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**Cluster algebras. IV: Coefficients.**
*(English)*
Zbl 1127.16023

This article is a substantial investigation into the properties of the coefficients of a cluster algebra, as introduced by the authors in part I [J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)]. Since inception, cluster algebras have been shown to have connections to many different fields. Note that a connection between cluster algebras and Y-systems has been made by S. Fomin and A. Zelevinsky [Ann. Math. (2) 158, No. 3, 977-1018 (2003; Zbl 1057.52003)].

The authors show that, in particular, the role of a particular choice of coefficients, known as the principal coefficients, is very important. For example, they show that the exchange graph in the principal coefficient case covers the exchange graph of any cluster algebra which possesses the same exchange matrix. Formulas for cluster variables in terms of the initial seed are given in the principal coefficient case.

The paper shows that there is a nice duality between the behaviour of the coefficients of a cluster algebra and the cluster variables themselves in which both are expressed in terms of a certain family of polynomials, known as F-polynomials. They observe that this is close to the so-called cluster ensembles studied by V. V. Fock and A. B. Goncharov [Cluster ensembles, quantization and the dilogarithm, Preprint arXiv:math.AG/0311245].

The authors show further that the coefficient dynamics gives a generalised version of the Y-systems of Zamolodchikov. It is also shown that for cluster algebras of finite type, universal coefficients can be chosen, in the sense that any cluster algebra with the same initial exchange matrix can be obtained from the universal one by specialisation of coefficients.

The paper contains a wealth of fascinating detail and combinatorics as well as many interesting conjectures.

The authors show that, in particular, the role of a particular choice of coefficients, known as the principal coefficients, is very important. For example, they show that the exchange graph in the principal coefficient case covers the exchange graph of any cluster algebra which possesses the same exchange matrix. Formulas for cluster variables in terms of the initial seed are given in the principal coefficient case.

The paper shows that there is a nice duality between the behaviour of the coefficients of a cluster algebra and the cluster variables themselves in which both are expressed in terms of a certain family of polynomials, known as F-polynomials. They observe that this is close to the so-called cluster ensembles studied by V. V. Fock and A. B. Goncharov [Cluster ensembles, quantization and the dilogarithm, Preprint arXiv:math.AG/0311245].

The authors show further that the coefficient dynamics gives a generalised version of the Y-systems of Zamolodchikov. It is also shown that for cluster algebras of finite type, universal coefficients can be chosen, in the sense that any cluster algebra with the same initial exchange matrix can be obtained from the universal one by specialisation of coefficients.

The paper contains a wealth of fascinating detail and combinatorics as well as many interesting conjectures.

Reviewer: Robert Marsh (Leeds)

### MSC:

16S50 | Endomorphism rings; matrix rings |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

16G20 | Representations of quivers and partially ordered sets |

22E46 | Semisimple Lie groups and their representations |

17B20 | Simple, semisimple, reductive (super)algebras |

52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |

20G05 | Representation theory for linear algebraic groups |

14M15 | Grassmannians, Schubert varieties, flag manifolds |