Markus Kleinau

I am a third year PhD-student under the supervision of Jan Schröer and a member of the Algebra and Representation Theory group.

Contact

Email:
Office: Endenicher Allee 60 N2.003

Research interests

I am interested in the intersection between the representation theory of finite dimensional algebras and combinatorics. This includes

• Homological properties of representations of incidence algebras,
• Ringel-Hall algebras of gentle algebras,
• Quiver representations over $\mathbb{F}_1$.

Posters

2025: Lattices of torsion classes and 2-cluster categories
2024: Quiver representations over $\mathbb F_1$

Publications and Preprints

5. Classification of Auslander-Gorenstein monomial algebras: The acyclic case Preprint. arXiv:2604.02146
joint with Viktória Klász and René Marczinzik.
Abstract

We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix C has a Bruhat factorisation U1PU2 with U1 the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.

6. Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics Preprint. arXiv:2603.23354
with an appendix by René Marczinzik.
Abstract

Reading constructed a Cambrian lattice $C_\Gamma$ for each oriented finite type Coxeter diagram $\Gamma$. We show that the derived category of representations of $C_\Gamma$ is fractionally Calabi-Yau for any $\Gamma$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $\Gamma$ is crystallographic, then $C_\Gamma$ is given by the lattice of torsion classes of any hereditary algebra $\Lambda$ of type $\Gamma$. In this case we introduce and study a class of intervals in $C_\Gamma$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $\Lambda$. This allows us to compute the Calabi-Yau dimension of $C_\Gamma$.

7. Pure minimal injective resolutions and perfect modules for lattices Preprint. arXiv:2511.03385
joint with Tal Gottesmann, Viktória Klász and René Marczinzik.
Abstract

In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a complete classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution.

8. Scalar extensions of quiver representations over $\mathbb{F}_1$ arXiv:2403.04597
   Algebr. Represent. Theory 28 (2025), no. 2, 531–548.
Abstract

Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\DeclareMathOperator{\Hom}{Hom} \Hom_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.

9. The Aizenbud-Lapid binary operation for symmetrizable Cartan types Preprint. arXiv:2311.17036
Abstract

Aizenbud and Lapid recently introduced a binary operation on the crystal graph $B(-\infty)$ associated to a symmetric Cartan matrix. We extend their construction to symmetrizable Cartan matrices and strengthen a cancellation property of the binary operation.

Activities

• Talk in Paris "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 24.11.2025
• Talk in Cologne "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 21.10.2025
• Talk in Bonn "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 17.7.2025
• Talk in Trento "Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics" 25.6.2025
• Talk in Bonn "Representation theory over $\mathbb{F}_1$" 23.1.2025
• Research stay in Bochum with Tal Gottesman 14.1.-17.1.2025
• Talk in Versailles "Representation theory over $\mathbb{F}_1$" 11.12.2024
• Research stay in Versailles with Judith Marquardt 11.12-13.12.2024