@article{reading_2022, title={DOMINANCE PHENOMENA: MUTATION, SCATTERING AND CLUSTER ALGEBRAS}, volume={10}, ISSN={["1088-6850"]}, DOI={10.1090/tran/7888}, abstractNote={An exchange matrix B B dominates an exchange matrix B ′ B’ if the signs of corresponding entries weakly agree, with the entry of B B always having weakly greater absolute value. When B B dominates B ′ 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 B often refines the mutation fan for B ′ B’ ; the scattering (diagram) fan for B B often refines the scattering fan for B ′ B’ ; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B ′ B’ to the principal-coefficients cluster algebra for B B , preserving g \mathbf {g} -vectors and sending the set of cluster variables for B ′ B’ (or an analogous larger set) into the set of cluster variables for B 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.}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Reading, Nathan}, year={2022}, month={Oct} }
@article{reading_speyer_thomas_2021, title={The fundamental theorem of finite semidistributive lattices}, volume={27}, ISSN={["1420-9020"]}, DOI={10.1007/s00029-021-00656-z}, abstractNote={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 with some additional structure, such that L is isomorphic to the admissible subsets of ordered by inclusion; in this case, and its additional structure are uniquely determined by L.” The additional structure on 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.}, number={4}, journal={SELECTA MATHEMATICA-NEW SERIES}, author={Reading, Nathan and Speyer, David E. and Thomas, Hugh}, year={2021}, month={Sep} }
@article{an affine almost positive roots model_2020, url={http://dx.doi.org/10.4171/jca/37}, DOI={10.4171/jca/37}, abstractNote={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 $\Phi_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(\Phi)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(\Phi)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $\Phi_c$ 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.}, journal={Journal of Combinatorial Algebra}, year={2020}, month={Feb} }
@article{reading_2020, title={Scattering Fans}, volume={2020}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rny260}, abstractNote={Abstract 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, is 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.}, number={23}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Reading, Nathan}, year={2020}, month={Nov}, pages={9640–9673} }
@article{the action of a coxeter element on an affine root system_2020, url={http://dx.doi.org/10.1090/proc/14769}, DOI={10.1090/proc/14769}, abstractNote={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.}, journal={Proceedings of the American Mathematical Society}, year={2020}, month={Apr} }
@article{reading_2019, title={Lattice homomorphisms between weak orders}, volume={26}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85067350483&partnerID=MN8TOARS}, number={2}, journal={Electron. J. Combin.}, author={Reading, Nathan}, year={2019}, pages={Paper 2.23, 50} }
@article{barnard_reading_2018, title={Coxeter-biCatalan combinatorics}, volume={47}, ISSN={["1572-9192"]}, url={https://doi.org/10.1007/s10801-017-0775-1}, DOI={10.1007/s10801-017-0775-1}, abstractNote={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.}, number={2}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, publisher={Springer Science and Business Media LLC}, author={Barnard, Emily and Reading, Nathan}, year={2018}, month={Mar}, pages={241–300} }
@article{reading_stella_2018, title={INITIAL-SEED RECURSIONS AND DUALITIES FOR d-VECTORS}, volume={293}, ISSN={["0030-8730"]}, url={https://doi.org/10.2140/pjm.2018.293.179}, DOI={10.2140/pjm.2018.293.179}, abstractNote={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.}, number={1}, journal={PACIFIC JOURNAL OF MATHEMATICS}, author={Reading, Nathan and Stella, Salvatore}, year={2018}, month={Mar}, pages={179–206} }
@article{iyama_reading_reiten_thomas_2018, title={Lattice structure of Weyl groups via representation theory of preprojective algebras}, volume={154}, ISSN={["1570-5846"]}, url={https://doi.org/10.1112/s0010437x18007078}, DOI={10.1112/s0010437x18007078}, abstractNote={This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\Pi$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\tau$-rigid (respectively, $\tau^-$-rigid) modules and layers of $\Pi$. The lattice-theoretically natural labeling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\Pi$. We show that layers of $\Pi$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\tau^-$-rigid modules for type $A$ and $D$.}, number={6}, journal={COMPOSITIO MATHEMATICA}, author={Iyama, Osamu and Reading, Nathan and Reiten, Idun and Thomas, Hugh}, year={2018}, month={Jun}, pages={1269–1305} }
