Ag7Au6: A 13-Atom Alloy Quantum Cluster

2012 ◽  
Vol 51 (9) ◽  
pp. 2155-2159 ◽  
Author(s):  
Thumu Udayabhaskararao ◽  
Yan Sun ◽  
Nirmal Goswami ◽  
Samir K. Pal ◽  
K. Balasubramanian ◽  
...  
Keyword(s):  
Quantum Cluster

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 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

scholarly journals Comparison of two-quantum-cluster approximations

2002 ◽  
Vol 65 (4) ◽  
Author(s):  
Th. A. Maier ◽  
M. Jarrell
Keyword(s):  
Quantum Cluster

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

Crystal excitation: survey of many-electron Hartree-Fock calculations of self-trapped excitons in insulating crystals

1992 ◽  
Vol 341 (1661) ◽  
pp. 221-231 ◽  
Keyword(s):  
Electronic Structure ◽  
Alkali Halides ◽  
Oxide Materials ◽  
Theoretical Methods ◽  
Hartree Fock ◽  
Cluster Calculations ◽  
Quantum Cluster ◽  
Similarities And Differences

To model successfully the diversity of electronic structure exhibited by excitons in alkali halides and in oxide materials, it is necessary to use a variety or combination of theoretical methods. In this review we restrict our discussion to the results of embedded quantum cluster calculations. By considering the results of such studies, it is possible to recognize the general similarities and differences in detail between the various exciton models in these materials.

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