Cluster algebras of finite type via Coxeter elements and Demazure crystals of type B,C,D

2019 ◽  
Vol 137 ◽  
pp. 40-75
Author(s):  
Yuki Kanakubo
Keyword(s):  
Finite Type ◽  
Cluster Algebras ◽  
Type B ◽  
Demazure Crystals

scholarly journals Polynomial recognition of cluster algebras of finite type

2016 ◽  
Vol 457 ◽  
pp. 457-468
Author(s):  
Elisângela Silva Dias ◽  
Diane Castonguay
Keyword(s):  
Finite Type ◽  
Cluster Algebras

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 Compatibility fans realizing graphical nested complexes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Thibault Manneville ◽  
Vincent Pilaud
Keyword(s):  
Finite Type ◽  
Type B ◽  
Cluster Complexes ◽  
Explicit Constructions ◽  
International Audience ◽  
Connected Subgraphs ◽  
Generalized Associahedra

International audience Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.

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