@article{barnard_meehan_reading_viel_2018, title={Universal Geometric Coefficients for the Four-Punctured Sphere}, volume={22}, ISSN={0218-0006 0219-3094}, url={http://dx.doi.org/10.1007/s00026-018-0378-0}, DOI={10.1007/s00026-018-0378-0}, abstractNote={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.}, number={1}, journal={Annals of Combinatorics}, publisher={Springer Nature}, author={Barnard, Emily and Meehan, Emily and Reading, Nathan and Viel, Shira}, year={2018}, month={Feb}, pages={1–44} }
@article{reading_speyer_2017, title={Cambrian frameworks for cluster algebras of affine type}, volume={370}, ISSN={0002-9947 1088-6850}, url={http://dx.doi.org/10.1090/tran/7193}, DOI={10.1090/tran/7193}, abstractNote={We give a combinatorial model for the exchange graph and $\mathbf {g}$-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, âCombinatorial frameworks for cluster algebrasâ] and establish good properties of this framework. The framework (and in particular the $\mathbf {g}$-vector fan) is constructed by combining a copy of the Cambrian fan for $B$ with an antipodal copy of the Cambrian fan for $-B$.}, number={2}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Reading, Nathan and Speyer, David E.}, year={2017}, month={Sep}, pages={1429–1468} }
@article{reading_speyer_2016, title={Combinatorial Frameworks for Cluster Algebras}, volume={2016}, ISSN={["1687-0247"]}, url={https://doi.org/10.1093/imrn/rnv101}, DOI={10.1093/imrn/rnv101}, abstractNote={Journal Article Combinatorial Frameworks for Cluster Algebras Get access Nathan Reading, Nathan Reading 1Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA Correspondence to be sent to: reading@math.ncsu.edu Search for other works by this author on: Oxford Academic Google Scholar David E. Speyer David E. Speyer 2Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2016, Issue 1, 2016, Pages 109–173, https://doi.org/10.1093/imrn/rnv101 Published: 11 May 2015 Article history Received: 16 April 2014 Revision received: 10 March 2015 Accepted: 11 March 2015 Published: 11 May 2015}, number={1}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Reading, Nathan and Speyer, David E.}, year={2016}, pages={109–173} }
@article{ehrenborg_klivans_reading_2016, title={Coxeter arrangements in three dimensions}, volume={57}, ISSN={0138-4821 2191-0383}, url={http://dx.doi.org/10.1007/s13366-016-0286-6}, DOI={10.1007/s13366-016-0286-6}, abstractNote={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.}, number={4}, journal={Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry}, publisher={Springer Science and Business Media LLC}, author={Ehrenborg, Richard and Klivans, Caroline and Reading, Nathan}, year={2016}, month={Feb}, pages={891–897} }
@inbook{reading_2016, title={Finite Coxeter Groups and the Weak Order}, volume={2}, ISBN={9783319442358 9783319442365}, url={http://dx.doi.org/10.1007/978-3-319-44236-5_10}, DOI={10.1007/978-3-319-44236-5_10}, booktitle={Lattice Theory: Special Topics and Applications}, publisher={Springer International Publishing}, author={Reading, N.}, year={2016}, pages={489–561} }
@inbook{reading_2016, title={Lattice Theory of the Poset of Regions}, volume={2}, ISBN={9783319442358 9783319442365}, url={http://dx.doi.org/10.1007/978-3-319-44236-5_9}, DOI={10.1007/978-3-319-44236-5_9}, booktitle={Lattice Theory: Special Topics and Applications}, publisher={Springer International Publishing}, author={Reading, N.}, year={2016}, pages={399–487} }
@article{reading_speyer_2015, title={A Cambrian framework for the oriented cycle}, volume={22}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84951847857&partnerID=MN8TOARS}, number={4}, journal={Electron. J. Combin.}, author={Reading, Nathan and Speyer, David E.}, year={2015}, pages={Paper 4.46, 21} }
