Journal of Lie Theory 24 (2014), No. 4, 931--956
Copyright Heldermann Verlag 2014
Rook Placements in An and Combinatorics of B-Orbit Closures
Mikhail V. Ignatyev
Chair of Algebra and Geometry, Samara State University, Ak. Pavlova 1, Samara 443011, Russia
mihail.ignatev@gmail.com
Anton S. Vasyukhin
Chair of Algebra and Geometry, Samara State University, Ak. Pavlova 1, Samara 443011, Russia
safian.malk@gmail.com
[Abstract-pdf]
\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.
Keywords: Coadjoint orbits, Borel subgroup, root systems, rook placements, polarizations.
MSC: 22E25, 17B22
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