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Dylan Rupel

Publications and source records attributed to Dylan Rupel.

At least 19 recordsLinked to original sources
arXiv2025

Dominance regions for affine cluster algebras

We determine dominance regions associated to cluster algebras of affine type. In the most interesting cases, the dominance region is a line segment, which we describe explicitly. Motivations for this work include a project to determine all pointed bases for cluster algebras of affine type and a separate application that determines all theta functions in the affine case. The proofs draw on known results from the doubled Cambrian fan and almost-positive roots models, as well as a new tool that we develop: a detailed description of neighboring seeds of affine type (seeds that are, in some sense, as close as possible to the boundary of the g-vector fan).

math.RT↗
arXiv2025

Dilogarithm identities of higher degree and cluster $a$-variable periodicity

We use the periodicities of cluster groupoid mutations established by Li and the second author to prove that the dilogarithm identities of higher degree obtained by Nakanishi follow from the classical dilogarithm identities associated to a periodicity of cluster mutations when the exchange matrix is full rank.

math.RA↗
arXiv2020

Some Consequences of Categorification

Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to $\mathbf{d}$-vectors, $\mathbf{g}$-vectors, and $F$-polynomials.

math.RA↗
arXiv2019

Rank Two Non-Commutative Laurent Phenomenon and Pseudo-Positivity

We study polynomial generalizations of the Kontsevich automorphisms acting on the skew-field of formal rational expressions in two non-commuting variables. Our main result is the Laurentness and pseudo-positivity of iterations of these automorphisms. The resulting expressions are described combinatorially using a generalization of the combinatorics of compatible pairs in a maximal Dyck path developed by Lee, Li, and Zelevinsky. By specializing to quasi-commuting variables we obtain pseudo-positive expressions for rank 2 quantum generalized cluster variables. In the binomial case when all internal exchange coefficients are zero, this quantum specialization provides a positive combinatorial construction of counting polynomials for Grassmannians of submodules in exceptional representations of valued quivers with two vertices.

math.QA↗
arXiv2018

Symplectic groupoids for cluster manifolds

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and Mărcuţ. These groupoid charts and their analogues for the symplectic double and blow-up groupoids are glued by lifting the cluster mutations to groupoid comorphisms whose formulas are motivated by the Hamiltonian perspective of cluster mutations introduced by Fock and Goncharov.

math.SG↗
arXiv2018

Introduction to Cluster Algebras

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.

math.CO↗
arXiv2018

Cell Decompositions for Rank Two Quiver Grassmannians

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

math.RT↗
arXiv2017

Affine cluster monomials are generalized minors

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac-Moody groups. We prove that all cluster monomials with g-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero-Chapoton description via quiver representations. In type A_1^{(1)}, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac-Moody algebras.

math.RT↗
arXiv2017

On Generalized Minors and Quiver Representations

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

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arXiv2016

Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

We introduce and study a Hamiltonian formalism of mutations in cluster algebras using canonical variables, where the Hamiltonian is given by the Euler dilogarithm. The corresponding Lagrangian, restricted to a certain subspace of the phase space, coincides with the Rogers dilogarithm. As an application, we show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from the Hamiltonian/Lagrangian point of view.

math.RA↗
arXiv2016

String diagrams for traced and compact categories are oriented 1-cobordisms

We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob/O in the sense that there is an equivalence between (Cob/O)-algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2-category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.

math.CT↗
arXiv2016

SL_2-Tilings Do Not Exist in Higher Dimensions (mostly)

We define a family of generalizations of $\operatorname{SL}_2$-tilings to higher dimensions called $\boldsymbolε$-$\operatorname{SL}_2$-tilings. We show that, in each dimension 3 or greater, $\boldsymbolε$-$\operatorname{SL}_2$-tilings exist only for certain choices of $\boldsymbolε$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers.

math.CO↗
arXiv2015

The greedy basis equals the theta basis

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

math.QA↗
arXiv2015

Companion cluster algebras to a generalized cluster algebra

We study the $c$-vectors, $g$-vectors, and $F$-polynomials for generalized cluster algebras satisfying a normalization condition and a power condition recovering classical recursions and separation of additions formulas. We establish a relationship between the $c$-vectors, $g$-vectors, and $F$-polynomials of such a generalized cluster algebra and its (left- and right-) companion cluster algebras. Our main result states that the cluster variables and coefficients of the (left- and right-) companion cluster algebras can be recovered via a specialization of the $F$-polynomials.

math.RA↗
arXiv2015

The Feigin Tetrahedron

The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms. These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra. The second goal in this project is to better understand the dual canonical basis conjecture for skew-symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would establish this "KLR conjecture" for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras.

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arXiv2014

Quantum cluster characters of Hall algebras

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that if C was hereditary, then the assignments V-> X_{V,ii} define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q,d) and ii=(ii_0,ii_0), where ii_0 is a repetition-free source-adapted sequence, then we prove that the ii-character X_{V,ii} equals the quantum cluster character X_V introduced earlier by the second author in [29] and [30]. Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6].

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