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## Generic bases for cluster algebras and the Chamber Ansatz (2011)

Citations: | 21 - 2 self |

### Citations

518 | Triangulated categories in representation theory of finite dimensional algebras - Happel - 1988 |

265 | Kac-Moody Groups, their Flag Varieties and Representation Theory - Kumar - 2002 |

238 | Cluster algebras - Fomin, Zelevinsky |

198 |
Perverse sheaves, and quantized enveloping algebras
- Lusztig, Quivers
- 1991
(Show Context)
Citation Context ... of submodules of X such that Xk−1/Xk ∼= S ak ik for all 1 ≤ k ≤ r, where Sj denotes the one-dimensional Λ-module supported on the vertex j of Q. The varieties Fi,a,X were first introduced by Lusztig =-=[L1]-=- for his Lagrangian construction of U(n). Dualizing Lusztig’s construction, we can associate with X a regular function ϕX ∈ C[N ] satisfying ϕX(xi(t)) = ∑ a∈Nr χ(Fi,a,X)t a. Here t = (tr, . . . , t1) ... |

176 | Quivers with potentials and their representations II: applications to cluster algebras
- Derksen, Weyman, et al.
- 2010
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Citation Context ... above isomorphism of Cw-projective-injectives. 6. Cluster character identities 6.1. Quivers with potential and mutations. We review some material from [DWZ2, Section 4], which in turn is a review of =-=[DWZ1]-=-. Let P(Γ,W ) := C〈〈Γ〉〉/J(W ) be the Jacobian algebra associated to a quiver Γ = (Γ0,Γ1, s, t) and a potential W ∈ mcyc ⊂ C〈〈Γ〉〉. For k ∈ Γ0 we set Γ(−,k) := {b ∈ Γ1 | s(b) = k}, Γ(k,+) := {a ∈ Γ1 | t... |

171 | From triangulated categories to cluster algebras
- Caldero, Keller
(Show Context)
Citation Context ...ns of cluster variables of some cluster algebras were described as Euler characteristics of Grassmannians of submodules of quiver representations. This was first achieved for acyclic cluster algebras =-=[CK]-=-, later for cluster algebras admitting a 2-CalabiYau categorification [P, FK], and more recently for general antisymmetric cluster algebras of geometric type [DWZ2]. There is a posterior but essential... |

142 | Cluster-tilted algebras are Gorenstein and stably - Keller, Reiten - 2007 |

135 | Cluster algebras as Hall algebras of quiver representations - Caldero, Chapoton |

118 | Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l’institut Fourier
- Amiot
(Show Context)
Citation Context ....(GT ) the corresponding subcategory of perfect complexes resp. of complexes with total finite-dimensional cohomology of the derived category. The shift in Dperf(GT ) is denoted by Σ. Following Amiot =-=[A]-=- we have the generalized cluster category as the triangulated quotient CT := Dperf(GT )/Df.d.(GT ). It follows from [ART] that Cw ∼= CV as triangulated categories, and then from [BIRSm] and [KY] that ... |

105 | Cluster structures for 2-Calabi-Yau categories and unipotent groups
- Buan, Iyama, et al.
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Citation Context ... C[N ] satisfying ϕX(xi(t)) = ∑ a∈Nr χ(Fi,a,X)t a. Here t = (tr, . . . , t1) ∈ C r, ta := tarr · · · t a2 2 t a1 1 , and χ denotes the topological Euler characteristic. Buan, Iyama, Reiten, and Scott =-=[BIRS]-=- have attached to w a 2-Calabi-Yau Frobenius subcategory Cw of the category of finite-dimensional nilpotent Λ-modules. (The same categories were studied independently in [GLS4] for special elements w ... |

90 |
Parametrizations of canonical bases and totally positive matrices, preprint
- Berenstein, Fomin, et al.
- 1995
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Citation Context ... . , tr. Inverting this monomial transformation yields expressions of the tk’s as explicit rational functions on N w, a result originally called the Chamber Ansatz by Berenstein, Fomin and Zelevinsky =-=[BFZ]-=- in type An, because of a convenient description of these formulas in terms of chambers in a wiring diagram. To present these formulas in the general Kac-Moody setting, we need more notation. By const... |

88 | Rigid modules over preprojective algebras - Geiß, Leclerc, et al. |

84 | Derived equivalences from mutations of quivers with potential
- Keller, Yang
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Citation Context ... Amiot [A] we have the generalized cluster category as the triangulated quotient CT := Dperf(GT )/Df.d.(GT ). It follows from [ART] that Cw ∼= CV as triangulated categories, and then from [BIRSm] and =-=[KY]-=- that CV ∼= CT . Next, denote by F ⊆ Dperf(GT ) the subcategory which consists of the cones of maps in add(GT ). Then the canonical projection Dperf(GT )→ CT induces an equivalence of additive categor... |

67 | Semicanonical bases and preprojective algebras - Geiss, Leclerc, et al. - 2005 |

57 |
Total positivity in Schubert varieties
- Berenstin, Zelevinsky
- 1997
(Show Context)
Citation Context ...,k) , a Laurent monomial in the ϕWi,k (since add(Wi) contains all Cw-projectives). As will be explained in Section 1.6 below, the regular functions ϕ′Vi,k on N w are the twisted generalized minors of =-=[BZ]-=- corresponding to i (in the Dynkin case). Denote by q(i, j) the number of edges between two vertices i and j of the underlying unoriented graph of the quiver Q. For 1 ≤ k ≤ r, put (1.1) Ci,k := 1 ϕ′Vi... |

