The Swiss Cheese Theorem for Linear Operators with Two Invariant Subspaces

TL;DR

This paper classifies the possible dimension types of indecomposable systems with a nilpotent operator and two invariant subspaces, revealing a 'holes' phenomenon in the classification for operators with nilpotency index four.

Contribution

It determines the dimension types of indecomposable systems for nilpotency index up to four and uncovers the existence of gaps in the possible types, contrasting with Bongartz's No-Gap Theorem.

Findings

Classified indecomposable systems for nilpotency index ≤ 4.

Identified infinite families of dimension types within a cylinder.

Discovered gaps where certain dimension types cannot be realized.

Abstract

We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $(x,y,z)\in (3,1,3)+\mathbb N(2,2,2)$ can be realized, while each neighbor $(x\pm1,y,z), (x,y\pm1,z),(x,y,z\pm1)$ can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra $A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $A$-modules of finite dimension.