On the Laurent Phenomenon for Somos-4 and Somos-5 Sequences

2019 ◽  
Vol 53 (3) ◽  
pp. 220-223
Author(s):  
V. A. Bykovskii ◽  
A. V. Ustinov
Keyword(s):  
Laurent Phenomenon

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

scholarly journals Heronian Friezes

2020 ◽  
Author(s):  
Sergey Fomin ◽  
Linus Setiabrata
Keyword(s):  
Computational Geometry ◽  
Euclidean Plane ◽  
Type A ◽  
Cluster Algebras ◽  
Algebraic Condition ◽  
Related Family ◽  
Point Configurations ◽  
Laurent Phenomenon ◽  
Geometric Origin ◽  
Propagation Rule

Abstract Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type $A$, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter’s frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic) and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley–Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley–Menger friezes, as well as the corresponding periodicity results.

scholarly journals Laurent phenomenon algebras

2016 ◽  
Vol 4 (1) ◽  
pp. 121-162 ◽  
Author(s):  
Thomas Lam ◽  
Pavlo Pylyavskyy
Keyword(s):  
Laurent Phenomenon

scholarly journals Laurent phenomenon sequences

2015 ◽  
Vol 43 (3) ◽  
pp. 589-633 ◽  
Author(s):  
Joshua Alman ◽  
Cesar Cuenca ◽  
Jiaoyang Huang
Keyword(s):  
Laurent Phenomenon

scholarly journals Linear Laurent Phenomenon Algebras

2015 ◽  
Vol 2016 (10) ◽  
pp. 3163-3203 ◽  
Author(s):  
Thomas Lam ◽  
Pavlo Pylyavskyy
Keyword(s):  
Laurent Phenomenon

scholarly journals Friezes and continuant polynomials with parameters

2018 ◽  
Vol 36 (2) ◽  
pp. 57-81
Author(s):  
Véronique Bazier-Matte ◽  
David Racicot-Desloges ◽  
Tanna Sánchez McMillan
Keyword(s):  
Finite Type ◽  
Type A ◽  
Cluster Algebras ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Family ◽  
Laurent Phenomenon ◽  
Necessary And Sufficient ◽  
Positive Integers

Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c-friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c-friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c-friezes of integers and of positive integers. We also show some specific properties of c-friezes.

Sign in / Sign up

Export Citation Format

Share Document