Invariants and orbits of the standard \((\text{SL}_4 (\mathbb{C})\times \text{SL}_4 (\mathbb{C})\times \text{SL}_2 (\mathbb{C}))\)-module (Q2710717)

The invariants and orbits of the natural linear representation of the standard \((SL_4(C)\times SL_4(C)\times SL_2(C))\)-module (denoted as (4,4,2)) are considered. It is taken into account that the triple (4,4,2) is one of those few \((k,l,m)\)-modules for which a complete description of the orbits and invariants is possible (1980, Kac). To classify the orbits, the embedding of the original representation in the adjoint representation of the simple Lie algebra of a type \(E_7\) graded module 4 and the general results (1976, Vinberg) on the orbits and invariants of a linear group associated with a graded algebra are used. It is shown that in the case under consideration the algebra of invariants is generated by two generators of degrees 8 and 12, and an explicit form for these invariants is found. Some results related to the structure of the orbits and invariants for a general standard \((n_1,n_2,\dots,n_s)\)-module are also established.