Classification of graded cluster algebras generated by rank 3 quivers

2021 ◽  
Author(s):  
Thomas Booker-Price
Keyword(s):  
Direct Proof ◽  
Cluster Algebras ◽  
Symmetric Matrices

We consider gradings on cluster algebras generated by [Formula: see text] skew-symmetric matrices. We show that, except in one particular case, mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables. For mutation-acyclic matrices, we prove that all occurring degrees are associated with infinitely many variables. We also give a direct proof that the gradings are balanced in this case (i.e. that there is a bijection between the cluster variables of degree [Formula: see text] and [Formula: see text] for each occurring degree [Formula: see text]).

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

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 Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

2017 ◽  
Vol 153 (12) ◽  
pp. 2482-2533
Author(s):  
Alexander I. Bufetov ◽  
Yanqi Qiu
Keyword(s):  
Probability Measures ◽  
Symmetric Matrices ◽  
Infinite Matrices ◽  
Locally Compact ◽  
Natural Action ◽  
Ring Of Integers ◽  
Compact Field ◽  
Ergodic Measures

Let$F$be a non-discrete non-Archimedean locally compact field and${\mathcal{O}}_{F}$the ring of integers in$F$. The main results of this paper are the classification of ergodic probability measures on the space$\text{Mat}(\mathbb{N},F)$of infinite matrices with entries in$F$with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$and the classification, for non-dyadic$F$, of ergodic probability measures on the space$\text{Sym}(\mathbb{N},F)$of infinite symmetric matrices with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})$.

scholarly journals Logical and historical determination of the Arrow and Sen impossibility theorems

2007 ◽  
Vol 52 (172) ◽  
pp. 7-20 ◽  
Author(s):  
Branislav Boricic
Keyword(s):  
Social Choice ◽  
History Of Mathematics ◽  
Direct Proof ◽  
Mathematical Economics ◽  
General Classification ◽  
Logical Equivalence ◽  
History Of

General classification of mathematical statements divides them into universal, those of the form xA , and existential ?xA ones. Common formulations of impossibility theorems of K. J. Arrow and A. K. Sen are represented by the statements of the form "there is no x such that A". Bearing in mind logical equivalence of formulae ??xA and x?A, we come to the conclusion that the corpus of impossibility theorems, which appears in the theory of social choice, could make a specific and recognizable subclass of universal statements. In this paper, on the basis of the established logical and methodological criteria, we point to a sequence of extremely significant "impossibility theorems", reaching throughout the history of mathematics to the present days and the famous results of Arrow and Sen in field of mathematical economics. We close with specifying the context which makes it possible to formulate the results of Arrow and Sen accurately, presenting a new direct proof of Sen?s result, with no reliance on the notion of minimal liberalism. .

scholarly journals Characters of equivariant -modules on spaces of matrices

2016 ◽  
Vol 152 (9) ◽  
pp. 1935-1965 ◽  
Author(s):  
Claudiu Raicu
Keyword(s):  
Local Cohomology ◽  
Direct Proof ◽  
Vector Spaces ◽  
Symmetric Matrices ◽  
Subsequent Work ◽  
Pole Order ◽  
Local Cohomology Modules ◽  
Generic Matrix ◽  
Composition Factors ◽  
Cohomology Modules

We compute the characters of the simple $\text{GL}$-equivariant holonomic ${\mathcal{D}}$-modules on the vector spaces of general, symmetric, and skew-symmetric matrices. We realize some of these ${\mathcal{D}}$-modules explicitly as subquotients in the pole order filtration associated to the $\text{determinant}/\text{Pfaffian}$ of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the ${\mathcal{D}}$-module composition factors of local cohomology modules with determinantal and Pfaffian support.

scholarly journals Affine cluster monomials are generalized minors

2019 ◽  
Vol 155 (7) ◽  
pp. 1301-1326
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Lie Groups ◽  
Cluster Algebras ◽  
Quiver Representations ◽  
Group Representations ◽  
Finite Dimensional ◽  
Bruhat Cells ◽  
Level Zero ◽  
Coordinate Rings ◽  
Affine Type

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

scholarly journals Combinatorial properties of the enhanced principal rank characteristic sequence over finite fields

2021 ◽  
Vol 10 (1) ◽  
pp. 166-179
Author(s):  
Peter J. Dukes ◽  
Xavier Martínez-Rivera
Keyword(s):  
Finite Fields ◽  
Coding Theory ◽  
Ramsey Theory ◽  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Characteristic Sequence ◽  
Combinatorial Properties ◽  
Recent Classification

Abstract The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix B ∈ 𝔽 n×n is defined as ℓ1ℓ2· · · ℓ n , where ℓ j ∈ {A, S, N} according to whether all, some but not all, or none of the principal minors of order j of B are nonzero. Building upon the second author’s recent classification of the epr-sequences of symmetric matrices over the field 𝔽 = 𝔽2, we initiate a study of the case 𝔽= 𝔽3. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.

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