The Bruhat--Chevalley order on involutions of the hyperoctahedral group and combinatorics of $B$-orbit closures

TL;DR

This paper explores the order relations and dimensions of B-orbit closures associated with involutions in the Weyl group of the symplectic group, revealing a deep combinatorial structure linked to the Bruhat--Chevalley order.

Contribution

It establishes a precise correspondence between the Bruhat--Chevalley order on involutions and the inclusion relations of their associated B-orbit closures, along with a dimension formula.

Findings

Orbit closure inclusion corresponds exactly to Bruhat--Chevalley order.

Dimension of each orbit equals the length of the involution.

Provides combinatorial insights into orbit structure in symplectic groups.

Abstract

Let $G$ be the symplectic group, $\Phi=C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $\Phi$. To each involution $\sigma\in W$ one can assign the $B$-orbit $\Omega_{\sigma}$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that $\Omega_{\sigma}$ is contained in the Zariski closure of $\Omega_{\tau}$ if and only of $\sigma\leq\tau$ with respect to the Bruhat--Chevalley order. We also prove that $\dim\Omega_{\sigma}$ is equal to $l(\sigma)$, the length of $\sigma$ in $W$.