@article{barnard_reading_2015, title={Coxeter-biCatalan combinatorics}, url={https://doi.org/10.46298/dmtcs.2519}, DOI={10.46298/dmtcs.2519}, abstractNote={We consider several counting problems related to Coxeter-Catalan combinatorics and conjecture that the problems all have the same answer, which we call the $W$ -biCatalan number. We prove the conjecture in many cases. Nous considérons des problèmes énumératifs liés à la combinatoire de Coxeter-Catalan et conjecturons que tous les problèmes ont la même solution, que nous appelons le nombre $W$ -biCatalan. Nous prouvons la conjecture dans de nombreux cas.}, journal={Discrete Mathematics & Theoretical Computer Science}, author={Barnard, Emily and Reading, Nathan}, year={2015}, month={Jan} }
@article{reading_2015, title={NONCROSSING ARC DIAGRAMS AND CANONICAL JOIN REPRESENTATIONS}, volume={29}, ISSN={["1095-7146"]}, url={https://doi.org/10.1137/140972391}, DOI={10.1137/140972391}, abstractNote={We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each permutation $x$ has a canonical join representation: a unique lowest set of permutations joining to $x$. The second problem is to determine which sets of permutations appear as canonical join representations. The two problems turn out to be closely related because the noncrossing arc diagrams provide a combinatorial model for canonical join representations. The same considerations apply more generally to lattice quotients of the weak order. Considering quotients produces, for example, a new combinatorial object counted by the Baxter numbers and an analogous new object in bijection with generic rectangulations.}, number={2}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Reading, Nathan}, year={2015}, pages={736–750} }
@article{reading_2015, title={Universal geometric coefficients for the once-punctured torus}, volume={B71e}, url={http://math.univ-lyon1.fr/~slc/wpapers/s71reading.html}, journal={Séminaire Lotharingien de Combinatoire}, author={Reading, Nathan}, year={2015}, pages={Art. B71e, 29} }
@article{reading_2014, title={UNIVERSAL GEOMETRIC CLUSTER ALGEBRAS FROM SURFACES}, volume={366}, ISSN={["1088-6850"]}, url={https://doi.org/10.1090/S0002-9947-2014-06156-4}, DOI={10.1090/s0002-9947-2014-06156-4}, abstractNote={A universal geometric cluster algebra over an exchange matrix B B is a universal object in the category of geometric cluster algebras over B B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan F B \mathcal {F}_B called the mutation fan for B B . In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of F B \mathcal {F}_B for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces.}, number={12}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Reading, Nathan}, year={2014}, month={Dec}, pages={6647–6685} }
@article{reading_2014, title={Universal geometric cluster algebras}, volume={277}, ISSN={["1432-1823"]}, url={https://doi.org/10.1007/s00209-013-1264-4}, DOI={10.1007/s00209-013-1264-4}, abstractNote={We consider, for each exchange matrix $$B$$ , a category of geometric cluster algebras over $$B$$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $$R$$ , usually $$\mathbb {Z},\,\mathbb {Q}$$ , or $$\mathbb {R}$$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $$B$$ with universal geometric coefficients, or the universal geometric cluster algebra over $$B$$ . Constructing universal geometric coefficients is equivalent to finding an $$R$$ -basis for $$B$$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan $${\mathcal {F}}_B$$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between $${\mathcal {F}}_B$$ and $$\mathbf{g}$$ -vectors. We construct universal geometric coefficients in rank $$2$$ and in finite type and discuss the construction in affine type.}, number={1-2}, journal={MATHEMATISCHE ZEITSCHRIFT}, author={Reading, Nathan}, year={2014}, month={Jun}, pages={499–547} }
@article{reading_2012, title={Coarsening polyhedral complexes}, volume={140}, url={https://doi.org/10.1090/S0002-9939-2012-11194-3}, DOI={10.1090/s0002-9939-2012-11194-3}, abstractNote={Given a pure, full-dimensional, locally strongly connected polyhedral complex $\mathcal {C}$, we characterize, by a local codimension-$2$ condition, polyhedral complexes that coarsen $\mathcal {C}$. The proof of the characterization draws upon a general shortcut for showing that a collection of polyhedra is a polyhedral complex and upon a property of hyperplane arrangements which is equivalent, for Coxeter arrangements, to Titsâ solution to the Word Problem. The motivating special case, the case where $\mathcal {C}$ is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietzeâs convexity theorem.}, number={10}, journal={Proc. Amer. Math. Soc.}, author={Reading, Nathan}, year={2012}, pages={3593–3605} }
