Journal of Lie Theory

no. 4
Rook Placements in An and Combinatorics of B-Orbit Closures
Journal of Lie Theory, Volume 24 (2014) no. 4, pp. 931-956
\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.
DOI: 10.5802/jolt.813
Classification: 22E25, 17B22
Keywords: Coadjoint orbits, Borel subgroup, root systems, rook placements, polarizations 
@article{JOLT_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},
     year = {2014},
     volume = {24},
     number = {4},
     doi = {10.5802/jolt.813},
     zbl = {1316.05019},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.813/}
}
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%A A. S. Vasyukhin
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M. V. Ignatyev; A. S. Vasyukhin. Rook Placements in An and Combinatorics of B-Orbit Closures. Journal of Lie Theory, Volume 24 (2014) no. 4, pp. 931-956. doi: 10.5802/jolt.813

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