# ALGEBRAIC COMBINATORICS

Minimal inclusions of torsion classes
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 879-901.

Let $\Lambda$ be a finite-dimensional associative algebra. The torsion classes of $\mathrm{mod}\Lambda$ form a lattice under containment, denoted by $\mathrm{tors}\Lambda$. In this paper, we characterize the cover relations in $\mathrm{tors}\Lambda$ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of $\mathrm{tors}\Lambda$ in terms of representation theory. Finally, we show that, in general, the algebra $\Lambda$ is not characterized by its lattice $\mathrm{tors}\Lambda$. In particular, we study the torsion theory of a quotient of the preprojective algebra of type ${A}_{n}$. We show that its torsion class lattice is isomorphic to the weak order on ${A}_{n}$.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.72
Classification: 05E10,  06B15
Keywords: lattice theory, torsion classes, canonical join representations
Barnard, Emily 1; Carroll, Andrew 2; Zhu, Shijie 3

1 Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Suite 502 Chicago IL 60614, USA
2 3778 Keating St. San Diego CA 92110, USA
3 Mathematics Department University of Iowa 14 MacLean Hall Iowa City IA 52242, USA
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Barnard, Emily; Carroll, Andrew; Zhu, Shijie. Minimal inclusions of torsion classes. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 879-901. doi : 10.5802/alco.72. https://alco.centre-mersenne.org/articles/10.5802/alco.72/

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