Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\Bbb T(n)$

TL;DR

This paper studies the structure of nilpotent operators with a focus on invariant subspaces, using a geometric visualization in the projective space T(n), revealing symmetries and properties of indecomposable objects.

Contribution

It introduces the projective space T(n) to visualize categorical structures of nilpotent operators and uncovers geometric symmetries and properties of indecomposables within this framework.

Findings

The action of duality and Auslander-Reiten translation correspond to reflection and rotation in T(n).

Indecomposables with boundary support are characterized by their boundary distance and support location.

In T(6), all indecomposables lie on 12 central lines, revealing a structured geometric pattern.

Abstract

We consider the category $\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n = 0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$. Our main interest in an object $X=(U,V)$ are the three numbers $uX=\dim U$ (for the subspace), $wX=\dim V/U$ (for the factor) and $bX=\dim {\rm Ker} T$ (for the operator). Actually, instead of looking at the reference space $\Bbb R^3$ with the triples $(uX,wX,bX)$, we will focus the attention to the corresponding projective space $\Bbb T(n)$ which contains for a non-zero object $X$ the level-colevel pair {\bf pr}$X = (uX/bX,wX/bX)$ supporting the object $X$. We use $\Bbb T(n)$ to visualize part of the categorical structure of $\mathcal S(n)$: The action of the duality $D$ and the square $\tau_n^2$ of the Auslander-Reiten translation are represented on $\Bbb T(n)$ by a reflection and a rotation by $120^\circ$ degrees, respectively. Moreover for $n\geq 6$, each component of the Auslander-Reiten quiver of $\mathcal S(n)$ has support either contained in the center of $\Bbb T(n)$ or with the center as its only accumulation point. We show that the only indecomposable objects $X$ in $\mathcal S(n)$ with support having boundary distance smaller than 1 are objects with $bX=1$ which lie on the boundary, whereas any rational vector in $\Bbb T(n)$ with boundary distance at least 2 supports infinitely many indecomposable objects. At present, it is not clear at all what happens for vectors with boundary distance between 1 and 2. The use of $\Bbb T(n)$ provides even in the (quite well-understood) case $n = 6$ some surprises: In particular, we will show that any indecomposable object in $\mathcal S(6)$ lies on one of 12 central lines in $\Bbb T(6)$. The paper is essentially self-contained, all prerequisites which are needed are outlined in detail.