## Cluster algebras: Notes for the CDM-03 conference

Citations: | 17 - 5 self |

### BibTeX

@TECHREPORT{Fomin_clusteralgebras:,

author = {Sergey Fomin and Andrei Zelevinsky},

title = {Cluster algebras: Notes for the CDM-03 conference},

institution = {},

year = {}

}

### OpenURL

### Abstract

Abstract. This is an expanded version of the notes of our lectures

### Citations

337 |
associativity of H-Spaces I
- Stasheff
- 1963
(Show Context)
Citation Context ...lso connected: it is well known that any two triangulations of Pn+3 are related by a sequence of flips. This exchange graph is the 1-skeleton of an n-dimensional associahedron, or Stasheff’s polytope =-=[48]-=-. See Figure 3. A generalization of this construction, to be presented in Section 5.2, will identify exchange graphs of cluster algebras of “finite type” with the 1-skeleta of generalized associahedra... |

143 |
Algorithms in invariant theory
- Sturmfels
- 1993
(Show Context)
Citation Context ... sides and diagonals of T. Remark 3.2. The collections ˜x(T) have already appeared in classical 19 th century literature on invariant theory; for invariant-theoretic connections and applications, see =-=[30, 49]-=-. In particular, it is known [30, 49] that the monomials in the generators xa which do not involve diagonals crossing each other form a linear basis in An. We shall later discuss a far-reaching genera... |

120 |
Taubes, On the self-linking of knots
- Bott, H
- 1994
(Show Context)
Citation Context ...lips as edges). The type A3 case is illustrated by Figures 3 and 8. In types Bn and Cn, the cluster complex is isomorphic to the dual complex of the n-dimensional cyclohedron, or Bott-Taubes polytope =-=[7]-=-. The vertices of this polytope are labeled by the centrally symmetricCLUSTER ALGEBRAS: CDM-03 NOTES 29 triangulations of a regular (2n + 2)-gon P2n+2 , while each edge corresponds to a pair of centr... |

114 |
The decorated Teichmuller space of punctured surfaces
- Penner
- 1987
(Show Context)
Citation Context ...ted hyperbolic distances between horocycles drawn around vertices of a polygon with geodesic sides and cusps at the vertices (the “Penner coordinates” on the corresponding decorated Teichmüller space =-=[33]-=-). This observation leads to one of many constructions of cluster algebras arising in the context of Teichmüller theory [15, 28, 50]. The algebra An has the following closely related models: • the rin... |

110 | Non-crossing partitions for classical reflection groups
- Reiner
- 1997
(Show Context)
Citation Context ...1 n n n−1 Figure 11. The numbers N(Φ) The numbers N(Φ) given by (5.2) can be thought of as generalizations of the Catalan numbers to an arbitrary Cartan-Killing type. These numbers are known to count =-=[6, 12, 14, 29, 34, 35, 45]-=- a variety of combinatorial objects related to the root system Φ. In particular, N(Φ) appears as the number of: • ad-nilpotent ideals in a semisimple Lie algebra; • antichains in the root poset; • pos... |

109 | Cluster algebras II: Finite type classification
- Fomin, Zelevinsky
(Show Context)
Citation Context ...lized associahedron. Research supported by NSF (DMS) grants 0245385 (S.F.) and 0200299 (A.Z.). 12 SERGEY FOMIN AND ANDREI ZELEVINSKY 1. Introduction Cluster algebras, first introduced and studied in =-=[21, 23, 3]-=-, are a class of axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality.... |

101 |
Cluster algebras I
- Fomin, Zelevinsky
- 2002
(Show Context)
Citation Context ...lized associahedron. Research supported by NSF (DMS) grants 0245385 (S.F.) and 0200299 (A.Z.). 12 SERGEY FOMIN AND ANDREI ZELEVINSKY 1. Introduction Cluster algebras, first introduced and studied in =-=[21, 23, 3]-=-, are a class of axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality.... |

99 | Conjectures on the quotient ring by diagonal invariants
- Haiman
- 1994
(Show Context)
Citation Context ...1 n n n−1 Figure 11. The numbers N(Φ) The numbers N(Φ) given by (5.2) can be thought of as generalizations of the Catalan numbers to an arbitrary Cartan-Killing type. These numbers are known to count =-=[6, 12, 14, 29, 34, 35, 45]-=- a variety of combinatorial objects related to the root system Φ. In particular, N(Φ) appears as the number of: • ad-nilpotent ideals in a semisimple Lie algebra; • antichains in the root poset; • pos... |

84 | Quivers with relations arising from clusters (An case
- Caldero, Chapoton, et al.
(Show Context)
Citation Context ... systems based on rational recurrences [22, 11, 46]. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations =-=[8, 9, 10, 32]-=-. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16, 27, 28, 5]. In these lectures, we conce... |

