Multi-Centered Invariants, Plethysm and Grassmannians

Motivated by multi-centered black hole solutions of Maxwel l-Einstein theo- ries of (super)gravity in D = 4 space-time dimensions, we develop some general meth- ods, that can be used to determine all homogeneous invariant polynomials on the irre- ducible ( 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 la- belling the centers. The black hole electric and magnetic ch arges 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 geome tric analysis, involving Grassmannians, Pl ̈ucker 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 N = 2 supergravity, we explicitly construct a basis for the 10 l inearly independent degree-12 invariant polynomials of 3-centered black holes