi\). In particular, it is shown that for any smooth \(\mathcal{S}_\Psi\)-field \(K\) any smooth finitely generated \(K\)-semilinear representation of \(\mathcal{S}_\Psi\) is Noetherian. Moreover, for any \(\mathcal{S}_\Psi\)-invariant subfield \(K\) of \(k(\Psi)\), the object \(k(\Psi)\) is an injective cogenerator of the category of smooth \(K\)-semilinear representations of \(\mathcal{S}_\Psi\). Additional properties are proven concerning the subfield of \(k(\Psi)\) of rational homogeneous functions of degree 0, respectively the subfield generated over \(k\) by \(x-y\) for all \(x,y\in \Psi\).
Mathematics Subject Classification ID: 20C32
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 6904444
zbMATH Keywords: semilinear representation
zbMATH Keywords: smooth representation
zbMATH Keywords: symmetric group
author: Marat Rovinsky
reviewed by: Frauke Bleher
MaRDI profile type: Publication
full work available at URL: s00029-017-0356-0
OpenAlex ID: W2755119910
cites work: Q5594631
cites work: Noetherian property of infinite EI categories
cites work: Groups Covered By Permutable Subsets
cites work: Semilinear representations of PGL
cites work: Admissible semi-linear representations
cites work: On maximal proper subgroups of field automorphism groups
cites work: Stable birational invariants with Galois descent and differential forms
cites work: Q5485133
DOI: 10.1007/S00029-017-0356-0
description: scientific article; zbMATH DE number 6904444