On the exterior calculus of invariant theory
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 Laplace's expansion, and culminating in a distributive law which is very useful in invariant theory. A final section contains a discussion of universal algebras having operations of the same type as G(V). Fundamental questions on invariant theory and synthetic geometry can be restated in terms of the structure of such algebras.
multilinear nondegenerate alternating n-form
scientific article; zbMATH DE number 3937344
scientific article