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Root system; rook placement; Borel subgroup; coadjoint orbit; graded poset
A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in $A_{n-1}$ with respect to a slightly different order and prove that this poset is graded.
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