Ring of invariants of general linear group over local ring \(\mathbb Z_{p^{m} }\)
type: Item
instance of: scholarly article
title: Ring of invariants of general linear group over local ring \(\mathbb Z_{p^{m} }\)
zbMATH Open document ID: 1229.13007
author: Yin Chen
author: Jizhu Nan
published in: Frontiers of Mathematics in China
publication date: 2011-11-04T00:00:00Z
review text: Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, and \(G=\mathrm{GL}_n(\mathbb{F}_q)\) be the general linear group. The group \(G\) acts naturally on the polynomial ring \(\mathbb{F}_q[x_1, \dots, x_n]\). The ring of invariants \(\mathbb{F}_q[x_1, \dots, x_n]^G\) is a polynomial ring over \(\mathbb{F}_q\); in fact, it is generated by the Dickson invariants (see [\textit{L. E. Dickson}, American M. S. Trans. 12, 75--98 (1911; JFM 42.0136.01); \textit{C. Wilkerson}, Contemp. Math. 19, 421--434 (1983; Zbl 0525.55013)]. The main result of this paper is to find the ring of invariants for an analogous situation by replacing the finite field \(\mathbb{F}_q\) by the finite ring \(\mathbb{Z}_{p^m}\) of characteristic \(p^m\) where \(p\) is a prime number. Thus the authors determine the ring of invariants of \(\mathbb{Z}_{p^m}[x_1, \ldots, x_n]\) under the action of \(G=\mathrm{GL}_n(\mathbb{Z}_{p^m})\).
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 5968164
zbMATH Keywords: Dickson invariants
reviewed by: Ming-Chang Kang
MaRDI profile type: Publication
full work available at URL: s11464-011-0151-8
OpenAlex ID: W2038085266
cites work: Q4273960
cites work: Computational invariant theory
cites work: Efficient generation of the ring of invariants
cites work: Generators for the Algebra of Symmetric Polynomials
cites work: Rational invariants of certain classical similitude groups over finite fields
cites work: Explicit generators of the invariants of finite groups
cites work: Q4838154
cites work: Polynomial invariants of finite groups. A survey of recent developments
cites work: A primer on the Dickson invariants
DOI: 10.1007/S11464-011-0151-8
description: scientific article; zbMATH DE number 5968164