## Cluster Algebras I: Foundations

Venue: | Journal of the American Mathematical Society |

Citations: | 64 - 2 self |

### BibTeX

@ARTICLE{Fomin_clusteralgebras,

author = {Sergey Fomin and Andrei Zelevinsky},

title = {Cluster Algebras I: Foundations},

journal = {Journal of the American Mathematical Society},

year = {},

pages = {497--529}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras. Contents

### Citations

483 |
Introduction to Quantum Groups
- Lusztig
- 1993
(Show Context)
Citation Context ...ression in the variables of any given cluster. This positivity property is consistent with a remarkable connection between canonical bases and the theory of total positivity, discovered by G. Lusztig =-=[14, 15]-=-. Generalizing the classical concept of totally positive matrices, he defined totally positive elements in any reductive group G, and proved that all elements of the dual canonical basis in C[G] take ... |

336 |
Infinite Dimensional Lie Algebras, 3rd ed
- Kac
- 1990
(Show Context)
Citation Context ...−c 2 (see (4.6)). We will show that the denominators in (6.2) have a nice interpretation in terms of the root system associated to A. Let us recall some basic facts about this root system (cf., e.g., =-=[10]-=-). Let Q ∼ = Z 2 be a lattice of rank 2 with a fixed basis {α1, α2} of simple roots. The Weyl group W = W(A) is a group of linear transformations of Q generated by two simple reflections s1 and s2 who... |

218 |
Canonical bases arising from quantized enveloping algebras
- Lusztig
- 1990
(Show Context)
Citation Context ...ach of the components is spanned by a part of the above basis. Thus, this construction provides a “canonical” basis in every irreducible finite-dimensional representation of SL3. After Lusztig’s work =-=[13]-=-, this basis had been recognized as (the classical limit at q → 1 of) the dual canonical basis, i.e., the basis in the q-deformed algebra Cq[SL3/N] which is dual to Lusztig’s canonical basis in the ap... |

84 | Double Bruhat Cells and Total Positivity
- Fomin, Zelevinsky
- 1999
(Show Context)
Citation Context ...lly positive matrices, he defined totally positive elements in any reductive group G, and proved that all elements of the dual canonical basis in C[G] take positive values at them. It was realized in =-=[15, 5]-=- that the natural geometric framework for total positivity is given by double Bruhat cells, the intersections of cells of the Bruhat decompositions with respect to two opposite Borel subgroups. Differ... |

69 |
Parametrizations of canonical bases and totally positive matrices
- Berenstein, Fomin, et al.
- 1996
(Show Context)
Citation Context ... decompositions with respect to two opposite Borel subgroups. Different aspects of total positivity in double Bruhat cells were explored by the authors of the present paper and their collaborators in =-=[1, 3, 4, 5, 6, 7, 12, 17, 20]-=-. The binomial exchange relations of the form (1.1) played a crucial role in these studies. It was the desire to explain the ubiquity of these relations and to place them in a proper context that led ... |

68 | Tensor product multiplicities, canonical bases and totally positive varieties
- Berenstein, Zelevinsky
(Show Context)
Citation Context ... decompositions with respect to two opposite Borel subgroups. Different aspects of total positivity in double Bruhat cells were explored by the authors of the present paper and their collaborators in =-=[1, 3, 4, 5, 6, 7, 12, 17, 20]-=-. The binomial exchange relations of the form (1.1) played a crucial role in these studies. It was the desire to explain the ubiquity of these relations and to place them in a proper context that led ... |

62 |
Total Positivity in Reductive Groups,” in Lie Theory and Geometry
- Lusztig
- 1994
(Show Context)
Citation Context ...ression in the variables of any given cluster. This positivity property is consistent with a remarkable connection between canonical bases and the theory of total positivity, discovered by G. Lusztig =-=[14, 15]-=-. Generalizing the classical concept of totally positive matrices, he defined totally positive elements in any reductive group G, and proved that all elements of the dual canonical basis in C[G] take ... |

57 |
The invariant theory of binary forms
- Kung, Rota
- 1984
(Show Context)
Citation Context ...g is naturally identified with the ring of polynomial SL2-invariants of an (n+3)-tuple of points in C 2 . Under this isomorphism, the basis of cluster monomials corresponds to the basis considered in =-=[11, 18]-=-. (We are grateful to Bernd Sturmfels for bringing these references to our attention.) An essential feature of the exchange relations (1.1) is that the right-hand side does not involve subtraction. Re... |

