Collineations preserving the lattice of invariant subspaces of a linear transformation

TL;DR

This paper characterizes the group of invertible transformations that preserve the lattice of invariant subspaces of a linear transformation, reducing the problem to nilpotent cases and analyzing their structure.

Contribution

It provides a detailed description of the collineation group for linear transformations, especially nilpotent ones, and identifies conditions for when this group differs from the commutant.

Findings

The collineation group always contains the invertible elements of the commutant.

It is contained within the reflexive cover of the commutant.

The group is a proper subgroup of the inverse of the reflexive cover if and only if certain Jordan blocks are of size two or more.

Abstract

Given a linear transformation $A$ on a finite-dimensional complex vector space $\eV$, in this paper we study the group $\Col(A)$ consisting of those invertible linear transformations $S$ on $\eV$ for which the mapping $\Phi_S$ defined as $\Phi_S\colon \eM\mapsto S\eM$ is an automorphism of the lattice $\Lat(A)$ of all invariant subspaces of $A$. By using the primary decomposition of $A$, we first reduce the problem of characterizing $\Col(A)$ to the problem of characterizing the group $\Col(N)$ of a given nilpotent linear transformation $N$. While $\Col(N)$ always contains all invertible linear transformations of the commutant $(N)'$ of $N$, it is always contained in the reflexive cover $\Alg\Lat(N)'$ of $(N)'$. We prove that $\Col(N)$ is a proper subgroup of $(\Alg\Lat(N)')^{-1}$ if and only if at least two Jordan blocks in the Jordan decomposition of $N$ are of dimension $2$ or more. We also determine the group $\Col(\bdJ_2\oplus \bdJ_2)$.