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David E. Speyer

Publications and source records attributed to David E. Speyer.

13 recordsLinked to original sources
arXiv2026

A Cambrian framework for the oriented cycle

This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented n-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B. In this paper, we extend this "doubled Cambrian fan" construction to the oriented n-cycle, using a more general notion of sortable elements for quivers with cycles.

math.CO↗
arXiv2021

Cohomology of cluster varieties. II. Acyclic case

In previous work, we initiated the study of the cohomology of locally acyclic cluster varieties. In the present work, we show that the mixed Hodge structure and point counts of acyclic cluster varieties are essentially determined by the combinatorics of the independent sets of the quiver. We use this to show that the mixed Hodge numbers of acyclic cluster varieties of really full rank satisfy a strong vanishing condition.

math.AG↗
arXiv2021

Cohomology of cluster varieties. I. Locally acyclic case

We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For cluster varieties satisfying the Louise property and of full rank, we show that the cohomology satisfies the curious Lefschetz property of Hausel and Rodriguez-Villegas, and that the mixed Hodge structure is split over the rationals. We give a complete description of the highest weight part of the mixed Hodge structure of these cluster varieties, and develop the notion of a standard differential form on a cluster variety. We show that the point counts of these cluster varieties over finite fields can be expressed in terms of Dirichlet characters. Under an additional integrality hypothesis, the point counts are shown to be polynomials in the order of the finite field.

math.AG↗
arXiv2018

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with alternating signs into the Schur basis.

math.RT↗
arXiv2018

The Growth Rate of Tri-Colored Sum-Free Sets

Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group of order $q$. Let $θ= \min_{ρ>0} (1+ρ+\cdots + ρ^{q-1}) ρ^{-(q-1)/3}$. Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in $C_q^n$ has size at most $3 θ^n$. Between this paper and a paper of Pebody, we will show that, for any $δ> 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(θ-δ)^n$. Our construction also works when $q$ is not prime.

math.CO↗
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.

math.CO↗
arXiv2015

Cluster Algebras of Grassmannians are Locally Acyclic

Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and most fundamental examples of a cluster algebra is the homogenous coordinate ring of the Grassmannian. We show that the Grassmannian is locally acyclic. Morally, we are in fact showing the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have cluster structure, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. Moreover, we actually establish a slightly stronger property than local acyclicity, which we term the Louise property, that is designed to facilitate proofs involving the Mayer-Vietores sequence.

math.CO↗
arXiv2015

Computing Hermitian determinantal representations of hyperbolic curves

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation.

math.AG↗
arXiv2014

Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM

Scattering amplitudes in 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d $\mathcal{N}=6$ ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for $\mathcal{N}=4$ SYM.

hep-th↗
arXiv2009

Looping of the numbers game and the alcoved hypercube

We study in detail the so-called looping case of Mozes's game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. The strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These are selfdual and mutually isomorphic, and dual to the triangulation of the unit hypercube by reflecting hyperplanes, studied by many authors. Unlike the weak and Bruhat orders, the top degree is cubic in the number of vertices of the graph. We explicitly compute the Hilbert polynomial of the poset.

math.RT↗
arXiv2009

The Multidimensional Cube Recurrence

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon -- any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n-1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities.

math.CO↗
arXiv2006

A Broken Circuit Ring

Given a matroid M represented by a linear subspace L in n-space (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken circuit complex of M. We give a geometric interpretation of Spec R(L), as well as a stratification indexed by the flats of M.

math.CO↗