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Growth rate of cluster algebras

Anna Felikson, Michael Shapiro, Hugh Thomas, Pavel Tumarkin
2014 arXiv   pre-print
We complete the computation of growth rate of cluster algebras. In particular, we show that growth of all exceptional non-affine mutation-finite cluster algebras is exponential.  ...  It is a pleasure to thank the Hausdorff Research Institute for Mathematics whose hospitality the second author enjoyed in the summer of 2011, and the Banff Research Center for hosting a workshop on cluster  ...  Vainshtein for stimulating discussions. We thank the anonymous referee for valuable comments and suggestions. We also thank N. Reading for Remark 1.3.  ... 
arXiv:1203.5558v2 fatcat:ufxb2zq6b5cnppnbeipthsrvae

Cluster automorphisms and the marked exchange graphs of skew-symmetrizable cluster algebras [article]

John W. Lawson
2016 arXiv   pre-print
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras.  ...  Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.  ...  Acknowledgements The author would like to thank Pavel Tumarkin for his help and supervision, as well as Anna Felikson and Robert Marsh for highlighting the links between this work and maximal green sequences  ... 
arXiv:1602.05494v2 fatcat:gtdzq7odo5hbvomkfl7d725frq

Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

John W Lawson
2016 Electronic Journal of Combinatorics  
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras.  ...  Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.  ...  Acknowledgements The author would like to thank Pavel Tumarkin for his help and supervision, as well as Anna Felikson and Robert Marsh for highlighting the links between this work and maximal green sequences  ... 
doi:10.37236/6230 fatcat:otw5ekudpzd7zbw3zeacgzl5rm

Growth rate of cluster algebras

Anna Felikson, Michael Shapiro, Hugh Thomas, Pavel Tumarkin
2014 Proceedings of the London Mathematical Society  
Please consult the full DRO policy for further details. Abstract. We complete the computation of growth rate of cluster algebras.  ...  The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes  ...  It is a pleasure to thank the Hausdorff Research Institute for Mathematics whose hospitality the second author enjoyed in the summer of 2011, and the Banff Research Center for hosting a workshop on cluster  ... 
doi:10.1112/plms/pdu010 fatcat:4lbj5ljt4va5nlpkf5ygv7sckm

Direction Matters: On Influence-Preserving Graph Summarization and Max-Cut Principle for Directed Graphs

Wenkai Xu, Gang Niu, Aapo Hyvärinen, Masashi Sugiyama
2021 Neural Computation  
Conventional clustering approaches, based on Min-Cut-style criteria, compress both the vertices and edges of the graph into the communities, which lead to a loss of directed edge information.  ...  Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features, useful for efficient interventions of population behavior.  ...  As the directed graph deals with skew-symmetric matrices, no prior works have explicitly addressed this problem.  ... 
doi:10.1162/neco_a_01402 fatcat:xgztzd7sbzcmhnr2bczlqbb7zu

Handbook of Network Analysis [KONECT -- the Koblenz Network Collection] [article]

Jérôme Kunegis
2017 arXiv   pre-print
This is the handbook for the KONECT project, the Koblenz Network Collection, a scientific project to collect, analyse, and provide network datasets for researchers in all related fields of research, by  ...  the Namur Center for Complex Systems (naXys) at the University of Namur, Belgium, with web hosting provided by the Institute for Web Science and Technologies (WeST) at the University of Koblenz--Landau  ...  The list of contributors would be too long to list, so instead we refer to the online list of datasets for links to dataset sources and citations relevant to each network. 14 KONECT was also supported  ... 
arXiv:1402.5500v4 fatcat:dcvsh2ptorasdb6heos5zkoyei

CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES

MICHAEL BAROT, CHRISTOF GEISS, ANDREI ZELEVINSKY
2006 Journal of the London Mathematical Society  
We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type  ...  to skew-symmetrizable matrices.  ...  Acknowledgments The first solution of the recognition problem for cluster algebras of finite type was given by A.  ... 
doi:10.1112/s0024610706022769 fatcat:32bvzulrlfbcppiyxkpyfuym4q