56 | On cluster algebras with coefficients and 2-Calabi-Yau categories
- Fu, Keller
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Citation Context ...rojective algebra representations. This provides the desired link between the two types of Euler characteristics mentioned above, and it allows us to show that the cluster characters of Fu and Keller =-=[FK]-=- coincide after an appropriate change of variables with the ϕ-functions of [GLS2, GLS5]. Finally, our third aim is to exploit these results for studying natural bases of cluster algebras containing th... |

54 | Mutation of cluster-tilting objects and potentials
- Buan, Iyama, et al.
(Show Context)
Citation Context ...not a property of the scheme Z equipped with its GLd-action, since the definition uses additionally the representation theory of the algebra ET . Note also that ET is given by a quiver with potential =-=[BIRSm]-=-, and that dimHomET (τ −1 ET (U), U) = Einj(U) is the E-invariant defined in [DWZ2]. Let Irr(mod(ET ,d)) be the set of irreducible components of mod(ET ,d), and set Irr(ET ) := ⋃ d∈NR− Irr(mod(ET ,d))... |

51 | Finite representations type is open - Gabriel - 1975 |

49 | algebras via cluster categories with infinite-dimensional morphism spaces
- Plamondon, “Cluster
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Citation Context ...gebras admitting a 2-CalabiYau categorification [P, FK], and more recently for general antisymmetric cluster algebras of geometric type [DWZ2]. There is a posterior but essentially different proof in =-=[Pl]-=-, see also [N]. The first aim of this paper is to compare these two types of formulas for the large class of cluster algebras which can be realized as coordinate rings of unipotent cells of Kac-Moody ... |

46 | Cluster structures and semicanonical bases for unipotent groups
- Geiss, Leclere, et al.
(Show Context)
Citation Context ...yama, Reiten, and Scott [BIRS] have attached to w a 2-Calabi-Yau Frobenius subcategory Cw of the category of finite-dimensional nilpotent Λ-modules. (The same categories were studied independently in =-=[GLS4]-=- for special elements w called adaptable.) In [GLS5] we showed that the C-span of {ϕX | X ∈ Cw} is a subalgebra of C[N ], which becomes isomorphic to C[Nw] after localization at the multiplicative sub... |

42 | On the exceptional fibres of Kleinian singularities - Crawley-Boevey |

42 | Irreducible components of varieties of modules
- Crawley-Boevey, Schröer
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Citation Context ...ion of T -sheets, and each irreducible component contains a unique dense T -sheet. 48 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Remark 8.4. (1) Let us recall some definitions and results from =-=[CBS]-=-. We write Z = Z ′⊕Z ′′ for irreducible components, say Z ′ ⊆ Λwd′ and Z ′′ ⊆ Λwd′′ , if and only if Z ⊆ Λ w d′+d′′ is an irreducible component which contains a dense open subset U such that for all X... |

40 | Donaldson-Thomas theory and cluster algebras
- Nagao
(Show Context)
Citation Context ...g a 2-CalabiYau categorification [P, FK], and more recently for general antisymmetric cluster algebras of geometric type [DWZ2]. There is a posterior but essentially different proof in [Pl], see also =-=[N]-=-. The first aim of this paper is to compare these two types of formulas for the large class of cluster algebras which can be realized as coordinate rings of unipotent cells of Kac-Moody groups. To do ... |

26 | Total positivity in reductive groups - Lusztig - 1994 |

24 | Minimal singularities for representations of Dynkin quivers - BONGARTZ - 1994 |

22 | Generic variables in acyclic cluster algebras
- Dupont
(Show Context)
Citation Context ...the same bases in terms of module varieties of endomorphism algebras of cluster-tilting modules. In the special case when the cluster algebra is acyclic, this proves Dupont’s generic basis conjecture =-=[D]-=-. In general, the elements of these bases are generating functions of Euler characteristics of quiver Grassmannians, at generic points of some particular irreducible components of the module varieties... |

14 |
characters for triangulated 2–Calabi-Yau categories, Annales de l’Institut Fourier 58
- Palu, Cluster
(Show Context)
Citation Context ...matrix B, we denote the inverse of its transpose by B−t.) For a general 2-Calabi-Yau Frobenius category C with a cluster-tilting object, Fu and Keller [FK, Section 3] (extending previous work of Palu =-=[P]-=-) have attached to every object of C a Laurent polynomial called its cluster character. When applied to the category Cw and the cluster-tilting object T , the formula for this cluster character can be... |

12 | The ubiquity of generalized cluster categories,
- Amiot, Reiten, et al.
- 2011
(Show Context)
Citation Context ...f the derived category. The shift in Dperf(GT ) is denoted by Σ. Following Amiot [A] we have the generalized cluster category as the triangulated quotient CT := Dperf(GT )/Df.d.(GT ). It follows from =-=[ART]-=- that Cw ∼= CV as triangulated categories, and then from [BIRSm] and [KY] that CV ∼= CT . Next, denote by F ⊆ Dperf(GT ) the subcategory which consists of the cones of maps in add(GT ). Then the canon... |

12 |
Kac-Moody groups and cluster algebras
- Geiß, Leclerc, et al.
(Show Context)
Citation Context ...∈W , let Nw := N ∩ (B−wB−) be the corresponding unipotent cell in N , where B− denotes the standard negative Borel subgroup of the Kac-Moody group G attached to g. Here we use the same notation as in =-=[GLS5]-=-. For details on Kac-Moody groups we refer GENERIC BASES FOR CLUSTER ALGEBRAS AND THE CHAMBER ANSATZ 3 to [Ku, Sections 6 and 7.4]. Let xi(t) denote the one-parameter subgroup of N associated to the s... |

7 | Partial flag varieties and preprojective algebras - Geiß, Leclerc, et al. |

6 | Finite-dimensional algebras, Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab - Drozd, Kirichenko - 1994 |