Rooted Clusters for Graph LP Algebras

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of T -paths.

[1]  S. Franco.  Bipartite field theories: from D-brane probes to scattering amplitudes , 2012, 1207.0807.

[2]  Hugh Thomas,et al.  On Cluster Algebras Arising from Unpunctured Surfaces , 2007, 0712.4131.

[3]  T. Bridgeland.  Scattering diagrams, Hall algebras and stability conditions , 2016, Algebraic Geometry.

[4]  H. Thomas,et al.  Noncrossing partitions and representations of quivers , 2006, Compositio Mathematica.

[5]  Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner , 2007, math/0702398.

[6]  H. Thomas,et al.  ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras , 2018, 1808.09986.

[8]  Cluster Algebras and Poisson Geometry , 2002, math/0208033.

[9]  A. Fordy,et al.  Discrete Integrable Systems and Poisson Algebras From Cluster Maps , 2012, Communications in Mathematical Physics.

[10]  Gregg Musiker,et al.  Cluster expansion formulas and perfect matchings , 2008, 0810.3638.

[11]  Ralf Schiffler,et al.  On cluster algebras arising from unpunctured surfaces II , 2008, 0809.2593.

[12]  B. Keller.  The periodicity conjecture for pairs of Dynkin diagrams , 2010, 1001.1531.

[13]  L. Williams,et al.  Positivity for cluster algebras from surfaces , 2009, 0906.0748.

[14]  Gregg Musiker,et al.  T -path formula and atomic bases for cluster algebras of type D , 2014, 1409.3610.

[15]  S. Fomin,et al.  Cluster algebras I: Foundations , 2001, math/0104151.

[16]  M. Gross,et al.  Birational geometry of cluster algebras , 2013, 1309.2573.

[17]  Sergey Fomin,et al.  Polytopal Realizations of Generalized Associahedra , 2002, Canadian Mathematical Bulletin.

[18]  Cluster ensembles, quantization and the dilogarithm , 2003, math/0311245.

[19]  D. Thurston,et al.  Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths , 2012, Memoirs of the American Mathematical Society.

[20]  Alek Vainshtein,et al.  Cluster algebras and Weil-Petersson forms , 2003 .

[21]  Ralf Schier,et al.  A cluster expansion formula (A n case) , 2008 .

[22]  T. Lam,et al.  Linear Laurent phenomenon algebras , 2012, 1206.2612.