Schur Modules and Segre Varieties

نویسنده

  • NICHOLAS ERIKSSON
چکیده

This paper is an elementary introduction to the methods of Landsberg and Manivel, [3] for finding the ideals of secant varieties to Segre varieties. We cover only the most basic topics from [3], but hope that since this is a topic which is rarely made explicit, these notes will be of some use. We assume the reader is familiar with the basic operations of multilinear algebra: tensor, symmetric, and wedge products. For background on these topics, see [1, Appendix B]. For problems in algebraic statistics [2] and linear algebra [4] it is important to determine the ideals of secant varieties of Segre varieties. A secant variety is defined as follows. Let X be a projective variety. Then the r − 1-st secant variety to X, denoted σr(X), is the algebraic closure of the set of secant Pr−1’s to X. For example, if r = 2, this is the set of all lines passing through 2 points on X. We are interested in the case of the Segre variety, X = Seg(PA1×· · ·×PAk). (For simplicity of notation, since we are primarily concerned with the ideal of X, we will write X as the product of the dual spaces.) Notice that GL(A1)×· · ·×GL(Ak) acts on X as well as on the space of degree d polynomials, S(A1⊗· · ·⊗AK). Therefore we can decompose this space into irreducible representations of GL(A1) × · · · × GL(Ak). In order to do this, we shall first describe the irreps of GL(V ). Then we use these irreps to give a decomposition of the homogeneous parts of the ideals of secant varieties of Segre varieties. Finally, we show how to explicitly construct this decomposition for the case Seg(P × P).

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تاریخ انتشار 2004