arXiv subjects

David E Speyer

Publications and source records attributed to David E Speyer.

At least 37 records · Page 2Linked to original sources

arXiv2017

A Gröbner basis for the graph of the reciprocal plane

Given the complement of a hyperplane arrangement, let $Γ$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $Γ$ in two different-seeming ways, one due to Orlik and Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.

math.AC↗

arXiv2016

Frobenius split subvarieties pull back in almost all characteristics

Let $X$ and $Y$ be schemes of finite type over $\mathrm{Spec}\ \mathbb{Z}$ and let $α: Y \to X$ be a finite map. We show the following holds for all sufficiently large primes $p$: If $ϕ$ and $ψ$ are any splittings on $X \times \mathrm{Spec}\ F_p$ and $Y \times \mathrm{Spec}\ F_p$, such that the restriction of $α$ is compatible with $ϕ$ and $ψ$, and $V$ is any compatibly split subvariety of $(X \times \mathrm{Spec}\ F_p, ϕ)$, then the reduction $α^{-1}(V)^{\mathrm{red}}$ is a compatibly split subvariety of $(Y \times \mathrm{Spec}\ F_p, ψ)$. This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.

math.AC↗

arXiv2016

Some sums over irreducible polynomials

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_P h(P)$ where the sum is over monic irreducible polynomials $P$ in $\mathbb{F}_q[T]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.

math.NT↗

arXiv2016

The twist for positroid varieties

The purpose of this note is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plücker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plücker coordinates.

math.CO↗

arXiv2014

Links in the complex of weakly separated collections

Plabic graphs are interesting combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by $k$-element sets of positive integers, and a collection of such $k$-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show that if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common.

math.CO↗

arXiv2013

An Infinitely Generated Upper Cluster Algebra

We show that upper cluster algebras need not be finitely generated, answering a question of Berenstein, Fomin and Zelevinsky. Our counter-example is a cluster algebra with B-matrix $\begin{pmatrix} 0 & 3 & -3 \\ -3 & 0 & 3 \\ 3 & -3 & 0 \end{pmatrix}$ and coefficients obeying a genericity condition.

math.RT↗

arXiv2012

Schubert problems with respect to osculating flags of stable rational curves

Given a point z in P^1, let F(z) be the osculating flag to the rational normal curve at point z. The study of Schubert problems with respect to such flags F(z_1), F(z_2), ..., F(z_r) has been studied both classically and recently, especially when the points z_i are real. Since the rational normal curve has an action of PGL_2, it is natural to consider the points (z_1, ..., z_r) as living in the moduli space of r distinct point in P^1 -- the famous M_{0,r}. One can then ask to extend the results on Schubert intersections to the compactification \bar{M}_{0,r}. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family over \bar{M}_{0,r}, whose fibers over M_{0,r} are isomorphic to G(d,n) and, given partitions lambda_1, ..., lambda_r, we construct a flat Cohen-Macualay family over \bar{M}_{0,r} whose fiber over (z_1, ..., z_r) in M_{0,r} is the intersection of the Schubert varieties indexed by lambda_i with respect to the osculating flags F(z_i). In the second part of the paper, we investigate the topology of the real points of our family, in the case that sum |lambda_i| = dim G(d,n). We show that our family is a finite covering space of \bar{M}_{0,r}, and give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux.

math.AG↗

arXiv2011

Weak Separation and Plabic Graphs

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. On the other hand, Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally nonnegative Grassmannian into positroid strata, and constructed their parametrization using plabic graphs. In this paper we link the study of weak separation to plabeic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures.

math.CO↗

arXiv2011

Projections of Richardson Varieties

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, Cohen-Macaulay, have rational singularities, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem [Schwede, Kumar-Mehta] that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by [Hague], whose proof we simplify somewhat.) One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an EL-shellable ball [Bjorner-Wachs '82]. We prove that the projection of such a complex into the order complex of the Bruhat order on W/W_P is again a shellable ball. This requires extensive analysis of "P-Bruhat order", a generalization of the k-Bruhat order of [Bergeron-Sottile '98]. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Grobner degeneration takes each projected Richardson variety to the Stanley-Reisner scheme of its corresponding ball.

