On the closures of orbits of fourth order matrix pencils

In this work we state a simple criterion for nilpotentness of a square n×n matrix pencil with respect to the action of SLn(C)×SLn(C)×SL2(C). The orbits of matrix pencils are classified explicitly for n = 4 and the hierarchy of closures of nilpotent orbits is described. Also, we prove that the algebra of invariants of the action of SLn(C)× SLn(C)× SL2(C) on C ⊗ C ⊗ C is isomorphic to the algebra of invariants of binary forms of degree n with respect to the action of SL2(C).