Equivariant Euler characteristics of partition posets

格 poset 分割与 cd 指数的组合学

Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index

本文提出了一种新的欧拉 poset 类，称为 S-分割 poset，证明其具有非负的 cd 指数，并引入了 SE-分割 poset 的半欧拉版本。

The homotopy theory of equivariant posets

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model structure are not well understood calculationally. To illustrate, we exhibit some examples of cofibrant and fibrant posets and an example of a non-cofibrant finite poset.

Resolutions for Equivariant Sheaves over Toric Varieties

In this work we construct global resolutions for general coherent equivariant sheaves over toric varieties. For this, we use the framework of sheaves over posets. We develop a notion of gluing of posets and of sheaves over posets, which we apply to construct global resolutions for equivariant sheaves. Our constructions give a natural correspondence between resolutions for reflexive equivariant sheaves and free resolutions of vector space arrangements.

Promotion and Rowmotion

We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. W…

平面分拆与矩形和梯形偏序集上的行移动

Plane partitions and rowmotion on rectangular and trapezoidal posets

本文定义矩形与梯形偏序集标号间的双有理映射，热带化后给出平面分拆双射，证明其关于行移动等变，解决猜想并表明有限阶。

分次偏序集的序多面体是 γ-有效的

Order polytopes of graded posets are gamma-effective

本文利用等变 Ehrhart 理论证明分次偏序集的序多面体具有 $\gamma$-有效性，并通过符号分次偏序集理论给出其等变 Ehrhart 级数分子的饱和表达式。

Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem o…

Euler flag enumeration of Whitney stratified spaces

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…

Skew hook formula for $d$-complete posets

Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…

Poset pinball, the dimension pair algorithm, and type A regular nilpotent Hessenberg varieties

In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for …

The equivariant Euler characteristic of moduli spaces of curves

We give a formula for the S_n - equivariant Euler characteristics of the moduli spaces of genus g curves with n marked points

Euler characteristic reciprocity for chromatic, flow and order polynomials

The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.

What is $-Q$ for a poset $Q$?

In the context of combinatorial reciprocity, it is a natural question to ask what "$-Q$" is for a poset $Q$. In a previous work, the definition "$-Q:=Q\times\mathbb{R}$ with lexicographic order" was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley's reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of $Q$ is me…

Torus graphs and simplicial posets

For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if it…

Euler characteristic of the truncated order complex of generalized noncrossing partitions

The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

V型偏序集链上的Rowmotion与旋转动力学

Rowmotion on the chain of V's poset and whirling dynamics

该研究建立了$k$-有界$P$-划分与$P\times [k]$序理想间的等变双射，将旋转作用对应到rowmotion算子，推导了V型偏序集链上rowmotion的周期性与同均性结果，并部分推广到爪形偏序集的情形。

Equivariant Euler characteristics of discriminants of reflection groups

Let G be a finite, complex reflection group and f its discriminant polynomial. The fibers of f admit commuting actions of G and a cyclic group. The virtual $G\times C_m$ character given by the Euler characteristic of the fiber is a refinement of the zeta function of the geometric monodromy, calculated in a paper of Denef and Loeser. We compute the virtual character explicitly, in terms of the poset of normalizers of centralizers of regular elements of G, and of the subspace arrangement given by proper eigenspaces of elements of G. As a consequence, we compute orbifold Euler characteristics and…

Stanley's conjectures on the Stern poset

The Stern poset $\mathcal{S}$ is a graded infinite poset naturally associated to Stern's triangle, which was defined by Stanley analogously to Pascal's triangle. Let $P_n$ denote the interval of $\mathcal{S}$ from the unique element of row $0$ of Stern's triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $n\geq 1$ let \begin{align*} L_n(q)&=2\cdot\left(\sum_{k=1}^{2^n-1}A_{P_k}(q)\right)+A_{P_{2^n}}(q), \end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $n\geq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(…

Discrete Euler integration over functions on finite categories

This paper provides the theory of integration with respect to Euler characteristics of finite categories. As an application, we use sensors to enumerate the targets lying on a poset. This is a discrete analogue to Baryshnikov and Ghrist's work on integral theory using topological Euler characteristics.

A glimpse into continuous combinatorics of posets, polytopes, and matroids

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…

A Voronoi poset

Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry relations for the number of faces, number of vertices and number of circles of certain orders. These symmetry relations are independent of the position of the sites in S. As a consequence we show that the reduced Euler characteristic of the poset of faces equals zero whenever n odd.

The Euler characteristic of a category

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula for the cardinality of the colimit of a diagram of sets is proved, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Mobius-Rota inversion from posets to categories.

An equivariant discrete model for complexified arrangement complements

We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…

New polytope decompositions and Euler-Maclaurin formulas for simple integral polytopes

We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.

$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

本文通过分解模空间、利用环面作用推导闭式公式，计算了Kontsevich模空间$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数，建立了其几何与对称函数、图染色计数的联系。

Euler characteristics of cominuscule quantum K-theory

We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce that the sum of the structure constants associated to any product of Schubert classes is equal to one. Equivalently, the sheaf Euler characteristic map extends to a ring homomorphism defined on the quantum K-theory ring.

刻画有限偏序集上的模型结构

Characterizing model structures on finite lattices

本文利用转移系统对有限格上的模型结构进行了完整刻画，建立了抽象同调理论与等变方法的新联系。