Arguesian identities in invariant theory
type: Item
instance of: scholarly article
title: Arguesian identities in invariant theory
zbMATH Open document ID: 0858.15019
published in: Advances in Mathematics
publication date: 1997-03-23T00:00:00Z
review text: The paper begins with a long introduction including the following observation. ``The key of classical invariant theory \dots begins with the evident remark that the facts of geometry, when expressed in terms of algebra, become dependent upon the choice of a coordinate system. Yet a statement about space can be viewed as a geometric fact only on the condition that such a fact express a property of space which is independent of the choice of a coordinate system. \dots The emphasis of classical invariant theory lies not in the description of geometric objects, such as algebraic varieties by coordinate-free devices, but in the discovery of a variety of new operations that can be defined among geometric entities in a coordinate-free manner.'' In this spirit, the remainder of the paper presents a technique for construction of geometric identities in a Grassmann-Cayley algebra. Each of these identities represents a projective invariant closely related to an \(n\)-dimensional Desargues Theorem and is proved by purely algebraic manipulation. The main theorem (Theorem 4.1) links the construction of such identities with calculations in a bipartite graph. The work was motivated in part by a paper of \textit{P. Doubilet}, \textit{G.-C. Rota} and \textit{J. Stein} [Stud. Appl. Math. 53, 185-216 (1974; Zbl 0426.05009)].
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 934302
zbMATH Keywords: Desargues theorem
zbMATH Keywords: invariant theory
zbMATH Keywords: geometric identities
zbMATH Keywords: Grassmann-Cayley algebra
zbMATH Keywords: projective invariant
zbMATH Keywords: bipartite graph
MaRDI profile type: Publication
full work available at URL: aima.1996.0056
OpenAlex ID: W2050891253
DOI: 10.1006/AIMA.1996.0056
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description: scientific article; zbMATH DE number 934302