Toric Topology. Chapter 1: Geometry and combinatorics of polytopes

Recent developments in algebraic combinatorics

A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of "Recent progress in algebraic combinatorics" (math.CO/0010218), which dealt with three other topics.

The combinatorics and topology of proper toric maps

We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the $f$-vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of $Y$, which is…

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study the combinatorics of Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterizatio…

The geometry and combinatorics of cographic toric face rings

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures of a graph: the cographic arrangement of hyperplanes, the Voronoi polytope, and the poset of totally cyclic orientations. We describe the properties of the cographic ring and, in particular, relate the invariants of the ring to the invariants of the corresponding graph. Our study of the cographic ring fits into a body of work on describing rings constructed …

Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials

The purpose of this paper is twofold. Firstly, we generalize the notion of characteristic polynomials of hyperplane and toric arrangements to those of certain abelian Lie group arrangements. Secondly, we give two interpretations for the chromatic quasi-polynomials and their constituents through subspace and toric viewpoints.

C型Schubert簇半环面退化的组合学

Combinatorics of semi-toric degenerations of Schubert varieties in type C

本文利用带自相交的pipe dream为C型辛Gelfand-Tsetlin多面体的Schubert类推广引入组合模型，并证明该推广可由斜有丝分裂算子构造。

复杂数最大行列式矩阵的组合学

Combinatorics of Complex Maximal Determinant Matrices

本论文研究复杂数最大行列式矩阵的构造与性质，探讨了数论工具、Butson型Hadamard矩阵、关联方案及设计理论在隐私保护中的应用。

Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomolo…

有向复合Du Val奇点的GIT商与二聚体组合学的变体

Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities

本文研究有向复合Du Val奇点的二聚体模型，利用锯齿路径序列确定稳定性参数空间的墙-腔结构，实现跨墙时稳定表示变化的追踪，为三维Gorenstein有向奇点的射影平展解析提供新视角。

Combinatorics of $(\ell,0)$-JM partitions, $\ell$-cores, the ladder crystal and the finite Hecke algebra

The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…

Schubert geometry and combinatorics

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.

置换多面体、结合多面体及其相关组合的探讨

Combinatorics of the Permutahedra, Associahedra, and Friends

本文概述了作者过去十年在代数、双射、计数和几何组合学领域的研究，重点探讨了置换多面体和结合多面体及其相关偏序集，并详细介绍了Tamari区间偏序集、q,t-计数、整数偏序Hopf代数、置换树格及s-弱序等结构的构造与性质。

论Arthur迹公式的组合学、凸多面体与环面簇

On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties

本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。

Linear Toric Fibrations

These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …

Small toric resolutions of toric varieties of string polytopes with small indices

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. For a reduced word $\bf i$ of the longest element in the Weyl group of $G$ and a dominant integral weight $λ$, one can construct the string polytope $Δ_{\bf i}(λ)$, whose lattice points encode the character of the irreducible representation $V_λ$. The string polytope $Δ_{\bf i}(λ)$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $\mathbf i$. In this paper, we study combinatorics of string polytopes when $G = SL_{n+1}(\mathbb{C})$, and present a sufficient condition on $\mathbf i$ such that …

环面切换中的Torsors与铺砌

Torsors and tilings from toric toggling

本文研究圈图独立集切换的环面类比，用网格编码轨道，通过交换双射构造蛇群与衔尾蛇群的torsors，并刻画圆柱与环面的平行四边形铺砌。

Toric networks, geometric $R$-matrices and generalized discrete Toda lattices

We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the $R$-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

Intersection cohomology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.

Toric Hyperkahler Varieties

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanle…

Geometry and Combinatorics of Crystal Melting

We survey geometrical and especially combinatorial aspects of generalized Donaldson-Thomas invariants (also called BPS invariants) for toric Calabi-Yau manifolds, emphasizing the role of plane partitions and their generalizations in the recently proposed crystal melting model. We also comment on equivalence with a vicious walker model and the matrix model representation of the partition function.

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide…

多面体的组合霍奇指标定理

Combinatorial Hodge Index Theorem for Polytopes

该研究将Maxim-Schuermann公式从顶点有理的多面体推广至任意凸多面体，依托BBFK的组合相交上同调框架，还探讨了多面体的霍奇指标定理。

通过混合 Hodge 模块研究环面簇的局部上同调与奇异上同调

Local cohomology and singular cohomology of toric varieties via mixed Hodge modules

本文利用混合 Hodge 模块技术研究了环面簇的局部上同调与奇异上同调，证明了真环面簇奇异上同调的混合 Hodge-Tate 型性质，并详细计算了特定子类的局部上同调与 Betti 数。

部分旗簇的环面退化与匹配场多面体的组合突变

Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes

本文通过匹配场多面体和组合突变，构造了部分旗簇的新环面退化，并证明其正规性及与Newton-Okounkov体的联系，最终应用于小规模格拉斯曼流形。

Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in t…

Matching polytopes, toric geometry, and the non-negative part of the Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Delta_G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data of P(G) to d…

Toric Topology

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discov…

Geometry, combinatorics, and algebra of inductively pierced codes

Convex neural codes are combinatorial structures describing the intersection pattern of a collection of convex sets. Inductively pierced codes are a particularly nice subclass of neural codes introduced in the information visualization literature by Stapleton et al. in 2011 and to the convex codes literature by Gross et al. in 2016. Here, we show that all inductively pierced codes are nondegenerate convex codes and nondegenerate hyperplane codes. In particular, we prove that a $k$-inductively pierced code on $n$ neurons has a convex realization with balls in $\mathbb R^{k+1}$ and with half spa…

关于$B$型与$C$型弦多面体的组合学研究

On combinatorics of string polytopes in types $B$ and $C$

本文通过给出B型与C型弦锥的显式描述（类比A型的Gleizer-Postnikov描述），研究了B、C型弦多面体的组合性质，并刻画了特定最高权下与C型Gelfand-Tsetlin多面体幺模等价的C型弦多面体。

Calabi-Yau超曲面在Toric变种中的Hilbert多项式与多面体边界的格点计数

Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries

本文研究了Calabi-Yau超曲面在Toric变种中的Hilbert多项式，通过多面体边界的格点计数和K-理论中的欧拉类计算，揭示了其与组合学中包含-排除原理的平行性。

Quaternionic Geometry of Matroids

Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central role in Stanley's proof for the necessity of McMullen's conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkaehler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we give two proofs that the injectivity part of the Hard Lefschetz theorem surv…

Matrix Models, Integral Polyhedra and Toric Geometry

We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on the well-known geometric interpretations of matrix model digram techniques, we construct a new model that enumerates triangulations, subdivisions, and numbers of integral points of integral polygons. This approach allows us to look at their combinatorics from a new perspective, motivated by knowledge about matrix models and their integrability. We show how an…

A survey of hypertoric geometry and topology

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.

Torus actions and combinatorics of polytopes

An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T…

博弈论中集合函数的几何性质：组合与计算层面

Geometry of Set Functions in Game Theory: Combinatorial and Computational Aspects

本论文从组合与计算角度研究博弈论集合函数的几何性质，定义基本多面体并推广Bondareva-Shapley定理，结合组合学理论，提出核心稳定性检验算法与新必要条件，推动合作博弈论跨领域发展。

OSCAR中的拟阵

Matroids in OSCAR

本文介绍了新型计算机代数系统OSCAR中处理拟阵的模块相关内容，聚焦拟阵的实现空间与周环计算，体现拟阵在多数学领域的关联作用。