Projective invariants of four subspaces
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 \textit{H. W. Turnbull} [Proc. Edinburgh Math. Soc. 7, No. 2, 55-72 (1942)] and \textit{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.
scientific article; zbMATH DE number 889816