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 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 An immanant formulation of the dual canonical basis of the quantum polynomial ring

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mark Skandera ◽  
Justin Lambright
Keyword(s):  
Polynomial Ring ◽  
Canonical Basis ◽  
Polynomial Rings ◽  
Dual Canonical Basis ◽  
International Audience ◽  
Quantum Polynomial ◽  
Lusztig Polynomials

International audience We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these. Nous démontrons que des éléments de la base canonique duale de l'anneau quantique des polynômes en $n^2$ variables peuvent s'exprimer en termes des spécialisations d'éléments de la base canonique duale des espaces de poids $0$ d'autres anneaux quantiques. Nos résultats dépendent fortement de l'apparition naturelle des polynômes de Kazhdan-Lusztig, des $R$-polynômes, et de certaines généralisations simplement et doublement paraboliques de ces polynômes dans l'anneau quantique.

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.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

scholarly journals Atomic Bases and $T$-path Formula for Cluster Algebras of Type $D$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emily Gunawan ◽  
Gregg Musiker
Keyword(s):  
Cluster Algebra ◽  
Cluster Algebras ◽  
Combinatorial Proof ◽  
Expansion Formula ◽  
Type D ◽  
International Audience ◽  
T Path ◽  
Nous Utilisons

International audience We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$. Nous généralisons une formule de développement en $T$-chemins pour les arcs sur une surface non-perforée aux arcs sur un polygone à une perforation. Nous utilisons cette formule pour donner une preuve combinatoire du fait que les monômes amassées constituent la base atomique d’une algèbre amassée de type $D$.

scholarly journals Cyclic Sieving, Promotion, and Representation Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Fixed Points ◽  
Canonical Basis ◽  
Dihedral Groups ◽  
Jeu De Taquin ◽  
Noncrossing Partitions ◽  
Dual Canonical Basis ◽  
International Audience ◽  
Cyclic Sieving

International audience We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.

scholarly journals Combinatorial Hopf algebra of supercharacters of type D

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Carolina Benedetti
Keyword(s):  
Hopf Algebra ◽  
Finite Field ◽  
Type A ◽  
The Other ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Type D ◽  
International Audience ◽  
Supercharacter Theory ◽  
Triangular Group

International audience We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types. Dotamos con una estructura de álgebra de Hopf la teoría de supercaracteres del grupo de matrices unipotentes triangulares superiores de tipo{D} sobre un cuerpo finito. Ademas, discutimos brevemente los tipos {B} y {C}. El tipo A fue explorado por M. Aguiar et al (2010), por lo tanto este resumen extendido es una contribución para entender combinatoriamente la teoría de supercaracteres de los otros tipos de Lie clásicos. Nous construisons une structure d'algèbre de Hopf sur la thérie des supercharactères du groupe de matrices unipotentes triangulaires supéieures de type {D}. Nous donnons aussi quelques commentaires à l'égard des types {B} et {C} . Le type {A} a été explorée par M. Aguiar et al. (2010), donc ce résumé étendu est une contribution à la théorie combinatoire des supercharactères pour les autres types de Lie classiques. \par

Multiplication Formulas and Canonical Bases for Quantum Affine gln

2018 ◽  
Vol 70 (4) ◽  
pp. 773-803 ◽  
Author(s):  
Jie Du ◽  
Zhonghua Zhao
Keyword(s):  
Quantum Group ◽  
Canonical Basis ◽  
Recursive Formula ◽  
Loop Algebra ◽  
Hall Algebra ◽  
Monomial Basis ◽  
Multiplication Formula ◽  
Canonical Bases ◽  
Theoretic Proof ◽  
Cyclic Quiver

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

scholarly journals Generalised cluster algebras and $q$-characters at roots of unity

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Anne-Sophie Gleitz
Keyword(s):  
Cluster Algebra ◽  
Integral Form ◽  
The Other ◽  
Cluster Algebras ◽  
Roots Of Unity ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Root Of Unity ◽  
International Audience

International audience Shapiro and Chekhov (2011) have introduced the notion of <i>generalised cluster algebra</i>; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the <i>restricted integral form</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$. Shapiro et Chekhov (2011) ont introduit la notion d'<i>algèbre amassée généralisée</i>; nous étudions un exemple en type $C_n$. Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la <i>forme entière restreinte</i> $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ d'une algèbre affine quantique $U_q(\widehat{\mathfrak{g}})$ où $q=ε$ est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle $C_{ε^\mathbb{z}}$ de représentations de $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ est une algèbre amassée généralisée de type $C_{l−1}$, où $l$ est l'ordre de $ε^2$. Nous conjecturons une propriété similaire pour $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$ et donnons un aperçu de la preuve pour $l=2$.

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