# On Indecomposable triples associated with nilpotent operators

## TL;DR

This paper classifies indecomposable triples consisting of a finite-dimensional space, a nilpotent operator, and an invariant subspace, based on nilpotency order and subspace dimensions, providing complete results for specific cases.

## Contribution

It offers a comprehensive classification of indecomposable triples related to nilpotent operators, extending previous work to cases with higher nilpotency orders and infinite indecomposables.

## Key findings

- Complete classification for arbitrary p with n_U=1
- Classification for n_U=2 and n_V ≤ 3
- Analysis of cases with p ≥ 6 where indecomposables are infinite

## Abstract

We consider in this paper the family of triples $(V, T, U),$ where $ V$ is a finite dimensional space, $T $ is a nilpotent linear operator on $V$ and $U $ is an invariant subspace of $T$. Denote $[U]= ker(T_{|U})$, and $n_U= dim([U] )$. Our main goal is to investigate possible classification of indecomposable triples. The obtained classification depends on the order of nilpotency $p$, on $n_U$ and on $n_V$. Complete classifications are given for arbitrary $p$, when $n_U=1$, and when $n_U=2$ and $n_V \le 3$. The case $ p \le 5$, treated in \cite{ring} is recaptured by using constructive proofs based on linear algebra tools. The case $p\ge 6$, where the number of indecomposable triples is infinite, is also investigated.