Geodesic
Rook Placements in An and Combinatorics of B-Orbit Closures
Journal of Lie theory, Tome 24 (2014) no. 4, pp. 931-956.

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\def\n{{\frak n}} 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 $\Phi$ be the root system of $G$, and let $\Phi^+$ be the set of positive roots with respect to $B$. A subset of $\Phi^+$ 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 $\Omega_D$ of $B$ in $\n^*$. By definition, $\Omega_D$ 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 $\Omega_D$ 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.
Classification : 22E25, 17B22
Mots-clés : Coadjoint orbits, Borel subgroup, root systems, rook placements, polarizations 
@article{JLT_2014_24_4_JLT_2014_24_4_a1,
     author = {M. V. Ignatyev and A. S. Vasyukhin },
     title = {Rook {Placements} in {A\protect\textsubscript{n}} and {Combinatorics} of {B-Orbit} {Closures}},
     journal = {Journal of Lie theory},
     pages = {931--956},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JLT_2014_24_4_JLT_2014_24_4_a1/}
}
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M. V. Ignatyev; A. S. Vasyukhin . Rook Placements in An and Combinatorics of B-Orbit Closures. Journal of Lie theory, Tome 24 (2014) no. 4, pp. 931-956. http://geodesic.mathdoc.fr/item/JLT_2014_24_4_JLT_2014_24_4_a1/