Rene Marczinzik
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Rene Marczinzik has a total of 20 co-authors. Here are the top co-authors by number of shared publications:
- Viktória Klász
Mathematical Institute of the University of Bonn, Bonn, Germany
6 shared publications - Aaron Chan
Nagoya University
2 shared publications Hugh Thomas
2 shared publications- Judith Marquardt
Mathematical Institute of the University of Bonn, Bonn, Germany
2 shared publications - Karin Erdmann
Oxford University, UK
2 shared publications - Osamu Iyama
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, Japan
2 shared publications
Latest Publications
- 18 April 2026
- Published by Springer Nature in Selecta Mathematica
- Vol. 32 (3) , 43
Dominant Auslander–Gorenstein algebras and mixed cluster tilting
- 18 April 2026
- Published by Springer Nature in Selecta Mathematica
- Vol. 32 (3) , 43
Dominant Auslander–Gorenstein algebras and mixed cluster tilting
- Open Access
- 30 March 2026
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 29 (2) , 721-723
On a Question of Auslander and Bridger on 2-Reflexive Modules
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
- 30 March 2026
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 29 (2) , 721-723
Open AccessOn a Question of Auslander and Bridger on 2-Reflexive Modules
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
- Open Access
On a question of Auslander and Bridger on 2-reflexive modules
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
Open AccessOn a question of Auslander and Bridger on 2-reflexive modules
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
- Open Access
A new formula for the classical dominant dimension using bimodules
We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata.Open AccessA new formula for the classical dominant dimension using bimodules
We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata. - Open Access
Rowmotion and Echelonmotion
Given a linear extension $σ$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $σ$. This yields a bijection $\mathrm{Ech}_σ\colon R\to R$ that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Klász, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when $R$ is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.Open AccessRowmotion and Echelonmotion
Given a linear extension $σ$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $σ$. This yields a bijection $\mathrm{Ech}_σ\colon R\to R$ that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Klász, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when $R$ is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions. - Open Access
Cluster tilting modules for local algebras
We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package QPA. We discuss related problems and conjectures for local finite-dimensional algebras.
Open AccessCluster tilting modules for local algebras
We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package QPA. We discuss related problems and conjectures for local finite-dimensional algebras.
- Open Access
On the global dimension of Nakayama algebras
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.
Open AccessOn the global dimension of Nakayama algebras
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.
- Open Access
Auslander regular algebras and Coxeter matrices
We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category $\mathcal{O}$ of semisimple Lie algebras.Open AccessAuslander regular algebras and Coxeter matrices
We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category $\mathcal{O}$ of semisimple Lie algebras. - Open Access
- 1 November 2024
- Published by Springer Nature in Annals of Combinatorics
- Vol. 30 (1) , 17-33
Nakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations
We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.
- 1 November 2024
- Published by Springer Nature in Annals of Combinatorics
- Vol. 30 (1) , 17-33
Open AccessNakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations
We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.
- Open Access
- 19 September 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Combinatorial Algebra
- Vol. 8 (3) , 359-374
On the interaction of the Coxeter transformation and the rowmotion bijection
Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.
- 19 September 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Combinatorial Algebra
- Vol. 8 (3) , 359-374
Open AccessOn the interaction of the Coxeter transformation and the rowmotion bijection
Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.