The hull metric on Coxeter groups

On a class of metrics related to graph layout problems

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization problems known as graph layout problems. We prove several results about the structure of these metrics. In particular, it is shown that their convex hull is not closed in general. We then show that certain linear inequalities define facets of the closure of the convex hull. Finally, we characterise th…

利用Coxeter复形对条形码空间进行分层

Stratifying the space of barcodes using Coxeter complexes

本研究运用几何群论工具，基于对称群抛物子群的带标记双陪集对含n条条带的条形码空间分层，得到推广现有不变量的新条形码不变量，并衍生出与修正瓶颈、Wasserstein度量一致的空间度量。

披萨与2-结构

Pizza and 2-structures

本文通过剖分证明和抽象剖分群，将广义披萨定理从体积推广到任意仿射等距不变的赋值（含内蕴体积），并利用作者关于Coxeter排列与2-结构的定理建立交替和恒等式。

Tropical Convex Hull Computations

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes.

Computing Linear Systems on Metric Graphs

The linear system $|D|$ of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute its extremal generators using the landmarks. We apply these methods to some examples - namely the canonical linear systems of some small trivalent graphs.

Shortcut Graphs and Groups

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated…

Young's lattice and dihedral symmetries revisited: Möbius strips and metric geometry

A cascade of dihedral symmetries is hidden in Young's lattice of integer partitions. In fact, for each integer N>2 the Hasse graph of the subposet consisting of the partitions with maximal hook length strictly less than N has the dihedral group of order 2N as its symmetry group. Here a new interpretation of those Hasse graphs is presented, namely as the 1-skeleta of the injective hulls of certain finite metric spaces.

Torus orbit closures in flag varieties and retractions on Weyl groups

A finite Coxeter group $W$ has a natural metric $d$ and if $\mathcal{M}$ is a subset of $W$, then for each $u\in W$, there is $q\in \mathcal{M}$ such that $d(u,q)=d(u,\mathcal{M})$. Such $q$ is not unique in general but if $\mathcal{M}$ is a Coxeter matroid, then it is unique, and we define a retraction $\mathcal{R}^m_{\mathcal{M}}\colon W\to \mathcal{M}\subset W$ so that $\mathcal{R}^m_{\mathcal{M}}(u)=q$. The $T$-fixed point set $Y^T$ of a $T$-orbit closure $Y$ in a flag variety $G/B$ is a Coxeter matroid, where $G$ is a semisimple algebraic group, $B$ is a Borel subgroup, and $T$ is a maxim…

Coxeter 群中的典范约化词与带符号下降长度枚举

Canonical reduced words and signed descent length enumeration in Coxeter groups

本文利用 Coxeter 群中典范约化词的下降构造对合，统一推导了对称群及 B、D 型 Coxeter 群中 drops、depth、exc 的带符号枚举公式，并借助 Foata-Zeilberger 双射得到连分数及应用，给出了交错群上 drops 的均值与方差。

具有相同峰值集合的有符号置换上的度量

Metrics on Signed Permutations with the Same Peak Set

本文将无符号置换峰值集合上的度量研究推广到 B 型 Coxeter 群的有符号置换，刻画了相同峰值集合子集上 Hamming、l∞ 和字度量的极值。

A "hidden" characterization of polyhedral convex sets

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$.

Convex Geometry of Orbits

We study metric properties of convex bodies B and their polars B^o, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G=S_n, the symmetric group), the set of non-negative polynomials in real algebraic geometry (G=SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G=SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, g…

Idempotent Analysis, Tropical Convexity and Reduced Divisors

We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact tropical convex sets. In this paper, we build the foundation of a theory of idempotent analysis using tropical projections and obtain a series of subsequent results, e.g. tropical retracts, construction of compact tropical convex sets and a set-theoretical characterization of tropical weak independence. In particular, we prove a tropical version of Mazur's Theor…

关于Moebius-Kantor图的注记

Remarks about the Moebius-Kantor graph

本文研究Moebius-Kantor图MK=G(8,3)的群论和拓扑性质，包括其作为三个非阿贝尔群的Cayley图、与Heawood数7和Tucker群的关系、Lefschetz数的计算、作为3-球面G的2-骨架复形的对偶、在Hopf纤维化中表示平坦Clifford环面，以及度量结构唯一确定Pauli群代数结构。