@inbook{reading_2012, place={New York}, series={Progress in Mathematics}, title={From the Tamari lattice to Cambrian lattices and beyond}, volume={299}, ISBN={9783034804042 9783034804059}, url={https://doi.org/10.1007/978-3-0348-0405-9_15}, DOI={10.1007/978-3-0348-0405-9_15}, abstractNote={We trace the path from the Tamari lattice, via lattice congruences of the weak order, to the definition of Cambrian lattices in the context of finite Coxeter groups, and onward to the construction of Cambrian fans. We then present sortable elements, the key combinatorial tool for studying Cambrian lattices and fans. The chapter concludes with a brief description of the applications of Cambrian lattices and sortable elements to Coxeter-Catalan combinatorics and to cluster algebras.}, booktitle={Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift}, publisher={Birkhauser}, author={Reading, N.}, editor={Mueller-Hoissen, F. and Pallo, J. and Stasheff, J.Editors}, year={2012}, pages={293–322}, collection={Progress in Mathematics} }
@article{reading_2012, title={Generic rectangulations}, volume={33}, ISSN={0195-6698}, url={http://dx.doi.org/10.1016/j.ejc.2011.11.004}, DOI={10.1016/j.ejc.2011.11.004}, abstractNote={A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analogous to the definition of the twisted Baxter permutations.}, number={4}, journal={European Journal of Combinatorics}, publisher={Elsevier BV}, author={Reading, Nathan}, year={2012}, month={May}, pages={610–623} }
@article{law_reading_2012, title={The Hopf algebra of diagonal rectangulations}, volume={119}, ISSN={["1096-0899"]}, url={https://doi.org/10.1016/j.jcta.2011.09.006}, DOI={10.1016/j.jcta.2011.09.006}, abstractNote={We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a Hopf subalgebra of the Malvenuto–Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.}, number={3}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Law, Shirley and Reading, Nathan}, year={2012}, month={Apr}, pages={788–824} }
@article{reading_2011, title={Noncrossing partitions and the shard intersection order}, volume={33}, ISSN={["1572-9192"]}, url={https://doi.org/10.1007/s10801-010-0255-3}, DOI={10.1007/s10801-010-0255-3}, abstractNote={We define a new lattice structure $(W,\preceq)$ on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC (W) as a sublattice. The new construction of NC (W) yields a new proof that NC (W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of $(W,\preceq)$ , like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC (W). There is a natural dimension-preserving bijection between simplices in the order complex of $(W,\preceq)$ (i.e. chains in $(W,\preceq)$ ) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC (W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The lattice $(W,\preceq)$ is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.}, number={4}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, publisher={Assoc. Discrete Math. Theor. Comput. Sci., Nancy}, author={Reading, Nathan}, year={2011}, month={Jun}, pages={483–530} }
@article{reading_spoyer_2011, title={SORTABLE ELEMENTS IN INFINITE COXETER GROUPS}, volume={363}, ISSN={["0002-9947"]}, url={https://doi.org/10.1090/S0002-9947-2010-05050-0}, DOI={10.1090/s0002-9947-2010-05050-0}, abstractNote={In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.}, number={2}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Reading, Nathan and Spoyer, David E.}, year={2011}, month={Feb}, pages={699–761} }
@article{reading_2010, title={Noncrossing partitions, clusters and the Coxeter plane}, volume={63}, ISSN={1286-4889}, url={https://www.mat.univie.ac.at/~slc/wpapers/s63reading.html}, journal={Séminaire Lotharingien de Combinatoire}, author={Reading, N.}, year={2010}, pages={B63b} }