Cluster algebras and triangulated orbifolds

Anna Felikson, Michael Shapiro, Pavel Tumarkin
2012 Advances in Mathematics  
We construct geometric realizations for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston [FT] to skew-symmetrizable case.  ...  Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds.  ...  This conjecture was proved for skew-symmetric cluster algebras by Derksen, Weyman and Zelevinsky [DWZ] , and for a large class of skew-symmetrizable algebras by Demonet [D] .  ... 
doi:10.1016/j.aim.2012.07.032 fatcat:cpidconavfbxvaco4boctv3x7u

The Decomposition Algorithm of Skew-symmetrizable Exchange Matrices [article]

Weiwen Gu
2012 arXiv   pre-print
Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices.  ...  As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.  ...  Mutation and mutation class are also defined for skew-symetrizable matrices.  ... 
arXiv:1202.0529v2 fatcat:4xos2i54eva5lpy6r3iffhfky4

Cluster automorphism groups and automorphism groups of exchange graphs [article]

Wen Chang, Bin Zhu
2020 arXiv   pre-print
For a coefficient free cluster algebra A, we study the cluster automorphism group Aut(A) and the automorphism group Aut(E_A) of its exchange graph E_A.  ...  We show that these two groups are isomorphic with each other, if A is of finite type excepting types of rank two and type F_4, or if A is of skew-symmetric finite mutation type.  ...  Acknowledgements Both authors thank for anonymous reviewer's careful reading and many valuable suggestions. In particular, the reference [6] is pointed out by the reviewer.  ... 
arXiv:1506.02029v3 fatcat:2ougwxrpbng47e4khegff4y6ua

Cluster algebras and triangulated orbifolds [article]

Anna Felikson, Michael Shapiro, Pavel Tumarkin
2014 arXiv   pre-print
We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case.  ...  Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds.  ...  This conjecture was proved for skew-symmetric cluster algebras by Derksen, Weyman and Zelevinsky [DWZ] , and for a large class of skew-symmetrizable algebras by Demonet [D] .  ... 
arXiv:1111.3449v4 fatcat:oczjlhj2zzfe5ekzqgjfaolwmi

Poisson Structures Compatible with the Cluster Algebra Structure in Grassmannians

M. Gekhtman, M. Shapiro, A. Stolin, A. Vainshtein
2012 Letters in Mathematical Physics  
The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.  ...  We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian G_k(n) and show that any such bracket endows G_k(n) with a structure  ...  Thus it is convenient to describe B by a directed graph Γ( B).  ... 
doi:10.1007/s11005-012-0547-8 fatcat:ajr5cx6wdjh3plruw7shncpo5u

Graphs with Non-unique Decomposition and Their Associated Surfaces [article]

Weiwen Gu
2011 arXiv   pre-print
The ideal (tagged resp.) triangulation of bounded surface with marked points are associated with skew-symmetric (skew-symmetrizable) exchange matrices.  ...  An algo- rithm is established to decompose the graph associated to such matrix. There are finite many graph with non-unique decomposition. We find all such graphs and their decompositions.  ...  The n × n matrix B is skew-symmetric, and all its entries b ij are equal to 0, 1, −1, 2, or −2. A quiver is defined as a finite oriented multi-graph without loops and 2-cycles. Definition 2.  ... 
arXiv:1112.1008v1 fatcat:rxp4oih5sjenrprgah6ug5hmoe

Cluster algebras II: Finite type classification

Sergey Fomin, Andrei Zelevinsky
2003 Inventiones Mathematicae  
The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex.  ...  Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters.  ...  We thank Tara Holm and Egon Schulte for bibliographical and terminological guidance, and Brian Conrad and Peter Magyar for useful discussions.  ... 
doi:10.1007/s00222-003-0302-y fatcat:cou7kroq3nhjje2myq2vam4sru

Asymmetric Multiresolution Matrix Factorization [article]

Pramod Kaushik Mudrakarta, Shubhendu Trivedi, Risi Kondor
2019 arXiv   pre-print
While effective for such matrices, there is plenty of data that is more naturally represented as nonsymmetric matrices (e.g. directed graphs), but nevertheless has similar hierarchical structure.  ...  Using ideas from multiresolution analysis (MRA), MMF teased out hierarchical structure in symmetric matrices by constructing a sequence of wavelet bases.  ...  MMF for skew-symmetric matrices For skew-symmetric matrices, the MRA axioms, especially MRA1, are no longer applicable.  ... 
arXiv:1910.05132v1 fatcat:6frjc32ivjdn3jvnygfaavrlmi
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