\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.
1
Chair of Algebra and Geometry, Samara State University, Ak. Pavlova 1, Samara 443011, Russia
Mikhail V. Ignatyev; Anton 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/JOLT_2014_24_4_a1/
@article{JOLT_2014_24_4_a1,
author = {Mikhail V. Ignatyev and Anton 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},
url = {http://geodesic.mathdoc.fr/item/JOLT_2014_24_4_a1/}
}
TY - JOUR
AU - Mikhail V. Ignatyev
AU - Anton S. Vasyukhin
TI - Rook Placements in An and Combinatorics of B-Orbit Closures
JO - Journal of Lie Theory
PY - 2014
SP - 931
EP - 956
VL - 24
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2014_24_4_a1/
ID - JOLT_2014_24_4_a1
ER -
%0 Journal Article
%A Mikhail V. Ignatyev
%A Anton S. Vasyukhin
%T Rook Placements in An and Combinatorics of B-Orbit Closures
%J Journal of Lie Theory
%D 2014
%P 931-956
%V 24
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2014_24_4_a1/
%F JOLT_2014_24_4_a1