@article{reading_speyer_2010, title={Sortable elements for quivers with cycles}, volume={17}, url={http://www.combinatorics.org/Volume_17/Abstracts/v17i1r90.html}, DOI={10.37236/362}, abstractNote={Each Coxeter element $c$ of a Coxeter group $W$ defines a subset of $W$ called the $c$-sortable elements. The choice of a Coxeter element of $W$ is equivalent to the choice of an acyclic orientation of the Coxeter diagram of $W$. In this paper, we define a more general notion of $\Omega$-sortable elements, where $\Omega$ is an arbitrary orientation of the diagram, and show that the key properties of $c$-sortable elements carry over to the $\Omega$-sortable elements. The proofs of these properties rely on reduction to the acyclic case, but the reductions are nontrivial; in particular, the proofs rely on a subtle combinatorial property of the weak order, as it relates to orientations of the Coxeter diagram. The $c$-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed; the ultimate motivation behind this paper is to extend this connection beyond the acyclic case.}, number={1}, journal={The Electronic Journal of Combinatorics}, author={Reading, Nathan and Speyer, David}, year={2010}, pages={R90} }
@article{reading_speyert_2010, title={Sortable elements for quivers with cycles}, volume={17}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77955640004&partnerID=MN8TOARS}, number={1}, journal={Electronic Journal of Combinatorics}, author={Reading, N. and Speyert, D.E.}, year={2010}, pages={1–19} }
@article{reading_speyer_2009, title={Cambrian fans}, volume={11}, url={https://doi.org/10.4171/JEMS/155}, DOI={10.4171/jems/155}, abstractNote={For a finite Coxeter group\~$W$ and a Coxeter element\~$c$ of $W,$ the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of\~$W!$. Its maximal cones are naturally indexed by the $c$-sortable elements of\~$W!$. The main result of this paper is that the known bijection $\cl\_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for\~$W!$. The rays of the $c$-Cambrian fan are generated by certain vectors in the $W$-orbit of the fundamental weights, while the rays of the $c$-cluster fan are generated by certain roots. For particular (\`\`bipartite'') choices of\~$c$, we show that the $c$-Cambrian fan is linearly isomorphic to the $c$-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map $\cl\_c$, on $c$-clusters by the $c$-Cambrian lattice. We give a simple bijection from $c$-clusters to $c$-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric context for $\mathbf{g}$-vectors and quasi-Cartan companions.}, number={2}, journal={J. Eur. Math. Soc. (JEMS)}, author={Reading, Nathan and Speyer, David E.}, year={2009}, pages={407–447} }
@article{reading_2009, title={Noncrossing partitions and the shard intersection order}, url={https://doi.org/10.46298/dmtcs.2709}, DOI={10.46298/dmtcs.2709}, journal={Discrete Mathematics & Theoretical Computer Science}, author={Reading, Nathan}, year={2009}, month={Jan} }
@inproceedings{reading_2009, title={Noncrossing partitions and the shard intersection order}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-79954628162&partnerID=MN8TOARS}, booktitle={FPSAC'09 - 21st International Conference on Formal Power Series and Algebraic Combinatorics}, author={Reading, N.}, year={2009}, pages={745–756} }
@article{reading_2008, title={Chains in the noncrossing partition lattice}, volume={22}, ISSN={["1095-7146"]}, url={https://doi.org/10.1137/07069777X}, DOI={10.1137/07069777X}, abstractNote={We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e., without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice.}, number={3}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Reading, Nathan}, year={2008}, pages={875–886} }
@article{reading_2007, title={Clusters, coxeter-sortable elements and noncrossing partitions}, volume={359}, ISSN={["0002-9947"]}, url={https://doi.org/10.1090/S0002-9947-07-04319-X}, DOI={10.1090/s0002-9947-07-04319-x}, abstractNote={We introduce Coxeter-sortable elements of a Coxeter group W . W. For finite W , W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical cases, in terms of permutations.}, number={12}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Reading, Nathan}, year={2007}, pages={5931–5958} }
@inproceedings{reading_2007, title={Clusters, noncrossing partitions and the Coxeter plane}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84860725128&partnerID=MN8TOARS}, booktitle={FPSAC'07 - 19th International Conference on Formal Power Series and Algebraic Combinatorics}, author={Reading, N.}, year={2007} }
@inbook{fomin_reading_2007, title={Root systems and generalized
associahedra}, volume={13}, ISBN={9780821837368 9781470439125}, ISSN={1079-5634 2472-5064}, url={http://dx.doi.org/10.1090/pcms/013/03}, DOI={10.1090/pcms/013/03}, booktitle={Geometric Combinatorics}, publisher={American Mathematical
Society}, author={Fomin, Sergey and Reading, Nathan}, year={2007}, month={Oct}, pages={63–131} }
