On the depth of the invariants of the symmetric power representations of \(SL_2(\mathbb{F}_p)\)
type: Item
instance of: scholarly article
title: On the depth of the invariants of the symmetric power representations of \(SL_2(\mathbb{F}_p)\)
zbMATH Open document ID: 0944.13004
author: R. J. Shank
author: David Wehlau
published in: Journal of Algebra
publication date: 1999-12-20T00:00:00Z
review text: The authors study the depth of the invariant ring of \(\text{SL}_2(p)\) acting on symmetric powers of the natural representation. In other words, the topic of this paper is the modular version of ``invariants of binary forms'', which is a recurrent theme in classical invariant theory. The main result is that the depth is exactly 3 if \(\text{SL}_2(p)\) acts on the \(n\)-th symmetric power, where \(2 \leq n < p\) and gcd\((n,p-1) \leq 2\). The authors also determine the depth for \(n = 3\) or 4 and \(p\) arbitrary. The result is always 3, except if \((n,p) \in \{(3,7),(4,5)\}\), where the invariant ring is Cohen-Macaulay. To obtain their results, the authors first reduce the question to the Borel subgroup \(B\) of \(\text{SL}_2(p)\), using a result of the reviewer [\textit{G. Kemper}, J. Algebra 215, No. 1, 330-351 (1999; Zbl 0934.13003)]. Then they explicitly give a homogeneous system of parameters for the invariants of \(B\) and prove that the first three polynomials of this system form a maximal regular sequence. The proof is achieved by using Gröbner bases, among other methods. In view of the fact that previously the cyclic groups were essentially the only ones for which the depth of the modular invariant ring was known [see \textit{G. Ellingsrud} and \textit{T. Skjelbred}, Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)], this paper constitutes a very significant progress.
reviewed by: Gregor Kemper
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 1348176
zbMATH Keywords: depth of invariant rings
zbMATH Keywords: invariants of binary forms
zbMATH Keywords: Gröbner basis
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: jabr.1999.7878
OpenAlex ID: W1971244073
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DOI: 10.1006/JABR.1999.7878
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description: scientific article; zbMATH DE number 1348176