ALGEBRAIC COMBINATORICS

no. 5
Minimal inclusions of torsion classes
Barnard, Emily1; Carroll, Andrew2; Zhu, Shijie3
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
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 879-901.

Let Λ be a finite-dimensional associative algebra. The torsion classes of modΛ form a lattice under containment, denoted by torsΛ. In this paper, we characterize the cover relations in torsΛ 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 torsΛ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice torsΛ. 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 .

Received:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

[1] Adachi, Takahide; Iyama, Osamu; Reiten, Idun τ-tilting theory, Compos. Math., Volume 150 (2014) no. 3, pp. 415-452 | DOI | MR | Zbl

[2] Asai, Sota Semibricks (2018) (to appear in Int. Math. Res. Not, https://arxiv.org/abs/1610.05860)

[3] Assem, Ibrahim; Skowroński, Andrzej Iterated tilted algebras of type A ˜ n , Math. Z., Volume 195 (1987) no. 2, pp. 269-290 | DOI | MR | Zbl

[4] Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997, xiv+425 pages (Corrected reprint of the 1995 original) | MR

[5] Barnard, Emily The canonical join complex, Electronic J. Combinatorics, Volume 26 (2019) no. 1, Paper no. P1.24, 25 pages | MR | Zbl

[6] Brüstle, Thomas; Smith, David; Treffinger, Hipolito Wall and Chamber Structure for finite-dimensional Algebras (2018) (Preprint available: https://arxiv.org/abs/1805.01880) | Zbl

[7] Butler, Michael C. R.; Ringel, Claus Michael Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, Volume 15 (1987) no. 1-2, pp. 145-179 | DOI | MR | Zbl

[8] Caspard, Nathalie; Le Conte de Poly-Barbut, Claude; Morvan, Michel Cayley lattices of finite Coxeter groups are bounded, Adv. in Appl. Math., Volume 33 (2004) no. 1, pp. 71-94 | DOI | MR | Zbl

[9] Crawley-Boevey, William Maps between representations of zero-relation algebras, J. Algebra, Volume 126 (1989) no. 2, pp. 259-263 | DOI | MR | Zbl

[10] Demonet, Laurent; Iyama, Osamu; Jasso, Gustavo τ-tilting finite algebras, bricks, and g-vectors, Int. Math. Res. Not. IMRN (2019) no. 3, pp. 852-892 | DOI | MR | Zbl

[11] Demonet, Laurent; Iyama, Osamu; Reading, Nathan; Reiten, Idun; Thomas, Hugh Lattice theory of torsion classes (2018) (Preprint available: https://arxiv.org/abs/1711.01785)

[12] Garver, Alexander; McConville, Thomas Lattice Properties of Oriented Exchange Graphs and Torsion Classes, Algebr. Represent. Theory, Volume 22 (2019) no. 1, pp. 43-78 | DOI | MR | Zbl

[13] Iyama, Osamu; Reading, Nathan; Reiten, Idun; Thomas, Hugh Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math., Volume 154 (2018) no. 6, pp. 1269-1305 | DOI | MR | Zbl

[14] Iyama, Osamu; Reiten, Idun; Thomas, Hugh; Todorov, Gordana Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc., Volume 47 (2015) no. 4, pp. 639-650 | DOI | MR | Zbl

[15] Mizuno, Yuya Classifying τ-tilting modules over preprojective algebras of Dynkin type, Math. Z., Volume 277 (2014) no. 3-4, pp. 665-690 | DOI | MR | Zbl

[16] Reading, Nathan Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., Volume 29 (2015) no. 2, pp. 736-750 | DOI | MR | Zbl

[17] Smalø, Sverre O. Torsion theories and tilting modules, Bull. London Math. Soc., Volume 16 (1984) no. 5, pp. 518-522 | DOI | MR | Zbl

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