49 |
On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys
- Zamolodchikov
- 1991
(Show Context)
Citation Context ...in terms of u1 and u2. □ Remark 6.3. Periodicity of the recurrence (6.9) is a very special case of the periodicity phenomenon for Y -system recurrences in the theory of the thermodynamic Bethe ansatz =-=[19]-=-. We plan to address the case of arbitrary rank in a forthcoming paper. We have shown that in a finite normalized case, any coefficient rm can be written as a monomial in qm+2, . . . , qm+h. There is ... |

48 |
The Laurent phenomenon
- Fomin, Zelevinsky
(Show Context)
Citation Context ... method developed here, it is possible to establish the Laurent phenomenon in many different situations spreading beyond the cluster algebra framework. We explore these situations in a separate paper =-=[8]-=-.4 SERGEY FOMIN AND ANDREI ZELEVINSKY The paper is organized as follows. Section 2 contains an axiomatic definition, first examples and the first structural properties of cluster algebras. One of the... |

42 | Total Positivity: tests and parametrizations
- Fomin, Zelevinsky
(Show Context)
Citation Context ... decompositions with respect to two opposite Borel subgroups. Different aspects of total positivity in double Bruhat cells were explored by the authors of the present paper and their collaborators in =-=[1, 3, 4, 5, 6, 7, 12, 17, 20]-=-. The binomial exchange relations of the form (1.1) played a crucial role in these studies. It was the desire to explain the ubiquity of these relations and to place them in a proper context that led ... |

42 |
Algorithms in invariant theory. Texts and Monographs in Symbolic Computation
- Sturmfels
- 1993
(Show Context)
Citation Context ...g is naturally identified with the ring of polynomial SL2-invariants of an (n+3)-tuple of points in C 2 . Under this isomorphism, the basis of cluster monomials corresponds to the basis considered in =-=[11, 18]-=-. (We are grateful to Bernd Sturmfels for bringing these references to our attention.) An essential feature of the exchange relations (1.1) is that the right-hand side does not involve subtraction. Re... |

41 |
Total positivity in Schubert varieties
- Berenstein, Zelevinsky
- 1997
(Show Context)
Citation Context |

36 |
String Bases for Quantum Groups of Type Ar
- Berenstein, Zelevinsky
- 1993
(Show Context)
Citation Context ... algebra (a.k.a. quantum group). The dual canonical basis in the space C[G/N] was later constructed explicitly for a few other classical groups G of small rank: for G = Sp4 in [16] and for G = SL4 in =-=[2]-=-. In both cases, C[G/N] can be seen to be a cluster algebra: there are 6 clusters of size 2 for G = Sp4, and 14 clusters of size 3 for for G = SL4. We conjecture that the above examples can be extensi... |

30 | Zelevinsky: Quasicommuting families of quantum Plücker coordinates
- Leclerc, A
- 1998
(Show Context)
Citation Context ..., for all i < j < k < l. It is convenient to identify the indices 1, . . .,n + 3 with the vertices of a convex (n + 3)-gon, and the Plücker coordinates with its sides and diagonals. We view the sides =-=[12]-=-, [23], . . ., [n + 2, n + 3], [1, n + 3] as scalars, and the diagonals as cluster variables. The clusters are the maximal families of pairwiseCLUSTER ALGEBRAS I 3 non-crossing diagonals; thus, they ... |

23 |
Connected components of real double Bruhat cells
- Zelevinsky
(Show Context)
Citation Context |

8 |
Totally nonnegative and oscillatory elements in semisimple groups, preprint math.RT/9811100
- Fomin, Zelevinsky
- 1998
(Show Context)
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8 | Simply-laced Coxeter groups and groups generated by symplectic transvections
- Shapiro, Shapiro, et al.
(Show Context)
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6 |
A.Zelevinsky, Canonical basis in irreducible representations of gl 3 and its application
- Gelfand
- 1986
(Show Context)
Citation Context ...C[SL3/N] is closely related to the choice of a linear basis in it consisting of all monomials in the six Plücker coordinates which are not divisible by x2x13. This basis was introduced and studied in =-=[9]-=- under the name “canonical basis.” As a representation of SL3, the space C[SL3/N] is the multiplicity-free direct sum of all irreducible finite-dimensional representations, and each of the components ... |

6 |
The base affine space and canonical bases in irreducible representations of the group
- RETAKH, ZELEVINSKY
- 1988
(Show Context)
Citation Context ...med universal enveloping algebra (a.k.a. quantum group). The dual canonical basis in the space C[G/N] was later constructed explicitly for a few other classical groups G of small rank: for G = Sp4 in =-=[16]-=- and for G = SL4 in [2]. In both cases, C[G/N] can be seen to be a cluster algebra: there are 6 clusters of size 2 for G = Sp4, and 14 clusters of size 3 for for G = SL4. We conjecture that the above ... |