Vector invariants of \(\mathsf{Syl}_p(\mathsf{GL}(n, \mathbb{F}_q))\) and their Hilbert ideals
type: Item
instance of: scholarly article
title: Vector invariants of \(\mathsf{Syl}_p(\mathsf{GL}(n, \mathbb{F}_q))\) and their Hilbert ideals
zbMATH Open document ID: 1326.13002
author: Chris Monico
author: Mara D. Neusel
published in: Advances in Mathematics
publication date: 2015-10-28T00:00:00Z
review text: Let \(\mathbb F=\mathbb F_q\) be a finite field with \(q=p^s\) elements for any prime \(p\). The upper unitriangular matrices in the general linear group \(GL(d,\mathbb F)\) constitute a \(p\)-Sylow subgroup \(P\) of \(GL(d,\mathbb F)\). In the present article are considered the \(n\)-fold invariants of \(P\), i.e., the embeddings of \(P\) in \(\mathrm{GL}(dn,\mathbb F)\). The Hilbert ideal of the ring of invariants \(\mathbb F[V]^P\) is defined as the ideal in the ambient ring of polynomials generated by all invariants of positive degree. The main result in the paper under review is that the Hilbert ideal is generated by the top orbit Chern classes of the standard dual basis of \(V^*\). This result actually follows from the second main theorem proved by the authors, which gives the generators of the ring of invariants \(\mathbb F[V]^P\).
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 6499704
zbMATH Keywords: invariant theory of finite groups
zbMATH Keywords: \(p\)-Sylow groups
zbMATH Keywords: vector invariants
zbMATH Keywords: generating set
zbMATH Keywords: transfer variety
zbMATH Keywords: Hilbert ideal
MaRDI profile type: Publication
full work available at URL: j.aim.2015.08.022
OpenAlex ID: W1552156930
cites work: Vector invariants of \(U_ 2(\mathbb{F}_ p)\): A proof of a conjecture of Richman
cites work: Vector invariants for the two-dimensional modular representation of a cyclic group of prime order
cites work: Vector invariants in arbitrary characteristic
cites work: Invariants of some abelian $p$-groups in characteristic $p$
cites work: Invariant theory of finite groups
cites work: On vector invariants over finite fields
cites work: Q4902916
DOI: 10.1016/J.AIM.2015.08.022
description: scientific article; zbMATH DE number 6499704