Continued fractions and orderings on the Markov numbers

Rabideau, Michelle; Schiffler, Ralf

doi:10.48550/arxiv.1801.07155

Cited by 2 publications

(3 citation statements)

References 6 publications

(11 reference statements)

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“…A Markov snake graph is a snake graph which can be built from the Christoffel path in a p by q grid, where p and q are relatively prime positive integers (see [32], [6], and [33]).…”

Section: Rank Symmetrymentioning

confidence: 99%

“…The definition of cluster algebras was motivated by observations in representation theory. Since then, cluster algebra structures have been recognized and studied in various other fields of mathematics, such as decorated Teichmüller theory and Poisson geometry (see [18] and [12], [13]), higher Teichmüller theory [9], rings of invariants (see [10] and [11]), elementary number theory [5] including diophantine equations [33], and knot theory [25], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

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Expansion Posets for Polygon Cluster Algebras

Claussen

2020

Preprint

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Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion formula. We introduce an involution on several of the interrelated combinatorial objects and constructions associated to type A surface cluster algebras, including certain classes of arcs, triangulations, and distributive lattices. We use these involutions to formulate a dual version of skein relations for arcs, and dual versions of three existing expansion posets. In particular, this leads to two new cluster expansion formulas, and recovers the lattice path expansion of Propp et al. We provide an explicit, structure-preserving poset isomorphism between an expansion poset and its dual version from the dual arc. We also show that an expansion poset and its dual version constructed from the same arc are dual in the sense of distributive lattices.We show that any expansion poset is isomorphic to a closed interval in one of the lattices L(m, n) of Young diagrams contained in an m × n grid, and that any L(m, n) has a covering by such intervals. In particular, this implies that any expansion poset is isomorphic to an interval in Young's lattice.We give two formulas for the rank function of any lattice path expansion poset, and prove that this rank function is unimodal whenever the underlying snake graph is built from at most four maximal straight segments. This gives a partial solution to a recent conjecture by Ovsienko and Morier-Genoud. We also characterize which expansion posets have symmetric rank generating functions, based on the shape of the underlying snake graph.

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“…A Markov snake graph is a snake graph which can be built from the Christoffel path in a p by q grid, where p and q are relatively prime positive integers (see [32], [6], and [33]).…”

Section: Rank Symmetrymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Expansion Posets for Polygon Cluster Algebras

Claussen

2020

Preprint

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“…For example, describing the mutations as quiver transformations or triangulations, we apply them to representation theory of quivers [1] or higher Teichmüller theory [4,9]. Also, considering the mutations of the Markov cluster algebra, a new combinatorial approach to solve the Unicity Conjecture about Markov numbers was given in number theory [3,23].…”

Section: Introduction and Main Theoremsmentioning

confidence: 99%

Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2

Gyoda

2019

Preprint

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We study the f -vectors, which are the maximal degree vectors of F -polynomials in cluster algebra theory. When a cluster algebra is of finite type or rank 2, we find that the positive f -vectors correspond with the d-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of d-vectors, we prove that cluster variables in a cluster are uniquely determined by their f -vectors when the cluster algebra is of finite type or rank 2.

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Continued fractions and orderings on the Markov numbers

Cited by 2 publications

References 6 publications

Expansion Posets for Polygon Cluster Algebras

Expansion Posets for Polygon Cluster Algebras

Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2

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