Extensions between Cohen–Macaulay modules of Grassmannian cluster categories

Baur, Karin; Bogdanic, Dusko

doi:10.1007/s10801-016-0731-5

Cited by 10 publications

(28 citation statements)

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“…where the numbers m i denote the lengths of the lateral sides of the trapezia used to compute the extension space Ext 1 (L I , L J ) [BB16]. If we choose a to be the minimal m i , then the proposition follows.…”

Section: Homological Propertiesmentioning

confidence: 99%

“…where the numbers m 1 and m 2 denote the lengths of the lateral sides of the trapezia used to compute the extension space (cf. [BB16]), as in the following picture, and that m = min{m 1 , m 2 }.…”

Section: Homological Propertiesmentioning

confidence: 99%

“…Theorem 2. (Sections 6.1 and 7.1) When (k, n) = (3,9) or (k, n) = (4,8), for every real root of degree 2 there are two rigid indecomposable modules. Moreover, if M is such a rigid indecomposable rank 2 module and if its filtration by rank 1 modules is L I |L J , then the rank 2 module with filtration L J |L I is also rigid indecomposable.…”

Section: §1 Introductionmentioning

confidence: 99%

“…It is not hard to see, given the triangulated structure and the Auslander-Reiten translation of CM(B k,n ), that nonprojective rank 1 modules are τ periodic (cf. [4,Proposition 2.7]). We use results from Demonet-Luo [7] to explain that this category is τ periodic (see Section 3.3).…”

Section: §1 Introductionmentioning

confidence: 99%

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Cluster Categories From Grassmannians and Root Combinatorics

Baur

Bogdanic²,

Elsener³

2019

Nagoya Math. J.

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The category of Cohen-Macaulay modules of an algebra B k,n is used in [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. In this paper, we find canonical Auslander-Reiten sequences and study the Auslander-Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases.Proposition 5.2. Let I and J be r-interlacing. Then there exist 0 ≤ a 1 ≤ a 2 ≤ . . . a r−1 such that, as Z-modules,Proof. We assume that we have drawn the rims I and J one above the other, say I above J, as in the proof of Theorem 3.1 in [BB16, Section 3]. For every i 2s ∈ J \ I we have that rims I and J are not parallel between points i 2s−1 and i 2s , yielding a left trapezium. Similarly, for every i 2s+1 ∈ I \ J we have that rims I and J are not parallel between points i 2s and i 2s+1 , yielding a right trapezium. Since I and J are r-interlacing, we have, in alternating order r-left and r-right trapezia, giving us in total r boxes. The statement now follows from the proof of Theorem 3.1 in [BB16] which says that Ext 1 (L I , L J ) is a product of r − 1 cyclic Z-modules. Corollary 5.3. Let k = 3 and I and J be rims. Then Ext 1 (L I , L J ) ∼ = C × C if and only if I and J are 3-interlacing.Proof. If I and J are 3-interlacing, then they are both unions of three one-element sets. Hence, all the lateral sides in the boxes from the above proof are of length 1 and the statement follows since a i from the previous proposition are strictly positive, but at most equal to the lengths of the boxes involved. Corollary 5.4. Let k = 3. If I and J are crossing but not 3-interlacing, then Ext 1 (L I , L J ) ∼ = C.Note that if in Corollary 5.4 we have L J = τ (L I ), then we are in the situation of Theorem 3.12.Proposition 5.5. Assume that (k, n) = (3, 9) or (k, n) = (4, 8). Let M ∈ CM(B k,n ) be a rigid indecomposable rank 2 module. Then M ∼ = L I | L J where I and J are 3-interlacing.

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Section: Homological Propertiesmentioning

confidence: 99%

Section: Homological Propertiesmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

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Cluster Categories From Grassmannians and Root Combinatorics

Baur

Bogdanic²,

Elsener³

2019

Nagoya Math. J.

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“…The modules have periodic projective resolutions, which can be computed using the combinatorics. Furthermore, the (higher) Ext-groups can be determined from the shapes of these modules, they either vanish or are isomorphic to (copies of) quotients of a polynomial ring in the central generator of B k,n , [1].…”

Section: A Module Category With Grassmannian Structurementioning

confidence: 99%

Dimer models and Grassmannian cluster categories

Baur

2016

Proc Appl Math and Mech

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A dimer model can be defined as a quiver embedded into a surface in such a way that the complement is a disjoint union of disks with oriented boundaries. Such models can also be considered in the case of a surface with boundary. The Postnikov diagrams used by J. Scott to describe the cluster structure of the homogeneous coordinate ring of the Grassmannian give rise to dimer models on a disk in this sense. We associate a natural algebra to such a dimer model. This algebra is a modified version of the corresponding Jacobian algebra, taking the boundary into account. Taking the sum of the idempotents corresponding to boundary vertices, we obtain an idempotent subalgebra, which we call the boundary algebra. We show that it is independent of the choice of dimer model and coincides with an algebra that B. Jensen, A. King and X. Su have used to model the cluster structure of the homogeneous coordinate ring of the Grassmannian categorically. This reports on joint work with A. King (Bath) and R. Marsh (Leeds) and with D. Bogdanic (Graz).

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Self-Injective Jacobian Algebras from Postnikov Diagrams

Pasquali

2019

Algebr Represent Theor

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We study a finite-dimensional algebra Λ constructed from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Λ is isomorphic to the stable endomorphism algebra of a cluster tilting module T ∈ CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of C[Gr k (C n )]. We show that Λ is self-injective if and only if D has a certain rotational symmetry. In this case, Λ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras. Proposition 2.3. If (Q, W ) is a quiver with finite potential such that ∂ a W | a ∈ Q 1 is an admissible ideal of CQ, then the canonical map ℘(Q, W ) →℘(Q, W ) is an isomorphism.Proof. Call I = ∂ a W | a ∈ Q 1 ⊆ CQ andÎ = ∂ a W | a ∈ Q 1 ⊆ CQ. Call J andĴ the arrow ideals of CQ and CQ respectively. By assumption we have that there exists N ≫ 0 such that J N ⊆ I and then J N ⊆Î. Observe that we have that CQ = CQ +Ĵ N , and that J N = CQ ∩Ĵ N . There is a commutative diagram J N

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Extensions between Cohen–Macaulay modules of Grassmannian cluster categories

Cited by 10 publications

References 17 publications

Cluster Categories From Grassmannians and Root Combinatorics

Cluster Categories From Grassmannians and Root Combinatorics

Dimer models and Grassmannian cluster categories

Self-Injective Jacobian Algebras from Postnikov Diagrams

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