Acyclic Cluster Algebras Revisited

Speyer, David E; Thomas, Hugh

doi:10.1007/978-3-642-39485-0_12

Cited by 32 publications

(23 citation statements)

References 23 publications

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“…We say that an EL‐labelling

λ

is an interpolating labelling if whenever we have

x, y, z

scriptL

with

x ⋖ y ⋖ z

, either

λ (x ⋖ y) < λ (y ⋖ z)

or if the increasing chain from

x

z

x = x_{0} ⋖ x_{1} ⋖ \dots ⋖ x_{r} = z

, then

λ false(x_{0} ⋖ x_{1} false) = λ false(y ⋖ z false)

and

λ false(x_{r - 1} ⋖ x_{r} false) = λ false(x ⋖ y false)

. Example We may describe Cambrian lattices using flips on 2‐colored factorizations of a Coxeter element . It is shown in that the natural labelling on the edges of a Cambrian lattice coming from these factorizations is an EL‐labelling; the flip definition allows us to easily see that this labelling also satisfies the interpolating condition.…”

Section: Lattices: Representations and Labellingsmentioning

confidence: 99%

Rowmotion in slow motion

Thomas

Williams

2019

Proc. London Math. Soc.

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Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and modA has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

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“…We say that an EL‐labelling

λ

is an interpolating labelling if whenever we have

x, y, z

scriptL

with

x ⋖ y ⋖ z

, either

λ (x ⋖ y) < λ (y ⋖ z)

or if the increasing chain from

x

z

x = x_{0} ⋖ x_{1} ⋖ \dots ⋖ x_{r} = z

, then

λ false(x_{0} ⋖ x_{1} false) = λ false(y ⋖ z false)

and

λ false(x_{r - 1} ⋖ x_{r} false) = λ false(x ⋖ y false)

Section: Lattices: Representations and Labellingsmentioning

confidence: 99%

Rowmotion in slow motion

Thomas

Williams

2019

Proc. London Math. Soc.

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“…Qq " I n and cp q Qq "´I n . For more details on c-vectors, we refer the reader to [20] where they were introduced and to [37,36,46,35,27] where they were studied.…”

Section: 3mentioning

confidence: 99%

On Maximal Green Sequences

Brüstle

Dupont

Pérotin³

2013

International Mathematics Research Notices

157

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Abstract. Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory.Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.

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“…we refer the reader to [35] where they were introduced and to [84,83,98,82,66] where they were studied.…”

Section: Much Of Combinatorics Of Clusters Is Encoded In C-matrices Amentioning

confidence: 99%

Ordered exchange graphs

Brüstle¹,

Yang²

2014

Advances in Representation Theory of Algebras

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Abstract. The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in nonpositive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support τ -tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.

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Acyclic Cluster Algebras Revisited

Cited by 32 publications

References 23 publications

Rowmotion in slow motion

Rowmotion in slow motion

On Maximal Green Sequences

Ordered exchange graphs

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