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Philipp Lampe

Publications and source records attributed to Philipp Lampe.

13 recordsLinked to original sources
arXiv2019

Snake graphs and their characteristic polynomials

The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyn's and Temperley--Fisher's ideas about spectral properties of weighted adjacency matrices of planar bipartite graphs to snake graphs. First we focus on snake graphs whose set of turning vertices is monochromatic. We provide recursive sequences to compute the characteristic polynomials; they are indexed by the upper or the lower boundary of the graph and are determined by a neighbour count. As an application, we compute the characteristic polynomials for L-shaped snake graphs and staircases in terms of Fibonacci product polynomials. Next, we introduce a method to compute the characteristic polynomials as convergents of continued fractions. Finally, we show how to transform a snake graph with turning vertices of two colours into a graph with the same number of perfect matchings to which we can apply the results above.

math.CO↗
arXiv2019

Exchange graphs for mutation-finite non-integer quivers of rank 3

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer weights of arrows. One can mutate such quivers according to usual rules of quiver mutation. Felikson and Tumarkin show that rank 3 mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows to define a notion of seeds (as Y-seeds), and hence, to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers in rank 3. The concept of finite and affine type generalises naturally to non-integer quivers. In particular, exchange graphs of finite type quivers are finite, while exchange graphs of affine quivers are finite modulo the action of a finite-dimensional lattice.

math.CO↗
arXiv2019

Cluster algebras and discrete integrability

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the context of cluster mutation. In particular, we give examples of birational maps that are integrable in the Liouville sense and arise from cluster algebras with periodicity, as well as examples of discrete Painlevé equations that are derived from Y-systems.

math.CO↗
arXiv2018

An expansion formula for type A and Kronecker quantum cluster algebras

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker case, the coefficients are q-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge's q=-1 phenomenon with respect to the weight function.

math.QA↗
arXiv2017

Factoriality and class groups of cluster algebras

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-$1$ prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss--Leclerc--Schr\"oer.

math.AC↗
arXiv2017

Variations on Baur--Marsh's determinant

Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from the squares of the cluster variables in Baur--Marsh's matrix. One such determinant admits an interpretation as a Cayley--Menger determinant. On the other hand, we wish to present a formula for the determinant of a matrix of cluster variables in a cluster algebra of type D. This cluster algebra is associated with a marked oriented surface. As in Baur--Marsh's setup the matrix is indexed by the marked points of the surface and an entry is given by the cluster variable corresponding to an arc between two marked points. Our formula asserts that the determinant may again be written as a product of cluster variables.

math.CO↗
arXiv2016

Maximum antichains in posets of quiver representations

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type A we construct a maximum antichain in the poset. Secondly, we consider the set of subrepresentations of a given quiver representation, again ordered by inclusion. It is a finite set if we restrict to linear representations over finite fields or to representations with values in the category of pointed sets. For particular situations we prove that this poset is Sperner.

math.RT↗
arXiv2016

Diophantine equations via cluster transformations

Motivated by Fomin and Zelevinsky's theory of cluster algebras we introduce a variant of the Markov equation; we show that all natural solutions of the equation arise from an initial solution by cluster transformations.

math.NT↗
arXiv2014

Quantisation Spaces of Cluster Algebras

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an explicit generating set to construct all quantisations.

math.QA↗
arXiv2013

Quantum cluster algebras of type A and the dual canonical basis

The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.

math.RT↗
arXiv2012

Acyclic cluster algebras from a ring theoretic point of view

The article gives a ring theoretic perspective on cluster algebras. Geiß-Leclerc-Schröer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster algebra to be a unique factorization domain, namely the irreducibility and the coprimality of the initial exchange polynomials. We present a sufficient condition for a cluster algebra to be a unique factorization domain in terms of primary decompositions of certain ideals generated by initial cluster variables and initial exchange polynomials. As an application, the criterion enables us to decide which coefficient-free cluster algebras of Dynkin type A, D or E are unique factorization domains. Moreover, it yields a normal form for irreducible elements in cluster algebras that satisfy the condition. Proof techniques include methods from commutative algebra. In addition, we state a conjecture about the range of application of the criterion.

math.RA↗
arXiv2010

A quantum cluster algebra of Kronecker type and the dual canonical basis

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U(n). We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(C_M) is the specialization of the dual of an appropriate canonical basis element. Therefore, U_q^+(w) is a quantum cluster algebra in the sense of Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized cluster variables and for expansions of products of dual canonical basis elements.

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