Invariant theory, orbits and non-decomposable quadruples of subspaces having non-zero defect Original Research Article

Let V be a vector space having dimension n greater-or-equal, slanted 3 over an algebraically closed field k of characteristic 0. Quadruples of subspaces of V were classified by Gelf and Ponomarev; more recently, Howe and Huang explicitly described the homogeneous generatorss Fi of the corresponding algebra of invariants. The purpose of this paper is to connect these two theories in the case where there is a non-decomposable quadruple, having non-zero defect, in the dimension class of the quadruples. We show that the non-decomposables form an open orbit characterized by the non-vanishing of each Fi; furthermore, the SL(V)-orbit of a non-decomposable is completely determined by the values of the Fi. The orbit structure for decomposable quadruples where some Fi does not vanish is more complicated but is completely described in three tables.