83 | Double Bruhat cells and total positivity
- Fomin, Zelevinsky
- 1999
(Show Context)
Citation Context ...things, Lusztig extended the subject by defining the totally positive variety G>0 and the totally nonnegative variety G≥0 inside every complex reductive group G. These ideas were further developed in =-=[2, 4, 18, 19]-=-. We are going to present some of the results obtained in those papers. For technical reasons, we restrict ourselves to the case where G is semisimple and simply connected. Classical total positivity ... |

70 | Parametrizations of Canonical Bases and Totally Positive Matrices - Berenstein, Fomin, et al. - 1996 |

70 | Moduli spaces of local systems and higher Teichmüller theory
- Fock, Goncharov
(Show Context)
Citation Context ...24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory =-=[15, 16, 27, 28, 5]-=-. In these lectures, we concentrate on the following aspects. Sections 2 and 3 set the groundwork for the future theory by supplying a family of motivating examples. Specifically, Section 2 1 is devot... |

67 | Cluster algebras III. Upper bounds and double Bruhat cells
- Berenstein, Fomin, et al.
(Show Context)
Citation Context ...lized associahedron. Research supported by NSF (DMS) grants 0245385 (S.F.) and 0200299 (A.Z.). 12 SERGEY FOMIN AND ANDREI ZELEVINSKY 1. Introduction Cluster algebras, first introduced and studied in =-=[21, 23, 3]-=-, are a class of axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality.... |

62 | Y -systems and generalized associahedra
- Fomin, Zelevinsky
(Show Context)
Citation Context ...with finite crystallographic root systems. We present no proofs, each time referring the reader to primary sources. Some of the material in these notes is based on the coverage in the earlier surveys =-=[17, 53]-=-. Pictures were borrowed from [24, 13, 17]. 1 This section contains material that was not covered in CDM-2003 lectures due to time constraints. We feel, however, that it is important to present in the... |

61 |
Total positivity in reductive Groups, in “Lie theory and geometry
- Lusztig
- 1994
(Show Context)
Citation Context ...ve matrices (with an emphasis on algebraic and combinatorial aspects) can be found in [20]. The interest in the subject intensified in the last decade due in large part to the discovery by G. Lusztig =-=[31]-=- of a surprising connection between total positivity and canonical bases for quantum groups. Among other things, Lusztig extended the subject by defining the totally positive variety G>0 and the total... |

57 |
The invariant theory of binary forms
- Kung, Rota
- 1984
(Show Context)
Citation Context ... sides and diagonals of T. Remark 3.2. The collections ˜x(T) have already appeared in classical 19 th century literature on invariant theory; for invariant-theoretic connections and applications, see =-=[30, 49]-=-. In particular, it is known [30, 49] that the monomials in the generators xa which do not involve diagonals crossing each other form a linear basis in An. We shall later discuss a far-reaching genera... |

56 | Generalized associahedra via quiver representations
- Marsh, Reineke, et al.
(Show Context)
Citation Context ... systems based on rational recurrences [22, 11, 46]. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations =-=[8, 9, 10, 32]-=-. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16, 27, 28, 5]. In these lectures, we conce... |

50 |
The dual braid monoid, Ann
- Bessis
(Show Context)
Citation Context ...1 n n n−1 Figure 11. The numbers N(Φ) The numbers N(Φ) given by (5.2) can be thought of as generalizations of the Catalan numbers to an arbitrary Cartan-Killing type. These numbers are known to count =-=[6, 12, 14, 29, 34, 35, 45]-=- a variety of combinatorial objects related to the root system Φ. In particular, N(Φ) appears as the number of: • ad-nilpotent ideals in a semisimple Lie algebra; • antichains in the root poset; • pos... |

49 | Polytopal realizations of generalized associahedra
- Chapoton, Fomin, et al.
(Show Context)
Citation Context ... applications: • Discrete dynamical systems based on rational recurrences [22, 11, 46]. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems =-=[24, 13]-=-. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16... |

48 | Cluster ensembles, quantization and the dilogarithm, preprint
- Fock, Goncharov
- 2003
(Show Context)
Citation Context ...24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory =-=[15, 16, 27, 28, 5]-=-. In these lectures, we concentrate on the following aspects. Sections 2 and 3 set the groundwork for the future theory by supplying a family of motivating examples. Specifically, Section 2 1 is devot... |

47 |
ad-nilpotent ideals of a Borel subalgebra
- Cellini, Papi
(Show Context)
Citation Context |

