scholarly journals An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras

2020 ◽  
Author(s):  
Fan Qin ◽  
Keyword(s):  
Cluster Algebras ◽  
Quantum Affine Algebras ◽  
Simple Modules ◽  
Canonical Bases ◽  
Affine Algebras ◽  
Quantum Cluster

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t-analogs of q-characters of simple modules of quantum affine algebras.

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 Cluster algebras and quantum affine algebras

2010 ◽  
Vol 154 (2) ◽  
pp. 265-341 ◽  
Author(s):  
David Hernandez ◽  
Bernard Leclerc
Keyword(s):  
Cluster Algebras ◽  
Quantum Affine Algebras ◽  
Affine Algebras

scholarly journals Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

2019 ◽  
Vol 373 (2) ◽  
pp. 655-705 ◽  
Author(s):  
So Young Cho ◽  
Hyuna Kim ◽  
Hyun Kyu Kim ◽  
Doeun Oh
Keyword(s):  
Canonical Bases ◽  
Quantum Cluster ◽  
Cluster Varieties

Quantum affine algebras

1991 ◽  
Vol 142 (2) ◽  
pp. 261-283 ◽  
Author(s):  
Vyjayanthi Chari ◽  
Andrew Pressley
Keyword(s):  
Quantum Affine Algebras ◽  
Affine Algebras
Sign in / Sign up

Export Citation Format

Share Document