arXiv subjects

Jon McCammond

Publications and source records attributed to Jon McCammond.

At least 19 recordsLinked to original sources

arXiv2026

Extended Weak Order for the Rank 3 Universal Coxeter Group

The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by problems in Kazhdan--Lusztig theory, Matthew Dyer introduced the extended weak order, a poset that contains a copy of the weak order as an order ideal, and he conjectured that the extended weak order for any Coxeter group is a lattice. We prove Dyer's conjecture for the rank $3$ universal Coxeter group. This is the first non-spherical, non-affine Coxeter group for which Dyer's conjecture has been proven.

math.CO↗

arXiv2026

The Geometry of Rectangular Multisets

This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra.

math.CO↗

arXiv2025

Continuous Noncrossing Partitions and Weighted Circular Factorizations

This article examines noncrossing partitions of the unit circle in the complex plane; we call these continuous noncrossing partitions. More precisely, we focus on the degree-$d$ continuous noncrossing partitions where unit complex numbers in the same block have identical $d$-th powers. We prove that the degree-$d$ continuous noncrossing partitions form a topological poset whose uncountable set of elements can be indexed by equivalence classes of objects we call weighted linear factorizations of factors of a $d$-cycle. Moreover, the maximal elements in this poset form a subspace homeomorphic to the dual Garside classifying space for the $d$-strand braid group. The degree-$d$ continuous noncrossing partitions of the unit circle are a special case of a more general construction. For every choice of Coxeter element $c$ in any Coxeter group $W$ we define a topological poset of equivalence classes of weighted linear factorizations of factors of $c$ in $W$ whose elements we call continuous $c$-noncrossing partitions. The maximal elements in this poset form a subspace homeomorphic to the one-vertex complex whose fundamental group is the corresponding dual Artin group.

math.GR↗

arXiv2024

Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched rectangle complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed rectangle. The vertices of the cell structure are indexed by the combinatorial "basketballs" studied by Martin, Savitt and Singer. Structurally, the branched rectangle complex is a full subcomplex of a direct product of two copies of the order complex of the noncrossing partition lattice. Topologically, it is homeomorphic to the closed $2n$-dimensional ball where $n=d-1$. Metrically, the simplices in each factor are orthoschemes. It can also be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$. We also introduce a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed annular region. We call this the branched annulus complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed annulus.It can be constructed from the branched rectangle complex as a cellular quotient by isometric face identifications. And it can be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$ with distinct roots. Finally, the branched annulus complex deformation retracts to the branched circle complex, which we identify with the dual braid complex. Our explicit embedding of the dual braid complex as a spine for the space of polynomials with distinct roots provides a direct proof that these two classifying spaces for the braid group are homotopy equivalent.

math.GT↗

arXiv2023

Fixed Points of Parking Functions

We define an action of words in $[m]^n$ on $\mathbb{R}^m$ to give a new characterization of rational parking functions -- they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani's zeta map on rational parking functions when m and n are coprime, and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington's sweep map on rational Dyck paths.

math.CO↗

arXiv2021

Geometric Combinatorics of Polynomials I: The Case of a Single Polynomial

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial $p$ with $d$ distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally CAT(0) metric that is locally Euclidean away from a finite set of interior points indexed by the critical points of $p$, and each of its 2-cells is a metric rectangle. From this planar rectangular 2-complex one can use metric graphs known as metric cacti and metric banyans to read off several pieces of combinatorial data: a chain in the partition lattice, a cyclic factorization of a d-cycle, a real noncrossing partition (also known as a primitive d-major), and the monodromy permutations for the polynomial. This article is the first in a series.

math.GT↗

arXiv2021

Factoring isometries of quadratic spaces into reflections

Let $V$ be a vector space endowed with a non-degenerate quadratic form $Q$. If the base field $\mathbb{F}$ is different from $\mathbb{F}_2$, it is known that every isometry can be written as a product of reflections. In this article, we detail the structure of the poset of all minimal length reflection factorizations of an isometry. If $\mathbb{F}$ is an ordered field, we also study factorizations into positive reflections, i.e., reflections defined by vectors of positive norm. We characterize such factorizations, under the hypothesis that the squares of $\mathbb{F}$ are dense in the positive elements (this includes Archimedean and Euclidean fields). In particular, we show that an isometry is a product of positive reflections if and only if its spinor norm is positive. As a final application, we explicitly describe the poset of all factorizations of isometries of the hyperbolic space.

math.GR↗

arXiv2019

Critical Points, Critical Values, and a Determinant Identity for Complex Polynomials

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $θ\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $θ$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $θ$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $θ$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture.

math.CV↗

arXiv2019

Connectivity at Infinity for the Braid Group of a Complete Bipartite Graph

The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the description of a hidden symmetry among the parameters, and known results from the literature to explicitly compute the exact degree to which these complexes and groups are connected at infinity.