41 | Total positivity: Tests and parametrizations
- Fomin, Zelevinsky
(Show Context)
Citation Context ...M. G. Krein. References to their papers and a discussion of further connections and applications of totally positive matrices (with an emphasis on algebraic and combinatorial aspects) can be found in =-=[20]-=-. The interest in the subject intensified in the last decade due in large part to the discovery by G. Lusztig [31] of a surprising connection between total positivity and canonical bases for quantum g... |

40 | Total positivity in Schubert varieties - Berenstein, Zelevinsky - 1997 |

38 | Cluster algebras and Poisson geometry
- Gekhtman, Shapiro, et al.
(Show Context)
Citation Context ...24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory =-=[15, 16, 27, 28, 5]-=-. In these lectures, we concentrate on the following aspects. Sections 2 and 3 set the groundwork for the future theory by supplying a family of motivating examples. Specifically, Section 2 1 is devot... |

37 | Positivity and canonical bases in rank 2 cluster algebras of finite and affine types
- Sherman, Zelevinsky
(Show Context)
Citation Context ... the sequence (yt)t∈Z has period 5 (resp., 6, 8). Conjectures 4.14, 4.16 and 4.17 are known to hold for any A(b, c). (The first one follows from the results in [21], while the last two were proved in =-=[44]-=-.) Conjecture 4.19 is still open even in rank 2; it was proved in [44] for the special case bc ≤ 4. Furthermore, it was shown in [44] that for bc ≤ 4, the indecomposable positive elements form a Zbasi... |

34 | Cluster algebras and Weil-Petersson forms
- Gekhtman, Shapiro, et al.
(Show Context)
Citation Context |

29 |
Grassmannians and cluster algebras
- Scott
(Show Context)
Citation Context ... Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues =-=[38, 47]-=-. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16, 27, 28, 5]. In these lectures, we concentrate on the following aspects. Sections 2 and 3 set the groundwork for the futur... |

28 |
The Laurent phenomenon’, Adv
- Fomin, Zelevinsky
(Show Context)
Citation Context ...simple algebraic groups. Since its inception, the theory of cluster algebras has developed several interesting connections and applications: • Discrete dynamical systems based on rational recurrences =-=[22, 11, 46]-=-. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configu... |

22 | The number of -sign types
- Shi
- 1997
(Show Context)
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22 |
Connected components of real double Bruhat cells
- Zelevinsky
(Show Context)
Citation Context ...s of the real part of Gu,v . An important special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in [41, 42]; see also related work [36, 37]. The general case was handled in =-=[51]-=- using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] utilized the following general approach, which goes back to [41]: try to ... |

21 | The multidimensional cube recurrence
- Henriques, Speyer
- 708
(Show Context)
Citation Context ...simple algebraic groups. Since its inception, the theory of cluster algebras has developed several interesting connections and applications: • Discrete dynamical systems based on rational recurrences =-=[22, 11, 46]-=-. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configu... |

21 |
personal communication
- Thurston
(Show Context)
Citation Context ... (the “Penner coordinates” on the corresponding decorated Teichmüller space [33]). This observation leads to one of many constructions of cluster algebras arising in the context of Teichmüller theory =-=[15, 28, 50]-=-. The algebra An has the following closely related models: • the ring of polynomial SL2-invariants of an (n + 3)-tuple of points in C2 ; • the homogeneous coordinate ring of the Grassmannian Gr2,n+3 o... |

20 | The tropical totally positive Grassmannian
- Speyer, Williams
(Show Context)
Citation Context ... Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configurations and their tropical analogues =-=[38, 47]-=-. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16, 27, 28, 5]. In these lectures, we concentrate on the following aspects. Sections 2 and 3 set the groundwork for the futur... |

17 | Quantum cluster algebras
- Berenstein, Zelevinsky
- 2005
(Show Context)
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17 | Explicit presentations for the dual braid monoids
- Picantin
(Show Context)
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16 | Connected components in the intersection of two open opposite Schubert cells
- Shapiro, Shapiro, et al.
(Show Context)
Citation Context ... Here is one such question: describe and enumerate the connected components of the real part of Gu,v . An important special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in =-=[41, 42]-=-; see also related work [36, 37]. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] uti... |

13 |
From Littlewood-Richardson coefficients to cluster algebras in three lectures, in: Symmetric Functions 2001: Surveys of Developments and Perspectives, S.Fomin (Ed.), 253-273
- Zelevinsky
- 2002
(Show Context)
Citation Context ...otivating examples. Specifically, Section 2 1 is devoted to total positivity and geometry of double Bruhat cells in semisimple groups (a closely related connection to canonical bases was discussed in =-=[52]-=-), while a more elementary setup of Section 3 involves Ptolemy relations, combinatorics of triangulations, and a Grassmannian of 2-dimensional subspaces. Section 4 introduces cluster algebras in earne... |

