Jacobians and invariants
type: Item
instance of: scholarly article
title: Jacobians and invariants
zbMATH Open document ID: 0791.32002
DOI: 10.1007/BF01232437
author: George R. Kempf
published in: Inventiones Mathematicae
publication date: 1993-12-15T00:00:00Z
full work available at URL: https://eudml.org/doc/144106
review text: Let \(f\) be a homogeneous polynomial of degree \(d\) on \(\mathbb{C}^ n\), the Jacobian \(J(f)\) is the span of all the partial derivatives in space of polynomials of degree \(d-1\). Let \(\rho\) be a representation of a connected semi-simple group \(G\) on \(\mathbb{C}^ n\). Then \(f\) is invariant if \(\rho(g)^*f=f\) for all \(g\) in \(G\). Now by the chain rule, one has \(J(\rho(g)^*f)=\rho(g)^*J(f)\) for all \(g \in G\). Hence \(J(f)\) is invariant under the \(\rho(g)^*\) action of \(G\) if \(f\) is invariant. In this paper, some converse result is obtained, namely Theorem: Assume that either a) \(f=0\) is projectively smooth or b) \(\rho\) is irreducible. Then \(f\) is invariant if \(J(f)\) is invariant and \(d \geq 3\).
reviewed by: Tan Vovan
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 447105
zbMATH Keywords: invariants
zbMATH Keywords: homogeneous polynomial of degree \(d\)
zbMATH Keywords: Jacobian
MaRDI profile type: Publication
cites work: On Jordan Algebras of Linear Transformations
cites work: The Wedderburn principal theorem for Jordan algebras
cites work: Toroidal Algebraic Groups
cites work: Q3671475
cites work: Classification of Jacobian ideals invariant by 𝑠𝑙(2,𝐶) actions
description: scientific article; zbMATH DE number 447105