Howe, Roger; Huang, Rosa Projective invariants of four subspaces. (English) Zbl 0852.15021 Adv. Math. 118, No. 2, 295-336 (1996). Let \(G_{n,k}\) be the Grassmann variety representing the \(k\)-flats in an \(n\)-dimensional vector space or, in other words, the \((k - 1)\)-dimensional subspaces of an \((n - 1)\)-dimensional projective space. The coordinate ring of \(G_{n,k}\) is introduced as the algebra of all polynomial functions on the affine variety formed by all pure \(k\)-fold wedge products lying in \(\Lambda^kV\). This coordinate ring carries the natural structure of a \(GL (V)\)-module.The authors discuss configurations of four subspaces with projective dimensions \(k_1 - 1\), \(k_2 - 1\), \(k_3 - 1\), \(k_4 - 1\) in terms of \(SL (V)\)-invariants in the tensor product of the corresponding coordinate rings. The special case of four self-dual subspaces \((2k_1 = \cdots = 2k_4 = n)\) has been treated by H. W. Turnbull [Proc. Edinburgh Math. Soc. 7, No. 2, 55-72 (1942)] and R. Huang [Proc. Natl. Acad. Sci. USA 87, No. 12, 4557-4560 (1990; Zbl 0717.15020)], but now the general case is described completely.It turns out that the number of invariants heavily depends on \(n\) and the \(k_i\)’s. Generators and relations for these invariants are described and the results are illustrated by several examples. Reviewer: H.Havlicek (Wien) Cited in 3 ReviewsCited in 9 Documents MSC: 15A72 Vector and tensor algebra, theory of invariants 15A69 Multilinear algebra, tensor calculus 16D40 Free, projective, and flat modules and ideals in associative algebras 51N15 Projective analytic geometry Keywords:flats; module; Grassmann variety; subspaces; projective space; wedge; coordinate ring; configurations; tensor product; invariants Citations:Zbl 0717.15020 PDFBibTeX XMLCite \textit{R. Howe} and \textit{R. Huang}, Adv. Math. 118, No. 2, 295--336 (1996; Zbl 0852.15021) Full Text: DOI