Unstable hyperplanes for Steiner bundles and multidimensional matrices

Abstract

We study some properties of the natural action of $SL(V_0) \times...\times SL(V_p)$ on nondegenerate multidimensional complex matrices $A\in¶(V_0\otimes...\otimes V_p)$ of boundary format(in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non stable ones,as the matrices which are in the orbit of a "triangular" matrix, and the matrices with a stabilizer containing $\C^*$, as those which are in the orbit of a "diagonal" matrix. For $p=2$ it turns out that a non degenerate matrix $A\in¶(V_0\otimes V_1\otimes V_2)$ detects a Steiner bundle $S_A$ (in the sense of Dolgachev and Kapranov) on the projective space $¶^{n}, n = dim (V_2)-1$. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SL(2) and that the SL(2)-invariant Steiner bundles are exactly the bundles introduced by Schwarzenberger [Schw], which correspond to "identity" matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of $Aut(¶^n)$, answering in the first nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of $S_A$ (counting multiplicities) produces an interesting discrete invariant of $A$, which can take the values $0, 1, 2, ... ,\dim~V_0+1$ or $ \infty$; the $\infty$ case occurs if and only if $S_A$ is Schwarzenberger (and $A$ is an identity). Finally, the Gale transform for Steiner bundles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.