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Outputs (28)

Exhange graphs for mutation-finite non-integer quivers (2023)
Journal Article
Felikson, A., & Lampe, P. (2023). Exhange graphs for mutation-finite non-integer quivers. Journal of Geometry and Physics, 188, Article 104811. https://doi.org/10.1016/j.geomphys.2023.104811

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with noninteger arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quiver... Read More about Exhange graphs for mutation-finite non-integer quivers.

Cluster algebras of finite mutation type with coefficients (2023)
Journal Article
Felikson, A., & Tumarkin, P. (in press). Cluster algebras of finite mutation type with coefficients. Journal of combinatorial algebra,

We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type. This completes the classification of all mutation-finite cluster algebras started in [FeSTu1].

Mutation-finite quivers with real weights (2023)
Journal Article
Felikson, A., & Tumarkin, P. (2023). Mutation-finite quivers with real weights. Forum of Mathematics, Sigma, 11, Article e9. https://doi.org/10.1017/fms.2023.8

We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrisable matrix has a geometric realisation by reflections. We also explore the structure of acyclic represe... Read More about Mutation-finite quivers with real weights.

Friezes for a pair of pants (2022)
Journal Article
Canakci, I., Garcia Elsener, A., Felikson, A., & Tumarkin, P. (2022). Friezes for a pair of pants. Séminaire lotharingien de combinatoire, 86B, Article 32

Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can be genera... Read More about Friezes for a pair of pants.

Cluster algebras from surfaces and extended affine Weyl groups (2021)
Journal Article
Felikson, A., Lawson, J., Shapiro, M., & Tumarkin, P. (2021). Cluster algebras from surfaces and extended affine Weyl groups. Transformation Groups, 26(2), 501-535. https://doi.org/10.1007/s00031-021-09647-y

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangul... Read More about Cluster algebras from surfaces and extended affine Weyl groups.

Infinite rank surface cluster algebras (2019)
Journal Article
Canakci, I., & Felikson, A. (2019). Infinite rank surface cluster algebras. Advances in Mathematics, 352, 862-942. https://doi.org/10.1016/j.aim.2019.06.008

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation... Read More about Infinite rank surface cluster algebras.

Geometry of mutation classes of rank 3 quivers (2019)
Journal Article
Felikson, A., & Tumarkin, P. (2019). Geometry of mutation classes of rank 3 quivers. Arnold Mathematical Journal, 5(1), 37-55. https://doi.org/10.1007/s40598-019-00101-2

We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by π-rotations for the cyclic ones. The geometr... Read More about Geometry of mutation classes of rank 3 quivers.

Acyclic cluster algebras, reflection groups, and curves on a punctured disc (2018)
Journal Article
Felikson, A., & Tumarkin, P. (2018). Acyclic cluster algebras, reflection groups, and curves on a punctured disc. Advances in Mathematics, 340, 855-882. https://doi.org/10.1016/j.aim.2018.10.020

We establish a bijective correspondence between certain non-self-intersecting curves in an n-punctured disc and positive c-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, we obtai... Read More about Acyclic cluster algebras, reflection groups, and curves on a punctured disc.

Bases for cluster algebras from orbifolds (2017)
Journal Article
Felikson, A., & Tumarkin, P. (2017). Bases for cluster algebras from orbifolds. Advances in Mathematics, 318, 191-232. https://doi.org/10.1016/j.aim.2017.07.025

We generalize the construction of the bracelet and bangle bases defined in [36] and the band basis defined in [43] to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive, and the bracelet basis for the affine cluste... Read More about Bases for cluster algebras from orbifolds.

Double Pants Decompositions Revisited (2017)
Journal Article
Felikson, A., & Natanzon, S. (2017). Double Pants Decompositions Revisited. Moscow mathematical journal, 17(1), 51-58

Double pants decompositions were introduced by the authors in 2011, together with a flip-twist groupoid acting on these decompositions. It was shown that flip-twist groupoid acts transitively on a certain topological class of the decompositions, howe... Read More about Double Pants Decompositions Revisited.

