Rook placements in Aₙ and combinatorics of B-orbit closures

الأصل · EN

Let G be a complex reductive group, B be a Borel subgroup of G, be the Lie algebra of the unipotent radical of B, and be its dual space. Let Φ be the root system of G, and Φ+ be the set of positive roots with respect to B. A subset of Φ+ 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 ΩD of B in. By definition, ΩD is the orbit of fD, where fD is the sum of root covectors corresponging to the roots from D. We find the dimension of ΩD and construct a polarization of at fD. We also study the partial order on the set of rook placements induced by the incidences among the orbits associated with rook placements.