On the exterior calculus of invariant theory
type: Item
instance of: scholarly article
title: On the exterior calculus of invariant theory
zbMATH Open document ID: 0585.15005
DOI: 10.1016/0021-8693(85)90043-2
author: Marilena Barnabei
author: Andrea Brini
author: Gian-Carlo Rota
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: This paper is rather unusual for a mathematical work in that it has a strong polemic content. The authors contend that the full significance of the work of Hermann Grassmann on the exterior calculus has not been appreciated as yet, and they have written this paper so that they can bring to light the geometrical content of the exterior calculus which is hidden by the usual treatments of the theory. The basic mathematical object which they consider is a ''Peano space'' V; that is, an n-dimensional vector space V over an arbitrary field K together with a ''bracket'' [,...,]: \(V^ n\to K\), which is a multilinear nondegenerate alternating n-form on V. Given a Peano space V, the exterior algebra G(V) is defined in the standard way, but the authors depart from usual practice by indicating the exterior product as \(''x_{\vee}y''\) rather than \(''x_{\wedge}y''\). The reason for this change is that the product in G(V) properly corresponds to the join of two subspaces. Given a subspace A of V with basis \(\{a_ 1,...,a_ k\}\), there is an associated ''extensor'' \(F=a_{1\vee}..._{\vee}a_ k\), which is an invariant of A to within nonzero scalar multiples. If B is another subspace of V with extensor G, then \((i)\quad F_{\vee}G=0\quad if\quad and\quad only\quad if\quad A_{\cap}B\neq 0,\) (ii) if \(A_{\cap}B=0\), the extensor \(F_{\vee}G\) is associated to the space spanned by \(A_{\cup}B.\) The geometrical significance of G(V) is drawn out by defining a ''cap'' product \(F_{\wedge}G\) that corresponds to the meet of subspaces: if \(A_{\cup}B\) spans V and \(A_{\cap}B\neq 0\), then \(F_{\wedge}G\) corresponds to \(A_{\cap}B\). It is shown that there is a ''Hodge Star Operator'' * on G(V) which exhibits the duality between the two products: with F as above, *F is an (n-k)-fold extensor, \(**F=(-1)^{k(n-k)}F\), and \(*(x_{\vee}y)=*x_{\wedge}*y\). The authors deduce a number of ingeneous identities for the join and meet, including many classical relations for determinants, such as …
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15-03
Mathematics Subject Classification ID: 01A55
zbMATH DE Number: 3937344
zbMATH Keywords: Hermann Grassmann
zbMATH Keywords: exterior calculus
zbMATH Keywords: Peano space
zbMATH Keywords: multilinear nondegenerate alternating n-form
zbMATH Keywords: exterior algebra
zbMATH Keywords: Hodge Star Operator
zbMATH Keywords: join
zbMATH Keywords: meet
zbMATH Keywords: determinants
zbMATH Keywords: Laplace's expansion
zbMATH Keywords: invariant theory
zbMATH Keywords: universal algebras
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90043-2
OpenAlex ID: W2018878283
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cites work: Q5816742
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cites work: Multilinear Algebra
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cites work: On the Foundations of Combinatorial Theory: IX Combinatorial Methods in Invariant Theory
cites work: Skew-symmetric invariant theory
cites work: Invariant theory, Young bitableaux, and combinatorics
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description: scientific article; zbMATH DE number 3937344