Coxeter groups, quiver mutations and geometric manifolds (2016)
Journal Article
Felikson, A., & Tumarkin, P. (2016). Coxeter groups, quiver mutations and geometric manifolds. Journal of the London Mathematical Society, 94(1), 38-60. https://doi.org/10.1112/jlms/jdw023

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh, and involves mutations of quivers and diagrams defined in the theory of... Read More about Coxeter groups, quiver mutations and geometric manifolds.

Coxeter groups and their quotients arising from cluster algebras (2015)
Journal Article
Felikson, A., & Tumarkin, P. (2016). Coxeter groups and their quotients arising from cluster algebras. International Mathematics Research Notices, 2016(17), 5135-5186. https://doi.org/10.1093/imrn/rnv282

In [1], Barot and Marsh presented an explicit construction of presentation of a finite Weyl group W by any initial seed of corresponding cluster algebra, that is, by any diagram mutation-equivalent to an orientation of a Dynkin diagram with Weyl grou... Read More about Coxeter groups and their quotients arising from cluster algebras.

Reflection subgroups of odd-angled Coxeter groups (2014)
Journal Article
Felikson, A., Fintzen, J., & Tumarkin, P. (2014). Reflection subgroups of odd-angled Coxeter groups. Journal of Combinatorial Theory, Series A, 126, 92-127. https://doi.org/10.1016/j.jcta.2014.04.008

We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.

Growth rate of cluster algebras (2014)
Journal Article
Felikson, A., Shapiro, M., Thomas, H., & Tumarkin, P. (2014). Growth rate of cluster algebras. Proceedings of the London Mathematical Society, 109(3), 653-675. https://doi.org/10.1112/plms/pdu010

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.

Essential hyperbolic Coxeter polytopes (2013)
Journal Article
Felikson, A., & Tumarkin, P. (2014). Essential hyperbolic Coxeter polytopes. Israel Journal of Mathematics, 199(1), 113-161. https://doi.org/10.1007/s11856-013-0046-3

We introduce a notion of an essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytope... Read More about Essential hyperbolic Coxeter polytopes.

Cluster algebras and triangulated orbifolds (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Cluster algebras and triangulated orbifolds. Advances in Mathematics, 231(5), 2953-3002. https://doi.org/10.1016/j.aim.2012.07.032

We construct geometric realizations for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston [10] to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hy... Read More about Cluster algebras and triangulated orbifolds.

Moduli via double pants decompositions (2012)
Journal Article
Felikson, A., & Natanzon, S. (2012). Moduli via double pants decompositions. Differential Geometry and its Applications, 30(5), 490-508. https://doi.org/10.1016/j.difgeo.2012.07.002

We consider (local) parameterizations of Teichmüller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6g−6+3n geodesics. We find a large family of suitable sets of 6g−6+3n geodesics, each set forming a special stru... Read More about Moduli via double pants decompositions.

Skew-symmetric cluster algebras of finite mutation type (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Skew-symmetric cluster algebras of finite mutation type. Journal of the European Mathematical Society, 14(4), 1135-1180. https://doi.org/10.4171/jems/329

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed... Read More about Skew-symmetric cluster algebras of finite mutation type.

Cluster algebras of finite mutation type via unfoldings (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Cluster algebras of finite mutation type via unfoldings. International Mathematics Research Notices, 2012(8), 1768-1804. https://doi.org/10.1093/imrn/rnr072

We complete the classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram characterizin... Read More about Cluster algebras of finite mutation type via unfoldings.

Labeled double pants decompositions (2011)
Journal Article
Felikson, A., & Natanzon, S. (2011). Labeled double pants decompositions. Moscow mathematical journal, 11(3), 505-519

Automorphism groups of root systems matroids (2011)
Journal Article
Dutour Sikirić, M., Felikson, A., & Tumarkin, P. (2011). Automorphism groups of root systems matroids. European Journal of Combinatorics, 32(3), 383-389. https://doi.org/10.1016/j.ejc.2010.11.003

Given a root system View the MathML source, the vector system View the MathML source is obtained by taking a representative v in each antipodal pair {v,−v}. The matroid View the MathML source is formed by all independent subsets of View the MathML so... Read More about Automorphism groups of root systems matroids.