Vector invariants of the groups \(\text{GL}(n,\mathbb{C}t)\) and \(\text{Sp}(2m,\mathbb{C}t)\)
type: Item
instance of: scholarly article
title: Vector invariants of the groups \(\text{GL}(n,\mathbb{C}t)\) and \(\text{Sp}(2m,\mathbb{C}t)\)
zbMATH Open document ID: 0820.22010
DOI: 10.1007/BF01075053
author: V. Pereyra
publication date: 1994-05-18T00:00:00Z
review text: Let a group \(G\) act on a finite dimensional space \(W\) over the complex field \(\mathbb{C}\). The main problem of invariant theory is to describe the ring of \(G\)-invariant polynomial functions on \(W\). Let an element \(w\) of the space \(W\) smoothly depend on a parameter \(t\). It is interesting to know polynomial functions of a finite number of derivatives of \(w\) at the point \(t = 0\) that are invariant under changes of coordinates from the group \(G\) also smoothly depending on \(t\). These functions are called local invariants. The coefficients of the Taylor series of a \(G\)- invariant are local invariants if we consider the \(G\)-invariant as a function of \(t\). The main result of this paper is that for \(\text{GL}(n,{\mathbb{C}}t)\) or \(\text{Sp}(2m,\mathbb{C} t)\) and for \(W\) described above, the coefficients of the Taylor series of classical basic variants generate the ring of local invariants. A similar description is obtained for relations between basic local invariants.
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 561000
zbMATH Keywords: ring of \(G\)-invariant polynomial functions
zbMATH Keywords: changes of coordinates
zbMATH Keywords: local invariants
zbMATH Keywords: Taylor series
zbMATH Keywords: basic variants
reviewed by: A. U. Klimyk
MaRDI profile type: Publication
cites work: Q4435456
full work available at URL: https://doi.org/10.1007/bf01075053
OpenAlex ID: W2091874902
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 561000