Rook placements in $A_n$ and combinatorics of $B$-orbit closures

TL;DR

This paper studies rook placements in root systems of type A, analyzing their associated coadjoint orbits, determining their dimensions, constructing polarizations, and exploring the partial order structure among these orbits.

Contribution

It introduces a new combinatorial model of rook placements for root systems and characterizes the geometry of the associated B-orbits, including dimensions and order relations.

Findings

Dimension formulas for the coadjoint orbits

Construction of polarizations at orbit points

Description of the partial order among rook placements

Abstract

Let $G$ be a complex reductive group, $B$ be a Borel subgroup of G, $\nt$ be the Lie algebra of the unipotent radical of $B$, and $\nt^*$ be its dual space. Let $\Phi$ be the root system of $G$, and $\Phi^+$ 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 $\nt^*$. By definition, $\Omega_D$ is the orbit of $f_D$, where $f_D$ is the sum of root covectors corresponging to the roots from $D$. We find the dimension of $\Omega_D$ and construct a polarization of $\nt$ at $f_D$. We also study the partial order on the set of rook placements induced by the incidences among the orbits associated with rook placements.