### Citations

355 | Tame algebras and integral quadratic form - Ringel - 1984 |

214 | Cluster algebras II: finite type classification
- Fomin, Zelevinsky
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Citation Context ...n that cluster monomials would be contained in Lusztig’s dual canonical basis for quantized coordinate rings of algebraic groups. Thus a serious attempt was made to describe the cluster monomials. In =-=[FZ2]-=-, they establish a simple bijection between the cluster variables and almost positive roots in an associated root system, thus proving the well known A−G finite-type classification for cluster algebra... |

179 |
Infinite root systems, representations of graphs and invariant theory
- Kac
- 1982
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Citation Context ...ction between the cluster variables and almost positive roots in an associated root system, thus proving the well known A−G finite-type classification for cluster algebras. It has been recognized [G],=-=[K]-=-,[Ri1] that the indecomposable representations of an associated (valued) quiver Q were also in bijection with a certain root datum, namely the (strictly) positive roots. To properly explain these bije... |

174 |
Unzerlegbare Darstellungen I
- Gabriel
- 1972
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Citation Context ...bijection between the cluster variables and almost positive roots in an associated root system, thus proving the well known A−G finite-type classification for cluster algebras. It has been recognized =-=[G]-=-,[K],[Ri1] that the indecomposable representations of an associated (valued) quiver Q were also in bijection with a certain root datum, namely the (strictly) positive roots. To properly explain these ... |

171 | From triangulated categories to cluster algebras
- Caldero, Keller
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Citation Context ...tion for cluster-tilting objects coincides with the seedmutation of Fomin and Zelevinsky, thus establishing a relationship between rigid objects of the cluster category and cluster monomials. Then in =-=[CK]-=- Caldero and Keller generalize these results to give cluster characters categorifying all cluster algebras with acyclic, skew-symmetric exchange matrix. Key words and phrases. quantum cluster algebra,... |

150 | Cluster algebras I: Foundations
- Fomin, Zelevinsky
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Citation Context ...tations of Exchange Matrices 23 7. Quantum Seeds Associated to Local Tilting Representations 30 References 31 1. Introduction and Main Results Cluster algebras were introduced by Fomin and Zelevinsky =-=[FZ1]-=- in anticipation that cluster monomials would be contained in Lusztig’s dual canonical basis for quantized coordinate rings of algebraic groups. Thus a serious attempt was made to describe the cluster... |

135 | F.: Cluster algebras as Hall algebras of quiver representations
- Caldero, Chapoton
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Citation Context ...he almost positive roots. Moreover, they observe that there is a bijection between the cluster-tilting objects of CQ and the clusters of the cluster algebra A(Q). Extending this, Caldero and Chapoton =-=[CC]-=- introduce cluster characters describing the initial cluster expansion of all cluster variables/monomials explicitly as generating functions of Euler characteristics of Grassmannians of submodules in ... |

71 | Quantum cluster algebras
- Berenstein, Zelevinsky
- 2005
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Citation Context ...ation and thus the isomorphism classes of rigid objects in the Grothendieck group K(Q,d) are independent of the choice of ground field F. Theorem 1.1 together with the quantum Laurent phenomenon from =-=[BZ]-=- imply the following result. Corollary 1.2. Let V be a rigid valued representation of Q. Then for any e ∈ K(Q,d) the Grassmannian GrVe has a counting polynomial, which we denote by P v e (q). For equa... |

62 |
Cluster algebras IV:
- Fomin, Zelevinsky
- 2007
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Citation Context ... 6= j = k; fij = δij if i 6= k; −1 if i = j = k; [bkj ]+ if i = k 6= j. Then we may compute the commutation matrix µkΛ of the cluster µkX as (4.3) µkΛ = E tΛE. A result of Fomin and Zelevinsky =-=[FZ4]-=- asserts that the cluster variables of Aq(B̃,Λ) are completely determined by the cluster variables of the principal coefficients quantum cluster algebra A(B̃P ,Λ′) where B̃P = ( B I ) with I the n × n... |

45 |
Representations of K-species and bimodules
- Ringel
- 1976
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Citation Context ...n between the cluster variables and almost positive roots in an associated root system, thus proving the well known A−G finite-type classification for cluster algebras. It has been recognized [G],[K],=-=[Ri1]-=- that the indecomposable representations of an associated (valued) quiver Q were also in bijection with a certain root datum, namely the (strictly) positive roots. To properly explain these bijections... |

