Rank two non-commutative Laurent phenomenon and pseudo-positivity

Dylan C. Rupel

doi:10.5802/alco.81

Laurent phenomenon and simple modules of quiver Hecke algebras

Compositio Mathematica ◽

10.1112/s0010437x19007565 ◽

2019 ◽

Vol 155 (12) ◽

pp. 2263-2295 ◽

Cited By ~ 1

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 $[\mathscr{C}]$.

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Heronian Friezes

International Mathematics Research Notices ◽

10.1093/imrn/rnaa057 ◽

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.

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