@article{reading_2007, title={Sortable elements and Cambrian lattices}, volume={56}, ISSN={["0002-5240"]}, url={https://doi.org/10.1007/s00012-007-2009-1}, DOI={10.1007/s00012-007-2009-1}, abstractNote={We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the Cambrian lattice is the weak order on Coxeter-sortable elements. These results exhibit W-Catalan combinatorics arising in the context of the lattice theory of the weak order on W.}, number={3-4}, journal={ALGEBRA UNIVERSALIS}, author={Reading, Nathan}, year={2007}, month={Jun}, pages={411–437} }
@article{reading_2006, title={Cambrian lattices}, volume={205}, ISSN={0001-8708}, url={http://dx.doi.org/10.1016/j.aim.2005.07.010}, DOI={10.1016/j.aim.2005.07.010}, abstractNote={For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a “cluster fan.” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.}, number={2}, journal={Advances in Mathematics}, publisher={Elsevier BV}, author={Reading, Nathan}, year={2006}, month={Oct}, pages={313–353} }
@inproceedings{reading_2006, title={Clusters, Coxeter-sortable elements and noncrossing partitions}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84860648750&partnerID=MN8TOARS}, booktitle={FPSAC 2006 - Proceedings: 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics}, author={Reading, N.}, year={2006}, pages={275–281} }
@article{fomin_reading_2005, title={Generalized cluster complexes and Coxeter combinatorics}, volume={2005}, url={https://doi.org/10.1155/IMRN.2005.2709}, DOI={10.1155/IMRN.2005.2709}, abstractNote={We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m = 1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C. A. Athanasiadis, suggesting links to a host of well-studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram.}, number={44}, journal={International Mathematics Research Notices}, author={Fomin, Sergey and Reading, Nathan}, year={2005}, pages={2709–2757} }
@article{reading_2005, title={Lattice congruences, fans and Hopf algebras}, volume={110}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2004.11.001}, DOI={10.1016/j.jcta.2004.11.001}, abstractNote={We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto–Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto–Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.}, number={2}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Reading, Nathan}, year={2005}, month={May}, pages={237–273} }
@article{reading_waugh_2005, title={The order dimension of Bruhat order on infinite Coxeter groups}, volume={11}, url={http://www.combinatorics.org/Volume_11/Abstracts/v11i2r13.html}, number={2}, journal={The Electronic Journal of Combinatorics}, author={Reading, Nathan and Waugh, Debra J.}, year={2005}, pages={Research Paper 13, 26} }
@article{reading_2004, title={Lattice Congruences of the Weak Order}, volume={21}, ISSN={0167-8094 1572-9273}, url={http://dx.doi.org/10.1007/s11083-005-4803-8}, DOI={10.1007/s11083-005-4803-8}, abstractNote={We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset $K\subseteq S$ , let ηK: w↦wK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1≡s for every s∈(S−K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.}, number={4}, journal={Order}, publisher={Springer Science and Business Media LLC}, author={Reading, Nathan}, year={2004}, month={Nov}, pages={315–344} }
@article{reading_2004, title={Non-negative cd-coefficients of Gorenstein∗ posets}, volume={274}, ISSN={0012-365X}, url={http://dx.doi.org/10.1016/j.disc.2003.07.001}, DOI={10.1016/j.disc.2003.07.001}, abstractNote={We give a convolution formula for cd-index coefficients. The convolution formula, together with the proof by Davis and Okun of the Charney–Davis Conjecture in dimension 3, imply that certain cd-coefficients are non-negative for all Gorenstein∗ posets. Additional coefficients are shown to be non-negative by interpreting them in terms of the top homology of certain Cohen–Macaulay complexes. In particular we verify, up to rank 6, Stanley's conjecture that the coefficients in the cd-index of a Gorenstein∗ ranked poset are non-negative.}, number={1-3}, journal={Discrete Mathematics}, publisher={Elsevier BV}, author={Reading, Nathan}, year={2004}, month={Jan}, pages={323–329} }