math.AT↗

arXiv2018

Boundary Braids

The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points remain pointwise fixed forms a subgroup isomorphic to a braid group with fewer strands. In this article, we generalize this phenomenon by introducing the notion of boundary braids. A boundary braid is a braid that has a representative where some specified subset of the points remains in the boundary cycle of the disk. Although boundary braids merely form a subgroupoid rather than a subgroup, they play an interesting geometric role in the piecewise Euclidean dual braid complex defined by Tom Brady and the second author. We prove several theorems in this setting, including the fact that the subcomplex of the dual braid complex determined by a specified set of boundary braids metrically splits as the direct metric product of a Euclidean polyhedron and a dual braid complex of smaller rank.

math.GR↗

arXiv2017

Computing reflection length in an affine Coxeter group

In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple formula that computes the reflection length of any element in any affine Coxeter group and we provide a simple uniform proof.

math.CO↗

arXiv2017

Artin groups of euclidean type

This article resolves several long-standing conjectures about Artin groups of euclidean type. In particular, we prove that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. We do this by showing that each of these groups is isomorphic to a subgroup of a group with an infinite-type Garside structure. The Garside groups involved are introduced here for the first time. They are constructed by applying semi-standard procedures to crystallographic groups that contain euclidean Coxeter groups but which need not be generated by the reflections they contain.

math.GR↗

arXiv2017

Undesired parking spaces and contractible pieces of the noncrossing partition link

There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.

math.CO↗

arXiv2017

A complex euclidean reflection group with an elegant complement complex

The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced depends on an initial choice of coordinates (Björner-Ziegler). In this article we consider the unique complex euclidean reflection group acting cocompactly by isometries on $\mathbb{C}^2$ whose linear part is the finite complex reflection group known as $G_4$ in the Shephard-Todd classification and we construct a choice-free deformation retraction from its hyperplane complement onto an elegant $2$-dimensional complex $K$ where every $2$-cell is a euclidean equilateral triangle and every vertex link is a Möbius-Kantor graph. Since $K$ is non-positively curved, the corresponding braid group is a CAT(0) group, despite the fact that there are non-regular points in the hyperplane complement, the action of the reflection group on $K$ is not free, and the braid group is not torsion-free.

math.GR↗

arXiv2017

Noncrossing hypertrees

Hypertrees and noncrossing trees are well-established objects in the combinatorics literature, but the hybrid notion of a noncrossing hypertree has received less attention. In this article I investigate the poset of noncrossing hypertrees as an induced subposet of the hypertree poset. Its dual is the face poset of a simplicial complex, one that can be identified with a generalized cluster complex of type $A$. The first main result is that this noncrossing hypertree complex is homeomorphic to a piecewise spherical complex associated with the noncrossing partition lattice and thus it has a natural metric. The fact that the order complex of the noncrossing partition lattice with its bounding elements removed is homeomorphic to a generalized cluster complex was not previously known or conjectured. The metric noncrossing hypertree complex is a union of unit spheres with a number of remarkable properties: 1) the metric subspheres and simplices in each dimension are both bijectively labeled by the set of noncrossing hypertrees with a fixed number of hyperedges, 2) the number of spheres containing the simplex labeled by the noncrossing tree $τ$ is the same as the number simplices in the sphere labeled by the noncrossing tree $τ$, and 3) among the maximal spherical subcomplexes one finds every normal fan of a metric realization of the simple associahedron associated to the cluster algebra of type $A$. In particular, the poset of noncrossing hypertrees and its metric simplicial complex provide a new perspective on familiar combinatorial objects and a common context in which to view the known bijections between noncrossing partitions and the vertices/facets of simple/simplicial associahedra.

math.CO↗

arXiv2014

Curvature and Computation

When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection between negative curvature and efficient computation. This theorem was one of the clearest early indications that applying a metric perspective to traditional group theory problems might lead to new and important insights.

math.GR↗

arXiv2014

The exceptional symmetry

This note gives an elementary proof that the symmetric groups possess only one exceptional symmetry. I am referring to the fact that the outer automorphism group of the symmetric group $S_n$ is trivial unless $n=6$ and the outer automorphism group of $S_6$ has a unique nontrivial element.

math.GR↗

arXiv2014

Dual euclidean Artin groups and the failure of the lattice property

The irreducible euclidean Coxeter groups that naturally act geometrically on euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three examples with three generators more than twenty years ago and François Digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these groups and showing that it forms an infinite-type Garside structure. In this article I establish that none of the remaining dual presentations for irreducible euclidean Artin groups corrspond to Garside structures because their factorization posets fail to be lattices. These are the first known examples of dual Artin presentations that fail to form Garside structures. Moreover, the results presented here about the cause of this failure form the foundation for a subsequent article in which the structure of all euclidean Artin groups is clarified.

math.GR↗