Michael Bjorklund, Alexander Engstrom
The McMullen Correspondence gives a linear dependence between M-sequences of length |d/2|+1 and f-vectors of simplical d-polytopes. Denote the transfer matrix between g and f by M_d. Recently, Bjorner proved that any 2x2-minor of M_d is nonnegative and conjectured that the same would be true for arbitrary minors. In this paper we answer the question in the affirmative.
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We give an alternative proof of a (former) conjecture of Björner stating that the matrix expressing face numbers in terms of g numbers is totally non-negative. We briefly discuss the case of simple flag polytopes.
A finite totally nonnegative Grassmannian
本文提出有限域上的全非负格拉斯曼流形，给出特殊情形的点计数并发现OEIS序列新解释，且与传统实数域理论对比。
We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide.
Postnikov constructed a decomposition of a totally nonnegative Grassmannian into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Pluecker coordinates as a totally positive base for the set of non-vanishing Pluecker coordinates for a given positroid cell.
Totally Nonnegative Tropical Flags and the Totally Nonnegative Flag Dressian
研究完全旗簇及其热带化的全非负部分，证明Lusztig的非负完全旗簇等价于Plücker坐标全非负的旗，并证明完全旗Dressian与热带完全旗簇的全非负部分一致。
Plücker inequalities for weakly separated coordinates in totally nonnegative Grassmannian
本文证明全非负 Grassmann 流形中弱分离 Plücker 坐标的长 Plücker 关系部分和围绕 0 振荡，并据此刻画弱可分离性，建立其与 Temperley–Lieb immanants 等对象的联系。
There is a cell decomposition of the nonnegative Grassmannian. For each cell, totally positive bases(TP-bases) is defined as the minimal set of Plücker variables such that all other nonzero Plücker variables in the cell can be expressed in those variables in a subtraction-free rational function. This is the generalization of the TP-bases defined for nonnegative part of $GL_k$ defined in \cite{FZ5}. For each cell, we have a $\LE$-diagram and a natural way to label the dots inside the diagram with Plücker variables. Those set of Plücker variables form a TP-bases of the cell. Using mutations comi…
The space of mxp totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegativ…
In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties.
We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonem…
Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of $\{1,\dotsc,n\}$ and its complement $I^c$, we have $\det(A) \leq \det(A_{I,I})\det(A_{I^c,I^c})$. Barrett and Johnson (Linear Multilinear Algebra 34, 1993) extended these to state inequalities for sums of products of principal minors whose orders are given by nonincreasing integer sequences $(λ_1,\dotsc,λ_r)$, $(μ_1,\dotsc,μ_s)$ summing to $n$. Specifically, if $λ_1+\cdots…
We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite perm…
We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
Totally nonnegative matrices, chain enumeration and zeros of polynomials
本文证明了任何下三角单位矩阵且完全非负的矩阵都产生只含实零点的多项式家族，从而在多个数学领域中产生影响。作者发展了一种在偏序集中链计数和链多项式零点的一般理论，并扩展和统一了Brenti、Welker和Athanasiadis等人的结果。此外，作者还定义了一种广义的h向量概念，用于一大类偏序集，以及通过方法对Förgács和Tran提出的关于某些双变量有理函数产生的多项式实根性的问题进行了回答。
Totally nonnegative Peterson variety and strongly dominant weight polytope
本文研究了Peterson变种的完全非负部分，证明其与强主导权重多面体的同胚关系，并验证了Rietsch的猜想。
We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety Crit$^{\ge0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of Crit$^{\ge0}_f$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety Crit$^{\ge0}_{k,n}$ is homeomorphic to the second hypersimplex $Δ_{2,n}$.
We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.
We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F.R.Gantmacher, M.G.Krein, A.Whitney, C.Loewner, M.Gasca, and J.M.Pena to the case of an arbitrary complex semisimple Lie group.
Chow polynomials of totally nonnegative matrices and posets
本文通过建立偏序集与矩阵的Chow函数新关系及多项式交错序列技术，证明全非负矩阵的Chow多项式仅有实零点，从而解决了几何格及多类重要偏序集上的实零点猜想，并研究了Toeplitz矩阵Chow多项式与二项式、Sheffer偏序集及广义Eulerian多项式的联系。
The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we p…
Regularity theorem for totally nonnegative flag varieties
本文证明了偏旗流形$G/P$的全非负部分是正则CW复形，证实了Williams的猜想，并特别证实了全非负Grassmannian中positroid胞腔闭包同胚于球的Postnikov猜想。
Regularity of Totally Nonnegative Wonderful Compactification
针对伴随型分裂实半单群的奇妙紧化全非负部分，证明其胞腔分解为正则CW复形，证实了Galashin等人的猜想，所用方法结合了Bruhat atlas与CL-标号。
Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible …
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.