@article{reading_2004, title={The cd-index of Bruhat intervals}, volume={11}, url={http://www.combinatorics.org/Volume_11/Abstracts/v11i1r74.html}, DOI={10.37236/1827}, abstractNote={We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of $x$ and $y$, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals $[x,y]$. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i. e. simplices).}, number={1}, journal={The Electronic Journal of Combinatorics}, author={Reading, Nathan}, year={2004}, pages={R74} }
@article{reading_2004, title={The cd-index of Bruhat intervals}, volume={11}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-8744221555&partnerID=MN8TOARS}, number={1 R}, journal={Electronic Journal of Combinatorics}, author={Reading, N.}, year={2004}, pages={1–25} }
@article{reading_2003, title={Lattice and order properties of the poset of regions in a hyperplane arrangement}, volume={50}, ISSN={0002-5240 1420-8911}, url={http://dx.doi.org/10.1007/s00012-003-1834-0}, DOI={10.1007/s00012-003-1834-0}, abstractNote={We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k - 1. An explicit description of the doublings leads to a proof that the order dimension of the poset of regions (again with respect to a canonical base region) of a supersolvable hyperplane arrangement is equal to the rank of the arrangement. In particular, the order dimension of the weak order on a finite Coxeter group of type A or B is equal to the number of generators. The result for type A (the permutation lattice) was proven previously by Flath [11]. We show that the poset of regions of a simplicial arrangement is a semi-distributive lattice, using the previously known result [2] that it is a lattice. A lattice is congruence uniform (or “bounded” in the sense of McKenzie [18]) if and only if it is semi-distributive and congruence normal [7]. Caspard, Le Conte de Poly-Barbut and Morvan [4] showed that the weak order on a finite Coxeter group is congruence uniform. Inspired by the methods of [4], we characterize congruence normality of a lattice in terms of edge-labelings. This leads to a simple criterion to determine whether or not a given simplicial arrangement has a congruence uniform lattice of regions. In the case when the criterion is satisfied, we explicitly characterize the congruence lattice of the lattice of regions.}, number={2}, journal={Algebra Universalis}, publisher={Springer Science and Business Media LLC}, author={Reading, Nathan}, year={2003}, month={Dec}, pages={179–205} }
@article{reading_2003, title={The order dimension of the poset of regions in a hyperplane arrangement}, volume={104}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2003.08.002}, DOI={10.1016/j.jcta.2003.08.002}, abstractNote={We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified approach which, in particular, leads to a simpler treatment of the previously known cases, types A and B. The result for weak orders follows from an upper bound on the dimension of the poset of regions of an arbitrary hyperplane arrangement. In some cases, including the weak orders, the upper bound is the chromatic number of a certain graph. For the weak orders, this graph has the positive roots as its vertex set, and the edges are related to the pairwise inner products of the roots.}, number={2}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Reading, Nathan}, year={2003}, month={Nov}, pages={265–285} }
@phdthesis{reading_2002, title={On the structure of Bruhat order}, url={http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3041950}, note={Thesis (Ph.D.)–University of Minnesota}, school={ProQuest LLC, Ann Arbor, MI}, author={Reading, Nathan Paul}, year={2002}, month={Apr}, pages={82} }
@article{reading_2002, title={Order Dimension, Strong Bruhat Order and Lattice Properties for Posets}, volume={19}, url={https://doi.org/10.1023/A:1015287106470}, DOI={10.1023/A:1015287106470}, number={1}, journal={Order}, author={Reading, Nathan}, year={2002}, pages={73–100} }
@article{reading_1999, title={Nim-Regularity of Graphs}, volume={6}, url={http://www.combinatorics.org/Volume_6/Abstracts/v6i1r11.html}, number={1}, journal={The Electronic Journal of Combinatorics}, author={Reading, Nathan}, year={1999}, pages={Research Paper 11, 8} }
@article{gibbon_kennedy_reading_quieroz_1992, title={The thermodynamics of home-made ice cream}, volume={69}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-27744592214&partnerID=MN8TOARS}, number={8}, journal={Journal of Chemical Education}, author={Gibbon, D.L. and Kennedy, K. and Reading, N. and Quieroz, M.}, year={1992}, pages={658–661} }