Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions

TL;DR

This paper provides a combinatorial description of the order structure on involutions in the symmetric group induced by Borel group orbits, linking rook placements with Bruhat--Chevalley order and orbit closures.

Contribution

It introduces a rook placement-based combinatorial model for the orbit closure order on involutions, connecting it with Bruhat--Chevalley order on the symmetric group.

Findings

The partial order on involutions matches the Bruhat--Chevalley order restriction.

Rook placements describe orbit closure relations.

The results extend Melnikov's and Incitti's work on B-orbits and involutions.

Abstract

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $\sigma$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $\Omega_{\sigma}\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of rook placements and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$. Using results of F. Incitti, we also prove that this partial order coincides with the restriction of the Bruhat--Chevalley order to the set of involutions.