These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes.
The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of tot…
In this paper we study the partially ordered set Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety P^J_{\geq 0}. Our goal is to understand the geometry of P^J_{\geq 0}: Lusztig has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P^J_{\geq 0} is homeomorphic to a ball. The order complex |Q^J| is a simplicial complex which can be thought of as a combinatorial approximation of P^J_{\geq 0}. Using combinatorial tools such as Bjorner's EL-labellings and Dyer's reflection o…
Rotationally symmetric plabic graphs and the Lagrangian Grassmannian
本文引入了旋转对称的plabic图来表示总非负拉格朗日格拉斯曼流形，并发展了处理非简化的plabic图的新技术。
From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections
本文研究单位区间序反邻接矩阵的永久不等式，构造双射证明特定情形，并猜想对所有全非负矩阵成立。
Totally nonnegative maximal tori and opposed Bruhat intervals
本文验证了Lusztig关于完全正部分到完全正最大环形群的映射猜想，并研究了完全非负布雷特子群与完全非正布雷特子群的对立关系，同时将完全正最大环形群与幅度hedron联系起来。
Hook immanantal inequalities for totally nonnegative matrices
本文将厄米半正定矩阵上成立的hook型不可约特征标immanant不等式推广，证明其对全非负矩阵同样有效，完善了矩阵不等式理论。
Permanental Inequalities and Unit Interval Orders
研究单位区间序反邻接矩阵的永久不等式，证明连续主子式永久乘积与奇偶选择主子式永久乘积的不等式，并构造置换间的双射，猜想对所有n成立。
Zero-free sector of the Wronski map on the totally nonnegative Grassmannian
本文受Shapiro--Shapiro猜想启发，将非负系数多项式无零点扇区的经典结果推广到全非负格拉斯曼流形上多项式的Wronskian行列式，证明其在扇区$|\text{arg}(z)| < \fracπ{n}$内无复零点且界紧。
Cactus networks were introduced by Lam as a generalization of planar electrical networks. He defined a map from these networks to the Grassmannian Gr($n+1,2n$) and showed that the image of this map, $\mathcal X_n$ lies inside the totally nonnegative part of this Grassmannian. In this paper, we show that $\mathcal X_n$ is exactly the elements of Gr($n+1,2n$) that are both totally nonnegative and isotropic for a particular skew-symmetric bilinear form. For certain classes of cactus networks, we also explicitly describe how to turn response matrices and effective resistance matrices into points o…
Cylindrical Networks and Total Nonnegativity
该研究证明具有有限对角支撑的无限块Toeplitz矩阵完全非负等价于其为圆柱网络权重矩阵，推广了Brenti的经典定理，借助Temperley-Lieb immanants给出非方阵情形的独立新证明，并得到完全非负块Toeplitz矩阵与交错多项式相关定理的新证法。
A sequence $(a_n)_{n \geq 0}$ is Stieltjes moment sequence if it has the form $a_n = \int_0^\infty x^n dμ(x)$ for $μ$ is a nonnegative measure on $[0,\infty)$. It is known that $(a_n)_{n \geq 0}$ is a Stieltjes moment sequence if and only if the matrix $H =[a_{i+j}]_{i,j \geq 0}$ is totally positive, i.e., all its minors are nonnegative. We define a sequence of polynomials in $x_1,x_2,\ldots,x_n$ $(a_n(x_1,x_2,\ldots,x_n))_{n \geq 0}$ to be a Stieltjes moment sequence of polynomials if the matrix $H =[a_{i+j} (x_1,x_2,\ldots,x_n)]_{i,j \geq 0}$ is $(x_1,x_2,\ldots,x_n)$-totally positive, i.e.,…
Total positivity in twisted flag varieties
本文研究了扭曲旗流形中完全非负部分的结构，证明其分解为细胞并具有正则CW复形性质，扩展了原有研究并解决了开放问题。
Fibers of maps to totally nonnegative spaces
本文研究Chevalley指数映射$f_{(i_1,\u2026,i_d)}$的纤维结构，证明其纤维的胞腔分层与子字复形内部对偶块复形面偏序集相同且可缩，并猜想纤维是同胚于该复形的正则CW复形，推论可得Fomin-Shapiro猜想新证明。
Totally positive Toeplitz matrices: classical and modern
本文研究无穷完全正Toeplitz矩阵作为有限矩阵的极限，得到了Schoenberg参数与flag variety量子上同调相关的两种渐近描述，并揭示了其与tropical analogue理论的联系。
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).
Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
本文从代数、几何与动力学视角研究偏旗流形全非负部分，定义了推广的扭转映射，探讨了三种黎曼度量下保持正性的梯度流，并应用于全非负旗流形与幅度多面体拓扑及对称 Toda 流的分析。
The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.
Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace
本文通过引入指标行/列运算，系统研究全非负矩阵的行列式不等式，揭示新的加法不等式并推广 Gantmacher–Krein 波动不等式，同时给出刻画乘法型不等式的算法，丰富其分类。
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. When k+m=n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k=1, the amplituhedron is a cyclic polytope. While the case m=4 is most relevant to physics, the amplituhedron is an interesting mathematical objec…
Acyclic matchings on Bruhat intervals and applications to total positivity
本文在任意Bruhat区间上基于反射序显式构造无环完全匹配，并借此证明全非负Springer纤维及全非负投影纤维可缩，从而验证Lusztig猜想并简化全非负部分旗簇正则性的证明。
This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles…
A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph $G$. Let $a_1,\dots,a_k,b_k,\dots,b_1$ be vertices placed in a counterclockwise order on the outer face of $G$. We show that the $k\times k$ matrix of the two-point spin correlation functions \[ M_{i,j} = \langle σ_{a_i} σ_{b_j} \rangle \] is totally nonnegative. Moreover, $\det M > 0$ if and only if there exist $k$ pairwise vertex-disjoint paths that connect $a_i$ with $b_i$. We also c…
Total positivity for matroid Schubert varieties
本文定义线性子空间的全非负拟阵舒伯特簇，证明其为正则 CW 复形且同胚于闭球，并由此给出实拟阵舒伯特簇的正则 CW 结构。
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V is totally nonnegative iff every vector in V, when viewed as a sequence of n numbers and ignoring any zeros, changes sign at most k-1 times. We generalize this result from the totally nonnegative Grassmannian to the entire Grassmannian, showing that if V is generic (i.e. has no zero Pluecker coordinates), then the vectors in V change sign at most m times iff certai…
Total positivity in twisted product of flag varieties
本文证明 Kac-Moody 群旗簇扭积的全非负部分具有胞腔分解和显式参数化，并证明双旗簇、辫子簇及双 Bruhat 胞腔链环均为同胚于闭球的正则 CW 复形，解决了 Fomin-Zelevinsky 公开问题。
Wronskians, total positivity, and real Schubert calculus
本文证明完全旗的全非负性等价于其各阶Wronskian多项式在正半轴非零，并由此将Eremenko猜想转化为关于Wronskian零点位置的全正性猜想，同时建立其与Sottile割线猜想全正版本的等价性。
Richardson tableaux and components of Springer fibers equal to Richardson varieties
本文定义了标准表的新子集Richardson表，从组合与几何层面进行刻画，证明其对应Springer纤维分量为Richardson簇且光滑，并发现其计数与Motzkin数相关。
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
On universal quadratic inequalities for minors of TNN matrices
本研究给出了所有n×n'全非负（TNN）矩阵均成立的子式二次不等式集合的组合刻画，该结论由研究者此前通过Lindström构造得到的平面图流生成矩阵子式的稳定二次恒等式结果推导而来。
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. In the case relevant to physics (m=4), there is a collection of recursively-defined 4k-dimensional BCFW cells in the totally nonnegative part of Gr(k,n), whose images conjecturally "triangulate" the amplituhedron--that is, their images are d…
Combinatorics of $m=1$ Grasstopes
本文研究了$m=1$情况下的Grasstope，将其刻画为满足特定符号变化条件的超平面布置的胞腔并集，推广了Karp和Williams的工作，并提出了由任意定向拟阵产生的Grasstope概念。
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