On band modules and $$\tau $$-tilting finiteness

Schroll, Sibylle; Treffinger, Hipolito; Valdivieso, Yadira

doi:10.1007/s00209-020-02687-2

Cited by 8 publications

(2 citation statements)

References 27 publications

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“…Furthermore, in the aforementioned paper, the conjecture was verified for gentle algebras, as well as all the min-rep-inf algebras treated there. In [STV], the authors have also come across the same conjecture and verified it for special biserial algebras.…”

Section: Introductionmentioning

confidence: 70%

Minimal (-)Tilting Infinite Algebras

Mousavand¹,

Paquette²

2022

Nagoya Math. J.

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Motivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $\tau $ -tilting infinite (min- $\tau $ -infinite, for short) algebras. In particular, we treat min- $\tau $ -infinite algebras as a modern counterpart of minimal representation-infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture, it is sufficient to treat those min- $\tau $ -infinite algebras where almost all bricks are faithful. Finally, we also prove that minimal extending bricks have open orbits, and consequently obtain a simple proof of the brick analogue of the first Brauer–Thrall conjecture, recently shown by Schroll and Treffinger using some different techniques.

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Section: Introductionmentioning

confidence: 70%

Minimal (-)Tilting Infinite Algebras

Mousavand¹,

Paquette²

2022

Nagoya Math. J.

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“…In the following, we use the terminology of string modules and band modules, for a general finite‐dimensional algebra [51]. However, as we use left modules, some care must be taken.…”

Section: Weight 2 Blocksmentioning

confidence: 99%

Schurian‐finiteness of blocks of type A$A$ Hecke algebras

Ariki,

Lyle,

Speyer

2023

Journal of London Math Soc

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For any algebra over an algebraically closed field , we say that an ‐module is Schurian if . We say that is Schurian‐finite if there are only finitely many isomorphism classes of Schurian ‐modules, and Schurian‐infinite otherwise. By work of Demonet, Iyama and Jasso, it is known that Schurian‐finiteness is equivalent to ‐tilting‐finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type Hecke algebras with quantum characteristic , all blocks of weight at least 2 are Schurian‐infinite in any characteristic. Weight 0 and 1 blocks are known by results of Erdmann and Nakano to be representation finite, and are therefore Schurian‐finite. This means that blocks of type Hecke algebras (when ) are Schurian‐infinite if and only if they have wild representation type if and only if the module category has finitely many wide subcategories. Along the way, we also prove a graded version of the Scopes equivalence, which is likely to be of independent interest.

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