math.AG↗

arXiv2010

K-classes of matroids and equivariant localization

To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.

math.CO↗

arXiv2010

Sortable elements in infinite Coxeter groups

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.

math.CO↗

arXiv2009

Positroid varieties I: juggling and geometry

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the {\em cyclic shifts} of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, and Brown-Goodearl-Yakimov. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call {\em bounded juggling patterns}. We adopt his terminology and call the strata {\em positroid varieties.} We show that positroid varieties are normal and Cohen-Macaulay, and are defined as schemes by the vanishing of Plucker coordinates. We compute their T-equivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the Hodge-Grobner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective Stanley-Reisner scheme of a shellable ball.

math.AG↗

arXiv2008

Cambrian fans

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 g-vectors and quasi-Cartan companions.

math.CO↗

arXiv2008

A non-crossing standard monomial theory

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.

math.CO↗

arXiv2007

Uniformizing Tropical Curves I: Genus Zero and One

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero case and by non-archimedean elliptic functions in the genus one case. For genus zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus one curves, show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.

math.AG↗

arXiv2007

Powers of Coxeter elements in infinite groups are reduced

Let W be an infinite irreducible Coxeter group with (s_1, ..., s_n) the simple generators. We give a simple proof that the word s_1 s_2 ... s_n s_1 s_2 >... s_n ... s_1 s_2 ... s_n is reduced for any number of repetitions of s_1 s_2 >... s_n. This result was proved for simply-laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof only using basic facts about Coxeter groups and the geometry of root systems.

math.CO↗

arXiv2007

A Kleiman-Bertini Theorem for sheaf tensor products

Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that, for elements g in a dense open subset of G, the sheaf Tor_i^X(E, g F) vanishes for all i > 0. When E and F are structure sheaves of smooth subschemes of X in characteristic zero, this follows from the Kleiman-Bertini theorem; our result has no smoothness hypotheses on the supports of E or F, or hypotheses on the characteristic of the ground field.

math.AG↗

arXiv2006

A matroid invariant via the K-theory of the Grassmannian

Let G(d,n) denote the Grassmannian of d-planes in C^n and let T be the torus (C^*)^n/diag(C^*) which acts on G(d,n). Let x be a point of G(d,n) and let \bar{Tx} be the closure of the T-orbit through x. Then the class of the structure sheaf of \bar{Tx} in the K-theory of G(d,n) depends only on which Plücker coordinates of x are nonzero -- combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from the K-theory of G(d,n) to Z[t]. Letting g_x(t) denote the image of (-1)^{n-dim Tx} [ O_{\bar{Tx}}], g_x behaves nicely under the standard constructions of matroid theory. Specifically, g_{x_1 \oplus x_2}(t)=g_{x_1}(t) g_{x_2}(t), g_{x_1 +_2 x_2}(t)=g_{x_1}(t) g_{x_2}(t)/t, g_x(t) = g_{x^{\perp}}(t) and g_x is unaltered by series and parallel extensions. Furthermore, the coefficients of g_x are nonnegative. The existence of this map implies bounds on (essentially equivalently) the complexity of Kapranov's Lie complexes, Hacking, Keel and Tevelev's very stable pairs and the author's tropical linear spaces when they are realizable in characteristic zero. Namely, in characteristic zero, a Lie complex or the underlying d-1 dimensional scheme of a very stable pair can have at most (n-i-1)! / (d-i)!(n-d-i)!(i-1)! strata of dimensions n-i and d-i respectively and a tropical linear space realizable in characteristic zero can have at most this many i-dimensional bounded faces.

math.AG↗