Reduction of the Semistability Condition for Tensors

In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle $E$ over a smooth irreducible projective variety and a morphism of vector bundles $\phi$. As for classical vector bundles, there exists a notion of stability for these objects given in terms of filtrations of the vector bundle $E$. The aim of the present paper is to prove that if a destabilizing filtration is “too” long then there exists a shorter subfiltration which destabilizes as well. Moreover, we describe some related combinatorial problems, which arise from the description of a tensor $(E,\phi )$ or, more precisely, a filtration of $E$ as a $a$-dimensional matrix. Eventually, as example we study semistable tensors on the projective line.