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 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 A Cluster Expansion Formula ($A_n$ case)

10.37236/788 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralf Schiffler
Keyword(s):  
Finite Type ◽  
Cluster Expansion ◽  
Regular Polygon ◽  
Cluster Algebra ◽  
Type A ◽  
Cluster Algebras ◽  
Expansion Formula ◽  
Cluster Variable

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\Sigma$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.

scholarly journals Diagrammatic Description of $c$-Vectors and $d$-Vectors of Cluster Algebras of Finite Type

10.37236/3106 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tomoki Nakanishi ◽  
Salvatore Stella
Keyword(s):  
Finite Type ◽  
Computer Algebra ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Computer Algebra Software ◽  
Exchange Matrix ◽  
General Cluster ◽  
Direct Inspection ◽  
Surface Realization

We provide an explicit Dynkin diagrammatic description of the $c$-vectors and the $d$-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types $A_n$ and $D_n$, then we apply the folding method to $D_{n+1}$ and $A_{2n-1}$ to obtain types $B_n$ and $C_n$. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of $c$-vectors for a general cluster algebra.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

scholarly journals Notes on cluster algebras and some all-loop Feynman integrals

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Qinglin Yang
Keyword(s):  
Configuration Space ◽  
Strong Evidence ◽  
Wilson Loop ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Gauge Invariant ◽  
Feynman Integrals ◽  
Cluster Function ◽  
Invariant Description ◽  
Cluster Configuration

Abstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

scholarly journals On Generalized Minors and Quiver Representations

2018 ◽  
Vol 2020 (3) ◽  
pp. 914-956 ◽  
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Finite Type ◽  
Dynkin Diagram ◽  
Cluster Algebra ◽  
Coordinate Ring ◽  
Coxeter Element ◽  
Quiver Representations ◽  
Group Representations ◽  
Weight Group ◽  
Highest Weight ◽  
Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

scholarly journals On classification of some surfaces of revolution of finite type

1993 ◽  
Vol 17 (1) ◽  
pp. 287-298 ◽  
Author(s):  
Bang-yen Chen ◽  
Susumu Ishikawa
Keyword(s):  
Finite Type ◽  
Surfaces Of Revolution

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

scholarly journals Universal geometric coefficients for the four-punctured sphere (Extended Abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emily Barnard ◽  
Emily Meehan ◽  
Shira Polster ◽  
Nathan Reading
Keyword(s):  
Cluster Algebra ◽  
Short List ◽  
Nous Permet ◽  
Rational Part ◽  
International Audience

International audience We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the $g$ -vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute these shear coordinates to obtain universal geometric coefficients. Nous construisons des coefficients géométriques universels pour l’algèbre amassée associée à la sphère privée de 4 points, et obtenons ce faisant les $g$-vecteurs des variables d’amas. Nous construisons aussi la partie rationnelle de l’éventail de mutation. Ces constructions reposent sur la classification des courbes admissibles (les courbes qui peuvent apparaître dans les quasi-laminations). Cette classification nous permet de prouver la “Null Tangle Property” pour la sphère privée de 4 points, ajoutant ainsi cette surface à la courte liste de surfaces pour lesquelles cette propriété est connue. La “Null Tangle Property” implique alors que les coordonnées de décalage des courbes admissibles sont les coefficients universels. Nous calculons ces coordonnées de décalage pour obtenir les coefficients géométriques universels.

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