Weak orders on symmetric groups and posets of support τ-tilting modules

Kase, Ryoichi

doi:10.1142/s0218196717500266

Cited by 6 publications

(6 citation statements)

References 19 publications

(13 reference statements)

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“…Note that all these classes of algebras contain algebras which are of infinite representation type. Further information can be found in [AAC15,EJR16,IZ16,Jas15,Kas16,MS16,Zha12].…”

Section: Introductionmentioning

confidence: 99%

$\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

Demonet

Iyama

Jasso

2017

International Mathematics Research Notices

154

179

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Abstract. The class of support τ -tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study τ -tilting finite algebras, i.e. finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ -rigid modules. We show that A is τ -tilting finite if and only if every torsion class in mod A is functorially finite. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ -tilting module form a simplicial complex ∆(A). We show that if A is τ -tilting finite, then ∆(A) is homeomorphic to an (n − 1)-dimensional sphere, and moreover the partial order on support τ -tilting modules can be recovered from the geometry of ∆(A). Finally we give a bijection between indecomposable τ -rigid A-modules and bricks of A satisfying a certain finiteness condition, which is automatic for τ -tilting finite algebras.

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“…Note that all these classes of algebras contain algebras which are of infinite representation type. Further information can be found in [AAC15,EJR16,IZ16,Jas15,Kas16,MS16,Zha12].…”

Section: Introductionmentioning

confidence: 99%

$\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

Demonet

Iyama

Jasso

2017

International Mathematics Research Notices

154

179

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“…Therefore, τ -tilting theory has attracted widespread attention. For the latest general results on τ -tilting theory, we refer to [DIJ,DIRRT,EJR,IJY,IZ1,IZ2,J,K,W] and references there.…”

Section: Introductionmentioning

confidence: 99%

Three results for $\tau $-rigid modules

Xie¹,

Zan²,

Zhang³

2019

Rocky Mountain J. Math.

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τ -rigid modules are essential in the τ -tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with selfinjective dimension at most one in terms of τ -rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is τ -rigid. Finally, we give a τ -tilting theorem on homological dimension which is an analog to that of classical tilting modules.

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“…Theorem 1.2 ( [5,18]). Assume that Γ = KQ ′ /I is either a tree quiver algebra or a preprojective algebra of type A.…”

Section: Questionmentioning

confidence: 99%

“…Then it is known that each indecomposable two-term silting object has the following form (see [14,Section 6.1], [18,Lemma 6.7] for example):…”

Section: 4mentioning

confidence: 99%

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From support $τ$-tilting posets to algebras

Kase

2017

Preprint

Self Cite

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The aim of this paper is to study a poset isomorphism between two support τ -tilting posets. We take several algebraic information from combinatorial properties of support τ -tilting posets. As an application, we treat a certain class of basic algebras which contains preprojective algebras of type A, Nakayama algebras, and generalized Brauer tree algebras. We provide a necessary condition for that an algebra Λ share the same support τ -tilting poset with a given algebra Γ in this class. Furthermore, we see that this necessary condition is also a sufficient condition if Γ is either a preprojective algebra of type A, a Nakayama algebra, or a generalized Brauer tree algebra.

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Weak orders on symmetric groups and posets of support τ-tilting modules

Cited by 6 publications

References 19 publications

$\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

$\boldsymbol{\tau}$ -Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

Three results for $\tau $-rigid modules

From support $τ$-tilting posets to algebras

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