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Ralf Schiffler

Publications and source records attributed to Ralf Schiffler.

At least 37 records · Page 2Linked to original sources
arXiv2020

Newton polytopes of rank 3 cluster variables

We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all $\mathbf{p}\in\mathbb{Z}^3$ such that the Laurent monomial ${\bf x}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster ${\bf x}$. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable. Along the way, we give a new proof of the fact that denominator vectors of non-initial cluster variables are non-negative in a cluster algebra of arbitrary rank.

math.AC↗
arXiv2020

On the Number of $τ$-Tilting Modules over Nakayama Algebras

Let $Λ^r_n$ be the path algebra of the linearly oriented quiver of type $\mathbb{A}$ with $n$ vertices modulo the $r$-th power of the radical, and let $\widetildeΛ^r_n$ be the path algebra of the cyclically oriented quiver of type $\widetilde{\mathbb{A}}$ with $n$ vertices modulo the $r$-th power of the radical. Adachi gave a recurrence relation for the number of $τ$-tilting modules over $Λ^r_n$. In this paper, we show that the same recurrence relation also holds for the number of $τ$-tilting modules over $\widetildeΛ^r_n$. As an application, we give a new proof for a result by Asai on recurrence formulae for the number of support $τ$-tilting modules over $Λ^r_n$ and $\widetildeΛ^r_n$.

math.RT↗
arXiv2020

Continued fractions and orderings on the Markov numbers

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper.

math.CO↗
arXiv2020

Tilting modules arising from knot invariants

We construct tilting modules over Jacobian algebras arising from knots. To a two-bridge knot $L[a_1,\ldots,a_n]$, we associate a quiver $Q$ with potential and its Jacobian algebra $A$. We construct a family of canonical indecomposable $A$-modules $M(i)$, each supported on a different specific subquiver $Q(i)$ of $Q$. Each of the $M(i)$ is expected to parametrize the Jones polynomial of the knot. We study the direct sum $M=\oplus_iM(i)$ of these indecomposables inside the module category of $A$ as well as in the cluster category. In this paper we consider the special case where the two-bridge knot is given by two parameters $a_1,a_2$. We show that the module $M$ is rigid and $τ$-rigid, and we construct a completion of $M$ to a tilting (and $τ$-tilting) $A$-module $T$. We show that the endomorphism algebra $\operatorname{End}_AT$ of $T$ is isomorphic to $A$, and that the mapping $T\mapsto A[1]$ induces a cluster automorphism of the cluster algebra $\mathcal{A}(Q)$. This automorphism is of order two. Moreover, we give a mutation sequence that realizes the cluster automorphism. In particular, we show that the quiver $Q$ is mutation equivalent to an acyclic quiver of type $T_{p,q,r}$ (a tree with three branches). This quiver is of finite type if $(a_1,a_2)=(a_1,2), (1,a_2),$ or $(2,3)$, it is tame for $(a_1,a_2)=(2,4)$ or $(3,3)$, and wild otherwise.

math.RT↗
arXiv2019

A geometric realization of socle-projective categories for posets of type $\mathbb{A}$

This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a geometric realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the realization of the cluster category of type $\mathbb{A}$ as the category of all diagonals by Caldero, Chapoton and the first author. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.

math.RT↗
arXiv2019

A note on sequential walks

This short note is devoted to motivate and clarify the notion of sequential walk introduced by the authors in a previous work. We also give some applications of this concept.

math.RT↗
arXiv2019

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.

math.CO↗
arXiv2019

A note on cluster automorphism groups

We conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with mutations as in the original definition of cluster automorphisms. We prove the conjecture in the case where at least one of the two clusters is bipartite.

math.RT↗
arXiv2019

Frieze varieties are invariant under Coxeter mutation

We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of cluster variables in the cluster algebra for this quiver. The original frieze variety is obtained when this specialization is (1, . . . , 1). The main result is that a generalized frieze variety is determined by any generic element of any component of that variety. We also show that the "Coxeter mutation" cyclically permutes these components. In particular, this shows that the frieze variety is invariant under the Coxeter mutation at a generic point. The paper contains many examples which are generated using a new technique which we call an invariant Laurent polynomial. We show that a symmetry of a mutation of a quiver gives such an invariant rational function.

math.RT↗
arXiv2019

A geometric $q$-character formula for snake modules

Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $\mathscr{C}_1$ are factorial for Dynkin types $\mathbb{A,D,E}$.

math.QA↗
arXiv2018

Group actions on cluster algebras and cluster categories

We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This generalized cluster structure is different from those introduced by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras from surfaces, we describe the generalized cluster structure of the orbit space in terms of a triangulated orbifold. In this case, we give a complete list of exchange polynomials, and we classify the algebras of rank 1 and 2. We also show that every admissible group action on a cluster category induces a precovering from the cluster category to the cluster category of orbits. Moreover this precovering is dense if the categories are of finite type.

math.RT↗
arXiv2018

Cluster automorphisms and quasi-automorphisms

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra $\mathcal{A}$. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of $\mathcal{A}$ is isomorphic to a subgroup of the cluster automorphism group of $\mathcal{A}_{triv}$, and the two groups are isomorphic if $\mathcal{A}$ has principal or universal coefficients; here $\mathcal{A}_{triv}$ is the cluster algebra with trivial coefficients obtained from $\mathcal{A}$ by setting all frozen variables equal to the integer 1. We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of $\mathcal{A}_{triv}$.

math.RA↗
arXiv2018

Frieze varieties : A characterization of the finite-tame-wild trichotomy for acyclic quivers

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is $0,1$, or $\ge2$, respectively.

math.RT↗
arXiv2017

Cluster algebras and Jones polynomials

We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each continued fraction a snake graph, and hence a cluster variable in a cluster algebra. We show that up to normalization by the leading term the Jones polynomial of the 2-bridge link is equal to the specialization of this cluster variable obtained by setting all initial cluster variables to 1 and specializing the initial principal coefficients of the cluster algebra as follows $y_1=t^{-2}$ and $ y_i=-t^{-1}$, for all $i> 1$. As a consequence we obtain a direct formula for the Jones polynomial of a 2-bridge link as the numerator of a continued fraction of Laurent polynomials in $q=-t^{-1}$. We also obtain formulas for the degree and the width of the Jones polynomial, as well as for the first three and the last three coefficients. Along the way, we also develop some basic facts about even continued fractions and construct their snake graphs. We show that the snake graph of an even continued fraction is ismorphic to the snake graph of a positive continued fraction if the continued fractions have the same value. We also give recursive formulas for the Jones polynomials.

math.GT↗
arXiv2016

Cluster algebras and continued fractions

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the quotient of the number of perfect matchings of $\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First, we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then, we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

math.CO↗
arXiv2016

Cluster-tilted and quasi-tilted algebras

In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections. We then give an algorithmic procedure for constructing the tubes of a cluster-tilted algebra of euclidean type. Our main result characterizes quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type.

math.RT↗