Multi-centered invariants, plethysm and grassmannians

Motivated by multi-centered black hole solutions of Maxwell-Einstein theories of (super)gravity in D = 4 space-time dimensions, we develop some general methods, that can be used to determine all homogeneous invariant polynomials on the irreducible (SL h (p, ℝ) ⊗ G 4)-representation (p, R), where p denotes the number of centers, and SL h (p, ℝ) is the "horizontal" symmetry of the system, acting upon the indices labelling the centers. The black hole electric and magnetic charges sit in the symplectic representation R of the generalized electric-magnetic (U -)duality group G 4. We start with an algebraic approach based on classical invariant theory, using Schur polynomials and the Cauchy formula. Then, we perform a geometric analysis, involving Grassmannians, Plücker coordinates, and exploiting Bott's Theorem. We focus on non-degenerate groups G 4 "of type E 7" relevant for (super)gravities whose (vector multiplets') scalar manifold is a symmetric space. In the triality-symmetric stu model of \mathcal{N}= 2 supergravity, we explicitly construct a basis for the 10 linearly independent degree-12 invariant polynomials of 3-centered black holes.