Projective invariants of four subspaces
type: Item
instance of: scholarly article
title: Projective invariants of four subspaces
zbMATH Open document ID: 0852.15021
author: Rosa Q. Huang
author: Roger Howe
published in: Advances in Mathematics
publication date: 1996-12-04T00:00:00Z
review text: 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.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 51N15
zbMATH DE Number: 889816
zbMATH Keywords: flats
zbMATH Keywords: module
zbMATH Keywords: Grassmann variety
zbMATH Keywords: subspaces
zbMATH Keywords: projective space
zbMATH Keywords: wedge
zbMATH Keywords: coordinate ring
zbMATH Keywords: configurations
zbMATH Keywords: tensor product
zbMATH Keywords: invariants
MaRDI profile type: Publication
full work available at URL: aima.1996.0023
OpenAlex ID: W1974101602
DOI: 10.1006/AIMA.1996.0023
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description: scientific article; zbMATH DE number 889816