scholarly journals Cluster algebras and symmetrizable matrices

2019 ◽  
Vol 147 (7) ◽  
pp. 2809-2814
Author(s):  
Ahmet I. Seven
Keyword(s):  
Cluster Algebras ◽  
Symmetrizable Matrices

Polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion

2021 ◽  
Vol 181 (4) ◽  
pp. 313-337
Author(s):  
Claudia Pérez ◽  
Daniel Rivera
Keyword(s):  
Finite Type ◽  
Equivalence Class ◽  
Polynomial Time ◽  
Essential Role ◽  
Cluster Algebra ◽  
Positive Definite ◽  
Cluster Algebras ◽  
Symmetrizable Matrices ◽  
Polynomial Class

Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Δ ∈ {𝔸n; 𝔻n; 𝔹n; ℂn; 𝔼6; 𝔼7; 𝔼8; 𝔽4; 𝔾2} of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B, if it has one.

scholarly journals Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

10.37236/6230 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
John W Lawson
Keyword(s):  
Mapping Class Groups ◽  
Cluster Algebras ◽  
Mapping Class ◽  
Symmetric Matrices ◽  
Class Groups ◽  
Exchange Graph ◽  
Symmetrizable Matrices

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.

scholarly journals Algorithms and Properties for Positive Symmetrizable Matrices

2016 ◽  
Vol 17 (2) ◽  
pp. 187 ◽  
Author(s):  
Elisângela Silva Dias ◽  
Diane Castonguay ◽  
Mitre Costa Dourado
Keyword(s):  
Diagonal Matrix ◽  
Cluster Algebras ◽  
Np Complete ◽  
Companion Matrices ◽  
Symmetrizable Matrices

Matrices are the most common representations of graphs. They are also used for representing algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding of symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices, and the problem of localizing them. Here, symmetrizable matrices are those which are symmetric when multiplied by a diagonal matrix with positive entries called symmetrizer matrix. We conjecture that this problem is NP-complete and we show that it is in NP by generalizing Sylvester's criterion for symmetrizable matrices. We straighten known coefficient limits for such matrices.

scholarly journals CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES

2006 ◽  
Vol 73 (03) ◽  
pp. 545-564 ◽  
Author(s):  
MICHAEL BAROT ◽  
CHRISTOF GEISS ◽  
ANDREI ZELEVINSKY
Keyword(s):  
Finite Type ◽  
Cluster Algebras ◽  
Symmetrizable Matrices

scholarly journals New Graphs of Finite Mutation Type

10.37236/863 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Harm Derksen ◽  
Theodore Owen
Keyword(s):  
Directed Graph ◽  
Symmetric Matrix ◽  
Cluster Algebras ◽  
Symmetric Matrices ◽  
Finite Graphs ◽  
Isomorphism Classes ◽  
Geometric Type ◽  
Symmetrizable Matrices ◽  
Marked Points ◽  
Mutation Type

To a directed graph without loops or $2$-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph $\Gamma$ is the set of all isomorphism classes of graphs that can be obtained from $\Gamma$ by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs "of geometric type". Besides graphs with $2$ vertices, and graphs of geometric type, there are only 9 other "exceptional" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

scholarly journals Positivity for cluster algebras

2015 ◽  
pp. 73-125 ◽  
Author(s):  
Kyungyong Lee ◽  
Ralf Schiffler
Keyword(s):  
Cluster Algebras

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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