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Nathan Reading

Publications and source records attributed to Nathan Reading.

At least 37 records · Page 2Linked to original sources
arXiv2025

Para-exceptional sequences for tame hereditary algebras and McCammond-Sulway lattices

Noncrossing partition posets in a Coxeter group $W$ can fail to be lattices when $W$ is not finite. When the lattice property fails for $W$ of affine type, McCammond and Sulway's construction provides a larger lattice that contains the noncrossing partition poset and that furthermore is a combinatorial Garside structure. We construct a lattice, isomorphic to McCammond and Sulway's lattice, using the representation theory of a corresponding connected tame hereditary algebra and give a representation-theoretic proof that it is a combinatorial Garside structure. To construct the lattice, we introduce para-exceptional sequences and para-exceptional subcategories in the module categories of tame hereditary algebras. Para-exceptional sequences are generalizations of exceptional sequences obtained by enlarging the set of allowed entries to include all non-homogeneous bricks. A para-exceptional subcategory is a subcategory obtained by applying a certain closure-like operator to the wide subcategory generated by a para-exceptional sequence.

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arXiv2025

Cluster scattering diagrams of acyclic affine type

We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost-positive roots model.

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arXiv2025

Noncrossing arc diagrams of type B

Noncrossing arc diagrams are combinatorial models for permutations that encode information about lattice congruences of the weak order and about the associated discrete geometry. In this paper, we consider two related, analogous models for signed permutations. One model features centrally symmetric noncrossing arc diagrams, while the other features their quotients modulo the central symmetry. We demonstrate the utility of the models by applying them to various questions about lattice quotients of the weak order.

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arXiv2024

Noncrossing partitions of an annulus

The noncrossing partition poset associated to a Coxeter group $W$ and Coxeter element $c$ is the interval $[1,c]_T$ in the absolute order on $W$. We construct a new model of noncrossing partititions for $W$ of classical affine type, using planar diagrams (affine types $\tilde A$ and $\tilde C$ in this paper and affine types $\tilde D$ and $\tilde B$ in the sequel). The model in type $\tilde A$ consists of noncrossing partitions of an annulus. In type $\tilde C$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $[1,c]_T$ to a lattice by factoring the translations in $[1,c]_T$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.

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arXiv2024

The fundamental theorem of finite semidistributive lattices

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.

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arXiv2024

Lattice theory of torsion classes: Beyond $τ$-tilting theory

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $Π$, for which $\operatorname{\mathsf{tors}} Π$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} Π$ are exactly its Hasse-regular lattice quotients.

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arXiv2024

Posets for $F$-polynomials in cluster algebras from surfaces

We prove a simple formula for arbitrary cluster variables in the marked surfaces model. As part of the formula, we associate a labeled poset to each tagged arc, such that the associated $F$-polynomial is a weighted sum of order ideals. Each element of the poset has a weight, and the weight of an ideal is the product of the weights of the elements of the ideal. In the unpunctured case, the weight on each element is a single $\hat{y}_i$, in the usual sense of principal coefficients. In the presence of punctures, some elements may have weights of the form $\hat{y}_i/\hat{y}_j$. Our search for such a formula was inspired by the Fundamental Theorem of Finite Distributive Lattices combined with work of Gregg Musiker, Ralf Schiffler, and Lauren Williams that, in some cases, organized the terms of the $F$-polynomial into a distributive lattice. The proof consists of a simple and poset-theoretically natural argument in a special case, followed by a hyperbolic geometry argument using a cover of the surface to prove the general case.

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arXiv2019

Dominance phenomena: mutation, scattering and cluster algebras

An exchange matrix $B$ dominates an exchange matrix $B'$ if the signs of corresponding entries weakly agree, with the entry of $B$ always having weakly greater absolute value. When $B$ dominates $B'$, interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for $B$ often refines the mutation fan for $B'$; the scattering (diagram) fan for $B$ often refines the scattering fan for $B'$; and there is often an injective homomorphism from the principal-coefficients cluster algebra for $B'$ to the principal-coefficients cluster algebra for $B$, preserving $\mathbf{g}$-vectors and sending the set of cluster variables for $B'$ (or an analogous larger set) into the set of cluster variables for $B$ (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena.

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arXiv2019

An affine almost positive roots model

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $Φ_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(Φ)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(Φ)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $Φ_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.

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arXiv2019

The action of a Coxeter element on an affine root system

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel.

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arXiv2018

Scattering fans

Scattering diagrams arose in the context of mirror symmetry, Donaldson-Thomas theory, and integrable systems. We show that a consistent scattering diagram with minimal support cuts the ambient space into a complete fan. A special class of scattering diagrams, the cluster scattering diagrams, are closely related to cluster algebras. We show that the cluster scattering fan associated to an exchange matrix $B$ refines the mutation fan for $B$ (a complete fan that encodes the geometry of mutations of $B$). We conjecture that, when $B$ is $n\times n$ for $n>2$, these two fans coincide if and only if $B$ is of finite mutation type.

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arXiv2017

Initial-seed recursions and dualities for d-vectors

We present an initial-seed-mutation formula for d-vectors of cluster variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.

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arXiv2017

Coxeter-biCatalan combinatorics

We pose counting problems related to the various settings for Coxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to count "twin" pairs of objects from a corresponding problem in Coxeter-Catalan combinatorics. We show that the problems all have the same answer, and, for a given finite Coxeter group W, we call the common solution to these problems the W-biCatalan number. We compute the W-biCatalan number for all W and take the first steps in the study of Coxeter-biCatalan combinatorics.

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arXiv2017

Cambrian frameworks for cluster algebras of affine type

We give a combinatorial model for the exchange graph and g-vector fan associated to any acyclic exchange matrix B of affine type. More specifically, we construct a reflection framework for B in the sense of [N. Reading and D. E. Speyer, "Combinatorial frameworks for cluster algebras"] and establish good properties of this framework. The framework (and in particular the g-vector fan) is constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B.

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arXiv2016

Universal geometric coefficients for the four-punctured sphere

We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the g-vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute shear coordinates explicitly to obtain universal geometric coefficients.

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arXiv2015

Universal geometric coefficients for the once-punctured torus

We construct universal geometric coefficients, over the integers, the rationals, and the reals, for cluster algebras arising from the once-punctured torus. We verify that the once-punctured torus has a property called the Null Tangle Property. The universal geometric coefficients over the integers and the rationals are then given by the shear coordinates of certain "allowable" curves in the torus. The universal geometric coefficients over the reals are given by the shear coordinates of allowable curves together with the normalized shear coordinates of certain other curves each of which is dense in the torus. We also construct the mutation fan for the once-punctured torus and recover a result of Nájera on g-vectors.

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arXiv2015

Coxeter arrangements in three dimensions

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric.

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