Fundamental Theorems of Morse Theory on posets

Groebner basis degree bounds on $\Tor^{k[Λ]}_\bullet(k,k)_\bullet$ and discrete Morse theory for posets

The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups $\Tor^{R}_\bullet(k,k)_\bullet$ in the case where $R = k[Λ]$ is an affine semi-group ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show that if the bounds hold for a field $k$ then they hold for $\field[Λ]$ and any field $\field$. Moreover, we provide a combinatorial construction for a free resolution of $\field$ over $\field[Λ]$ which achieves these bounds. 2. We extend the lexicographic discrete Morse fu…

Discrete Morse theory for the collapsibility of supremum sections

The Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and whic…

Moduli Spaces of Morse Functions for Persistence

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions o…

$G$-colorings of posets, coverings and presentations of the fundamental group

We introduce the notion of a coloring of a poset, which consists of a labeling of the edges in its Hasse diagram by elements in a given group $G$. We use $G$-colorings to describe the covering maps of posets and present a new method based on colorings to obtain concrete and simple presentations of the fundamental group of polyhedra.

The Fundamental Group of Balanced Simplicial Complexes and Posets

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

A Broken Circuit Model for Chromatic Homology Theories

Using the tools of algebraic Morse theory, and the thin poset approach to constructing homology theories, we give a categorification of Whitney's broken circuit theorem for the chromatic polynomial, and for Stanley's chromatic symmetric function.

On homology of finite topological spaces

We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding differentials. Our method proves to be powerful and involves far fewer computations than the standard one. We derive many applications of our technique which include a generalization of Hurewicz theorem for regular CW-complexes, results in homological Morse theory and formulas to compute the Möbius function of posets.

Relation between the Weyl group orbits of fundamental weights for multiply-laced finite dimensional simple Lie algebras and d-complete posets

It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.

通过生灭转换连接离散Morse函数

Connecting Discrete Morse Functions via Birth-Death Transitions

本文通过生灭转换研究有限单纯复形上离散Morse函数间的转换，证明了任意两函数均可由有限转换序列连接，并以此给出Forman基本结果的替代证明及探讨函数空间的拓扑结构。

对称化多面体与偏序集

Symmetrizing polytopes and posets

本文引入任意多面体关于任意反射群的对称化概念，证明其组合结构与法扇可由精细化基本扇恢复，并将对称偏序集的实现问题归结为生成子偏序集的精细化基本扇实现。

Poset splitting and minimality of finite models

We develop a novel technique, which we call poset splitting, that allows us to solve two open problems regarding minimality of finite models of spaces: the nonexistence of a finite model of the real projective plane with fewer than 13 points and the nonexistence of a finite model of the torus with fewer than 16 points. Indeed, we prove much stronger results from which we also obtain that there does not exist a finite model of the Klein bottle with fewer than 16 points and that the integral homology groups of finite spaces with fewer than 13 points are torsion-free, settling a conjecture of Har…

From symmetric fundamental expansions to Schur positivity

We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of $f$. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisym…

Weighted posets and the enriched monomial basis of QSym (extended abstract)

Gessel's fundamental and Stembridge's peak functions are the generating functions for (enriched) $P$-partitions on labelled chains. They are also the bases of two significant subalgebras of formal power series, respectively the ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao introduced the monomial peak functions, a basis of the algebra of peaks indexed by odd integer compositions whose relation to peak functions mimics the one between the monomial and fundamental bases of QSym. We show that the extension of monomial peaks to any composition is a new basis of QSym and g…

Steinberg 之后饱和融合系统轨道空间的可缩性

Contractibility of the orbit space of a saturated fusion system after Steinberg

本文扩展了 Steinberg 利用离散 Morse 理论证明有限群偏序集轨道空间可缩性的方法，进一步证明了饱和融合系统轨道空间的可缩性。

超可解偏序集与纤维型阿贝尔排列

Supersolvable posets and fiber-type abelian arrangements

本文对连通阿贝尔李群上广义构型空间的纤维丛进行组合分析，通过有限偏序集性质检测纤维丛存在性，并证明了 K(π,1) 排列的确定及 Poincaré 多项式的因式分解。

带约束实除子的实多项式 I. 基本群

Real polynomials with constrained real divisors. I. Fundamental groups

本研究拓展Arnold与Vassiliev的多项式空间拓扑研究，考虑实除子避开特定重数序列的实首一多项式空间，给出其基本群的显式表示、秩的上界与稳定性，并给出配边解释。

团宽与维数

Cliquewidth and dimension

本文证明有界团宽且维数足够大的偏序集必含标准例子或Kelly例子，并据此完整刻画了覆盖图在子图封闭图类中时偏序集维数有界的条件，同时证明有界团宽蕴含布尔维数有界。

Posets, Tensor Products and Schur positivity

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set $P(λ, k)$ of k-tuples of dominant weights which add up to λ. Let $P(λ, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if λis a multiple of a fundamental weight (and k is general) or if k=2 (and λis general), then $P(λ, k)/\sim$ coincides with the set of S_k-orbits in $P(λ,k)$, where S_k acts on $P(λ, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row …

Multidimensional extension of the Morse--Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $\ZZ^d$ definable by a first order formula in the Presburger arithmetic $<\ZZ;<,+>$. With this latter not…

摇摆表恒等式的新方法

New Approaches to Identities for Vacillating Tableaux

本文通过微分偏序集的结构性质统一证明摇摆表计数恒等式，并利用简单删除-插入过程给出杨氏格及其笛卡尔幂的新双射证明。

有限半分配格的基本定理

The fundamental theorem of finite semidistributive lattices

本文证明了有限半分配格的基本定理，通过构造带有额外结构的集合Sha，揭示了半分配格的结构特征及其与表示论中扭对的联系。

曲面型 cluster代数中$F$-多项式的偏序集

Posets for $F$-polynomials in cluster algebras from surfaces

本研究证明了带标记曲面模型中cluster变量的简洁公式，通过为带标记弧关联偏序集将$F$-多项式表示为序理想加权和，结合偏序集理论与双曲几何完成证明，推进了cluster代数中$F$-多项式的结构研究。

Non-embeddability of geometric lattices and buildings

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such d-dimensional complexes require (2d + 1)-dimensional Euclidean space for an embedding. (This dimension is in general always sufficient for any d-complex.) We develop a method to show non-embeddability for general order complexes of …

Incidence Geometry in a Weyl Chamber II: $SL_n$

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of $sl_n$ inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces. From the perspective of supersymmetric gauge theories with 8 supercharges in five dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces…