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Projective invariants of four subspaces. (English) Zbl 0852.15021

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)

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

Citations:

Zbl 0717.15020
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