Stabilizers for nondegenerate matrices of boundary format and Steiner bundles

Abstract

In this paper nondegenerate multidimensional matrices of boundary format in $V_0 \otimes ... \otimes V_p$ are investigated by their link with Steiner vector bundles on product of projective spaces. For any nondegenerate matrix $A$ the stabilizer for the $SL(V_0)\times ... \times SL(V_p)$-action, $Stab(A)$, is completely described. In particular we prove that there exists an explicit action of SL(2) on $V_0 \otimes ... \otimes V_p$ such that $Stab(A)^0\subseteq SL(2)$ and the equality holds if and only if $A$ belongs to a unique $SL(V_0)\times >... \times SL(V_p)$-orbit containing the identity matrices.