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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.

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.

SL(2)-tilings do not exist in higher dimensions (mostly) (2018)
Journal Article
Demonet, L., Plamondon, P., Rupel, D., Stella, S., & Tumarkin, P. (2018). SL(2)-tilings do not exist in higher dimensions (mostly). Séminaire lotharingien de combinatoire, 76, Article B76d

We define a family of generalizations of SL2-tilings to higher dimensions called ϵ-SL2-tilings. We show that, in each dimension 3 or greater, ϵ-SL2-tilings exist only for certain choices of ϵ. In the case that they exist, we show that they are essent... Read More about SL(2)-tilings do not exist in higher dimensions (mostly).

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.

Exchange relations for finite type cluster algebras with acyclic initial seed and principal coefficients (2016)
Journal Article
Stella, S., & Tumarkin, P. (2016). Exchange relations for finite type cluster algebras with acyclic initial seed and principal coefficients. Symmetry, integrability and geometry: methods and applications, 12, Article 067. https://doi.org/10.3842/sigma.2016.067

We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.

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.

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.

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.