34 | Quantum Cluster Variables via Serre Polynomials, eprint
- Qin
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Citation Context ...ite type valued quivers, for rank 2 valued quivers, and for those representations of acyclic valued quivers which can be obtained from simple representations by a sequence of reflection functors. Qin =-=[Q]-=- proved this conjecture for acyclic equally valued quivers (i.e. di = 1 for all i). Note that there is a unique isomorphism class for each exceptional valued representation and thus the isomorphism cl... |

32 | Acyclic cluster algebras via Ringel-Hall algebras, Preprint available at the author’s home - Hubery |

14 | representations respecting a quiver automorphism: A generalisation of a theorem of Kac
- Hubery
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Citation Context ... and form the tensor algebra ΓQ = TΓ0(Γ1) of Γ1 over Γ0. A module X over ΓQ is given by an Fdi-vector space Xi for each vertex i and an Fdj -linear map θ X ij : Xi ⊗Fdi Γij → Xj whenever bij > 0, see =-=[H2]-=- for more details. A morphism of ΓQ-modules f : X → Y is a collection {fi}i∈Q0 , with fi : Xi → Yi an Fdi-linear map, such that θ Y ij(fi ⊗ id) = fjθ X ij . Proposition 3.1. [Ru1] The categories Rep F... |

12 | Cluster-tilting theory
- Buan, Marsh
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Citation Context ...were also in bijection with a certain root datum, namely the (strictly) positive roots. To properly explain these bijections and fully understand the role of the negative simple roots, the authors of =-=[BMRRT]-=- introduce the cluster category CQ in which the indecomposable representations are exactly in bijection with the almost positive roots. Moreover, they observe that there is a bijection between the clu... |

12 | Quantum cluster variables via vanishing cycles
- Efimov
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Citation Context ...ns of the valued quiver (Q,d) over the finite field F. This character assigns to a valued representation V of Q the element XV in the quantum torus TΛ,|F|, given by (1.1) XV = ∑ E⊂V |F|− 1 2 〈E,V/E〉X−=-=[E]-=- ∗−∗[V/E] where Λ is compatible with (Q,d), TΛ,|F| is the quantum torus defined in section 4, 〈·, ·〉 denotes the RingelEuler form of the category rep F (Q,d), and ∗ denotes certain left and right dual... |

7 |
On a quantum analog of the Caldero-Chapoton formula
- Rupel
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Citation Context ...LUSTER CHARACTERS DYLAN RUPEL Abstract. Let F be a finite field and (Q,d) an acyclic valued quiver with associated exchange matrix B̃. We follow Hubery’s approach [H1] to prove our main conjecture of =-=[Ru1]-=-: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of Q to the set of noninitial quantum cluster variables for the quantum cluster alg... |

6 | Quantum cluster algebras. Oberwolfach talk, February 2005 tt arXiv:0502260 [math.QA]. PDF: Insitut de Physique Théorique du Commissariat à l’Energie Atomique, Unité de recherche associeée du
- Zelevinsky
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Citation Context ...skew-symmetrizable matrix Λ = (λij) such that BtQΛ = D and write Λ(·, ·) : Zn × Zn → Z for the associated skew-symmetric bilinear form. As in Zelevinsky’s Oberwolfach talk on quantum cluster algebras =-=[Z]-=-, we may always replace the quiver Q by Q̃, where we attach principal frozen vertices to the valued quiver Q, to guarantee that such a matrix Λ exists. Note that the compatibility condition for BQ and... |

4 | A quantum analogue of generic bases for affine cluster algebras
- Ding, Xu
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Citation Context ...y of Grassmannians in the category C. While this manuscript was in its final stages of preparation Ding and Sheng [DS] proved multiplication theorems for quantum cluster characters extending those of =-=[DX]-=-. In this work Ding and Sheng deduce results about bases of quantum cluster algebras of finite type or rank 2. We are curious what can be said about bases in more general acyclic quantum cluster algeb... |

3 | Counting Using Hall Algebras I - Fei - 2011 |

1 |
Multiplicative Properties of a Quantum Caldero-Chapoton Map Associated to Valued Quivers, Preprint: math/1109.5342v1
- Ding, Sheng
- 2011
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Citation Context ...onomials. In this case, the same machinery would imply polynomiality of the cardinality of Grassmannians in the category C. While this manuscript was in its final stages of preparation Ding and Sheng =-=[DS]-=- proved multiplication theorems for quantum cluster characters extending those of [DX]. In this work Ding and Sheng deduce results about bases of quantum cluster algebras of finite type or rank 2. We ... |