9 |
Intersections of Bruhat cells in real flag varieties
- Rietsch
- 1997
(Show Context)
Citation Context ...ribe and enumerate the connected components of the real part of Gu,v . An important special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in [41, 42]; see also related work =-=[36, 37]-=-. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] utilized the following general appr... |

9 | Skew-symmetric vanishing lattices and intersection of Schubert cells
- Shapiro, Shapiro, et al.
(Show Context)
Citation Context ... Here is one such question: describe and enumerate the connected components of the real part of Gu,v . An important special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in =-=[41, 42]-=-; see also related work [36, 37]. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] uti... |

8 |
On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups
- Djoković
- 1980
(Show Context)
Citation Context |

8 | Recognizing cluster algebras of finite type
- Seven
(Show Context)
Citation Context ... the Cartan matrix of type An. Remark 5.4. It is a non-trivial problem to decide, given a matrix B, whether the corresponding cluster algebra is of finite type. This problem was solved by A. Seven in =-=[40]-=-. An alternative criterion has been recently given in [1]. Remark 5.5. It would be interesting to classify the double Bruhat cells such that the corresponding cluster algebra is of finite type. Some e... |

7 | Totally nonnegative and oscillatory elements in semisimple groups, preprint math.RT/9811100 - Fomin, Zelevinsky - 1998 |

7 | The intersection of opposed big cells in real flag varieties
- Rietsch
- 1997
(Show Context)
Citation Context ...ribe and enumerate the connected components of the real part of Gu,v . An important special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in [41, 42]; see also related work =-=[36, 37]-=-. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] utilized the following general appr... |

7 | Simply-laced Coxeter groups and groups generated by symplectic transvections
- Shapiro, Shapiro, et al.
(Show Context)
Citation Context ...portant special case of this problem, with u = 1 and v = w◦ for G = SLr+1(C), was solved in [41, 42]; see also related work [36, 37]. The general case was handled in [51] using results and ideas from =-=[43]-=- and the earlier papers mentioned above. (For follow-ups see [26, 39].) The solution in [51] utilized the following general approach, which goes back to [41]: try to find a “simple” Zariski open subva... |

4 | Orbits of groups generated by transvections over F2
- Seven
(Show Context)
Citation Context ... = SLr+1(C), was solved in [41, 42]; see also related work [36, 37]. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see =-=[26, 39]-=-.) The solution in [51] utilized the following general approach, which goes back to [41]: try to find a “simple” Zariski open subvariety U ⊂ Gu,v such that the codimension in Gu,v of the complement of... |

3 |
Tilting theory and cluster
- Buan, Marsh, et al.
(Show Context)
Citation Context ... systems based on rational recurrences [22, 11, 46]. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations =-=[8, 9, 10, 32]-=-. • Grassmannians, projective configurations and their tropical analogues [38, 47]. • Quantum cluster algebras, Poisson geometry and Teichmüller theory [15, 16, 27, 28, 5]. In these lectures, we conce... |

2 | The number of connected components in double Bruhat cells for nonsimply-laced groups
- Gekhtman, Shapiro, et al.
(Show Context)
Citation Context ... = SLr+1(C), was solved in [41, 42]; see also related work [36, 37]. The general case was handled in [51] using results and ideas from [43] and the earlier papers mentioned above. (For follow-ups see =-=[26, 39]-=-.) The solution in [51] utilized the following general approach, which goes back to [41]: try to find a “simple” Zariski open subvariety U ⊂ Gu,v such that the codimension in Gu,v of the complement of... |

2 |
algebras: notes for 2004 IMCC (Chonju, Korea, August 2004), preprint math.RT/0407414, (2004) Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N7491
- Cluster
(Show Context)
Citation Context ...with finite crystallographic root systems. We present no proofs, each time referring the reader to primary sources. Some of the material in these notes is based on the coverage in the earlier surveys =-=[17, 53]-=-. Pictures were borrowed from [24, 13, 17]. 1 This section contains material that was not covered in CDM-2003 lectures due to time constraints. We feel, however, that it is important to present in the... |

1 | Cluster algebras IV: Classical types, in preparation - Fomin, Zelevinsky |

1 |
Perfect matchings and the octahedron recurrence, math.CO/ 0402452
- Speyer
(Show Context)
Citation Context ...simple algebraic groups. Since its inception, the theory of cluster algebras has developed several interesting connections and applications: • Discrete dynamical systems based on rational recurrences =-=[22, 11, 46]-=-. • Y -systems in thermodynamic Bethe Ansatz [24]. • Generalized associahedra associated with finite root systems [24, 13]. • Quiver representations [8, 9, 10, 32]. • Grassmannians, projective configu... |