Tensor diagrams and cluster algebras

  title={Tensor diagrams and cluster algebras},
  author={Sergey Fomin and Pavlo Pylyavskyy},
  journal={arXiv: Combinatorics},

Cluster Algebras and Classical Invariant Rings.

Let V be a k-dimensional complex vector space. The Plücker ring of polynomial SL(V ) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of

Webs on surfaces, rings of invariants, and clusters

Cluster algebra structures in classical rings of invariants for the special linear group SL3 are investigated and constructed using Kuperberg’s calculus of webs on marked surfaces with boundary.

Tensor diagrams and cluster combinatorics at punctures

Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SLk-local systems on a marked surface with extra decorations at marked points. We study this family from an

Cluster algebras and semi‐invariant rings I. Triple flags

We prove that each semi‐invariant ring of the complete triple flag of length n is an upper cluster algebra associated to an ice hive quiver. We find a rational polyhedral cone Gn such that the

Graded quantum cluster algebras and an application to quantum Grassmannians

We introduce a framework for Z ‐gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in

Braid group symmetries of Grassmannian cluster algebras

We define an action of the extended affine d-strand braid group on the open positroid stratum in the Grassmannian Gr(k,n), for d the greatest common divisor of k and n. The action is by

Tropical Fock-Goncharov coordinates for SL3-webs on surfaces I: construction

For a finite-type surface S, we study a preferred basis for the commutative algebra C[RSL3(C)(S)] of regular functions on the SL3(C)-character variety, introduced by Sikora-Westbury. These basis

Quasi-homomorphisms of cluster algebras



Imaginary vectors in the dual canonical basis of Uq(n)

Let n be a maximal nilpotent subalgebra of a simple complex Lie algebra g. We introduce the notion of imaginary vector in the dual canonical basis of Uq(n), and we give examples of such vectors for

Quantum Grothendieck rings and derived Hall algebras

We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain

Cluster algebras IV: Coefficients

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these

Positivity and canonical bases in rank 2 cluster algebras of finite and affine types

The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and

Cluster algebras II: Finite type classification

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many

Factorial cluster algebras

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an

Grassmannians and Cluster Algebras

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian G(k, n) is a cluster algebra of geometric

Triangulated Categories: Cluster algebras, quiver representations and triangulated categories

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on

The classical groups

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be