We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called balancing condition if and only if it is an indexed branched cover, i.e.~locally over connected sets the count with multiplicity of points in every fibre is a constant, which in particular gives a well-defined global degree when the target is connected. Given a balanced map $(Σ, m_Σ) \to Δ$, we lift several connectivity properties of $Δ$ to~$Σ$. Using these lifting results we determine whether a multiplicity $m_{\mathcal U}$ that is defined only on the interiors of maximal cells of $Σ$ can be extended to all $Σ$ in a balanced manner. This relies on a strong connectivity assumption; we give a counterexample when this is missing.
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of $\mathbb{R}^d$. This leads to natural cell decompositions of the tropical projective space $\mathbb{TP}_{\min}^{d-1}$. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information which we call a signed tropical matroid. We demonstrate the connections with the classical simplex method, mean payoff games and scheduling.
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
In recent work, the authors derived a tropical interpretation of monotone and strictly monotone double Hurwitz numbers. In this paper, we apply the technique of tropical flows to this interpretation in order to provide a new proof of the piecewise polynomiality of these enumerative invariants. Moreover, we derive new types of wall-crossing formulae.
Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $Δ_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $Δ_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $Δ_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $Δ_{g, w}$ in terms of the combinatorics of $w$.
We calculate the volume of the tropical Prym variety of a harmonic double cover of metric graphs having non-trivial dilation. We show that the tropical Prym variety behaves discontinuously under deformations of the double cover that change the number of connected components of the dilation subgraph.
Marion Scheepers在他对开覆盖的组合学的研究中，引入了性质Split（U，V），断言U型覆盖可以分裂成两个V型覆盖。在本文的第一部分中，我们给出了这种形式的所有性质的几乎完整的分类，其中U和V是文献中出现的覆盖的重要族（即，大覆盖，ω覆盖，τ覆盖，和伽马覆盖），使用这些属性的组合特征，在N上的超滤。在本文的第二部分中，我们考虑的问题是否，给定U和V，财产分裂（U，V）是保持下采取有限工会，任意子集，权力或产品。几个有趣的问题仍然悬而未决。
We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical matrices. In the case $n=2$ these identities are precisely those satisfied by the bicyclic monoid, whilst in the case $n=3$ they form a subset of the identities which hold in the plactic monoid of rank $3$. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for $\ut{n}$ if and only if their signatures a…
We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym-Brill-Noether locus for generic unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym-Brill-Noether Theorem for generic unramified double…
We give a combinatorial evaluation of Iwahori Whittaker functions for unramified genuine principal series representations on metaplectic covers of the general linear group over a non-archimedean local field. To describe the combinatorics, we introduce new combinatorial data that we call colored data: colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models. We show that all three are equivalent. To achieve the result, we give an explicit Iwahori decomposition for the maximal unipotent subgroup of a split reductive group which gives the parametrization of the generaliz…
Some of Philippe Flajolet's combinatorial contributions that he wrote between 1976 and 1995, say, are described. In most of Flajolet's papers, asymptotic/analytic considerations play a major role. To be true to the spirit of the journal ECA, the emphasis is on the \emph{combinatorial} part. Covered are Register function of binary trees, approximate counting, digital search trees and the Rice formula, probabilistic counting, adding a new slice, Ramanujan's influence, finite differences and harmonic numbers, Digits and the Mellin-Perron formula.
The moduli space $Δ_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $Δ_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $Δ_{0,w}$ is disconnected and for which the fundamental group of $Δ…
We prove an analogue of Kirchhoff's matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebast…
