scholarly journals GRASSMANNIANS AND CLUSTER ALGEBRAS

2006 ◽  
Vol 92 (2) ◽  
pp. 345-380 ◽  
Author(s):  
JOSHUA S. SCOTT
Keyword(s):  
Cluster Algebra ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Program Of Study ◽  
Geometric Type ◽  
Cluster Type

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

scholarly journals CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

2019 ◽  
Vol 240 ◽  
pp. 322-354 ◽  
Author(s):  
KARIN BAUR ◽  
DUSKO BOGDANIC ◽  
ANA GARCIA ELSENER
Keyword(s):  
Cluster Algebra ◽  
Explicit Construction ◽  
Cluster Algebras ◽  
Algebra Structure ◽  
Coordinate Ring ◽  
Moody Algebra ◽  
Indecomposable Modules ◽  
Real Roots ◽  
Rank 2 ◽  
Cluster Categories

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

scholarly journals Notes on cluster algebras and some all-loop Feynman integrals

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Qinglin Yang
Keyword(s):  
Configuration Space ◽  
Strong Evidence ◽  
Wilson Loop ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Gauge Invariant ◽  
Feynman Integrals ◽  
Cluster Function ◽  
Invariant Description ◽  
Cluster Configuration

Abstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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 On Generalized Minors and Quiver Representations

2018 ◽  
Vol 2020 (3) ◽  
pp. 914-956 ◽  
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Finite Type ◽  
Dynkin Diagram ◽  
Cluster Algebra ◽  
Coordinate Ring ◽  
Coxeter Element ◽  
Quiver Representations ◽  
Group Representations ◽  
Weight Group ◽  
Highest Weight ◽  
Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

scholarly journals Unistructurality of cluster algebras

2020 ◽  
Vol 156 (5) ◽  
pp. 946-958 ◽  
Author(s):  
Peigen Cao ◽  
Fang Li
Keyword(s):  
Cluster Algebra ◽  
Cluster Algebras

We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra ${\mathcal{A}}({\mathcal{S}})$ is just an automorphism of the ambient field ${\mathcal{F}}$ which restricts to a permutation of the cluster variables of ${\mathcal{A}}({\mathcal{S}})$.

scholarly journals CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES

2014 ◽  
Vol 56 (3) ◽  
pp. 705-720 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
VASILISA SHRAMCHENKO ◽  
RALF SCHIFFLER
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Cluster Algebras ◽  
Dynkin Type ◽  
Rank 2

AbstractIn this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if and only if any automorphism of the ambient field, which restricts to a permutation of cluster variables of $\mathcal A$, is a cluster automorphism. We also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.

scholarly journals On the Generalized Cluster Algebras of Geometric Type

2020 ◽  
Author(s):  
Liqian Bai ◽  
◽  
Xueqing Chen ◽  
Ming Ding ◽  
Fan Xu ◽  
...  
Keyword(s):  
Cluster Algebras ◽  
Geometric Type

scholarly journals Truncated cluster algebras and Feynman integrals with algebraic letters

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Qinglin Yang
Keyword(s):  
Differential Equations ◽  
Wilson Loop ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Square Root ◽  
Conformal Invariant ◽  
Feynman Integrals ◽  
Normal Vectors

Abstract We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) G+(4, n)/T for the n-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with n = 7, 8, 9, we find finite cluster algebras D4, D5 and D6 respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine D4 cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of G+(4, 8)/T with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of 38 rational letters and 5 algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight 4, by imposing last-entry conditions inspired by the n = 8 double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the n = 8 double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop d log forms, and we find a remarkable pattern about the appearance of algebraic letters.

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