Rook Placements in An and Combinatorics of B-Orbit Closures

Let G be a complex reductive group, B be a Borel subgroup in G, n be the Lie algebra of the unipotent radical of B , and n * be its dual space.Let be the root system of G, and let + be the set of positive roots with respect to B .A subset of + is called a rook placement if it consists of roots with pairwise non-positive inner products.To each rook placement D one can associate the coadjoint orbit D of B in n * .By definition, D is the orbit of f D , where f D is the sum of root covectors corresponding to the roots from D .We find the dimension of D and construct a polarization of n at f D .We also study the partial order on the set of rook placements induced by the incidences among the closures of orbits associated with rook placements.