scholarly journals A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis

2010 ◽  
Author(s):  
P. Lampe
Keyword(s):  
Cluster Algebra ◽  
Canonical Basis ◽  
Dual Canonical Basis ◽  
Quantum Cluster Algebra ◽  
Quantum Cluster

scholarly journals Bar-invariant bases of the quantum cluster algebra of type A 2 (2)

2011 ◽  
Vol 61 (4) ◽  
pp. 1077-1090 ◽  
Author(s):  
Xueqing Chen ◽  
Ming Ding ◽  
Jie Sheng
Keyword(s):  
Cluster Algebra ◽  
Type A ◽  
Quantum Cluster Algebra ◽  
Quantum Cluster

scholarly journals The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials. L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .

scholarly journals The cluster and dual canonical bases of Z[x(11), ..., x(33)] are equal

2010 ◽  
Vol Vol. 12 no. 5 (Combinatorics) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Quantum Group ◽  
Polynomial Ring ◽  
Geometric Model ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
The Other ◽  
Monomial Basis ◽  
Type D ◽  
Dual Canonical Basis ◽  
International Audience

Combinatorics International audience The polynomial ring Z[x(11), ..., x(33)] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U-q(sl(3) (C)). On the other hand, Z[x(1 1), ... , x(33)] inherits a basis from the cluster monomial basis of a geometric model of the type D-4 cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.

scholarly journals Braiding operator via quantum cluster algebra

2014 ◽  
Vol 47 (47) ◽  
pp. 474006 ◽  
Author(s):  
Kazuhiro Hikami ◽  
Rei Inoue
Keyword(s):  
Cluster Algebra ◽  
Quantum Cluster Algebra ◽  
Quantum Cluster

scholarly journals A quantum cluster algebra approach to representations of simply laced quantum affine algebras

2020 ◽  
Author(s):  
Léa Bittmann
Keyword(s):  
Cluster Algebra ◽  
Irreducible Representations ◽  
Algebra Structure ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Quantum Cluster Algebra ◽  
Algebra Approach ◽  
Quantum Cluster

AbstractWe establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

scholarly journals The cluster basis $\mathbb{Z}[x_{1,1},…,x_{3,3}]

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mark Skandera
Keyword(s):  
Cluster Algebra ◽  
Canonical Basis ◽  
Transition Matrices ◽  
Canonical Bases ◽  
Basis Element ◽  
Dual Canonical Basis ◽  
Nous Montrons ◽  
International Audience

International audience We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the $\mathbb{Z}$-module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables. Nous montrons que l'ensemble des monômes de l'algebre "cluster'' $D_4$ contient une base-$\mathbb{Z}$ pour le module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. Nous montrons aussi que les matrices transitoires qui relient cette base à la base canonique duale sont unitriangulaires. Ces résultats renforcent une conjecture de Fomin et de Zelevinsky sur l'égalité de ces deux bases. Si cette égalité s'avérait être vraie, notre résultat donnerait aussi une factorisation des éléments de la base canonique duale.

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