Unstable hyperplanes for Steiner bundles and multidimensional matrices

We study some properties of the natural action of SL(V-0) x...x SL(V-p) on non-degenerate multidimensional complex matrices A is an element of P(V-0 circle times...circle times 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 is an element of P(V-0 circle times V-1 circle times V-2) detects a Steiner bundle S-A (in the sense of Dolgachev and Kapranov) on the projective space P-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(P-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 infinity; the infinity 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.