Geodesic
Gradedness of the set of rook placements in $A_{n-1}$
Communications in Mathematics, Tome 29 (2021) no. 2, pp. 171-182.

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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.
Classification : 06A07, 17B08, 17B22
Keywords: Root system; rook placement; Borel subgroup; coadjoint orbit; graded poset 
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     title = {Gradedness of the set of rook placements in $A_{n-1}$},
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Ignatev, Mikhail V. Gradedness of the set of rook placements in $A_{n-1}$. Communications in Mathematics, Tome 29 (2021) no. 2, pp. 171-182. http://geodesic.mathdoc.fr/item/COMIM_2021__29_2_a1/