Date of Award
Fall 11-12-2025
Document Type
Dissertation
Publication Status
Version of Record
Submission Date
December 2025
Department
Mathematical Sciences
Degree Name
Doctor of Philosophy (PhD)
Thesis/Dissertation Advisor [Chair]
Markus Schmidmeier
Abstract
Let k be a field and n ∈ N. We consider the quadruples (V, U’, U”, T) where V is a finite-dimensional k-vector space, U’ and U” are subspaces of V , and T : V → V is a linear operator with Tn = 0, satisfying T(U’) ⊆ U’ and T(U”) ⊆ U”. We say U’ and U” are invariant under T and T acts nilpotently on V with nilpotent index n. These systems of quadruples form a category with the Krull-Remak-Schmidt property. Thus, classification is reduced to the classification of indecomposable systems.
By translating a seemingly intractable linear algebra problem to that of module theory and quiver representations, we are able to utilize Auslander-Reiten Theory to aid our analysis. We will give a complete classification in this work. The nilpotent index is crucial for classification of the category. For n ≤ 3, our problem is classifiable. For n > 3, our category is of wild representation type, and no complete classification can be done. For n = 1, 2 we will list the finitely many indecomposables, and for n = 3, we will give a complete description.
In particular, we demonstrate that S(3) is a tubular category of type E6 and explicitly construct its exceptional tubes. We provide a complete classification of dimension triples in N3 corresponding to indecomposable quadruples in S(n), together with visualizations. Furthermore, we identify all Gorenstein-projective modules in these categories, finding 9 such modules in S(2) and 27 in S(3). Finally, we present an application of these results in the context of linear time-invariant dynamical systems.
Recommended Citation
Snyder, David, "LINEAR OPERATORS WITH TWO INVARIANT SUBSPACES" (2025). Electronic Theses and Dissertations. 231.
https://digitalcommons.fau.edu/etd_general/231