Konstanze Rietsch
We explain how A. Givental's mirror symmetric family to the type A flag variety and its proposed generalization to partial flag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y in SL_n/B. We then use this theory to describe the totally nonnegative part of Y.
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Totally nonnegative part of the Peterson variety in Lie type A
本文利用与Peterson簇密切相关的环面几何，证明了Rietsch提出的李型A中Peterson簇全非负部分作为胞腔分解空间同胚于对应维数立方体的猜想。
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。
The totally nonnegative part of a partial flag variety G/P is known to have a decomposition into semi-algebraic cells. We show that the closure of a cell is again a union of cells and give a combinatorial description of the closure relations. The totally nonnegative cells are defined by intersecting the totally nonnegative part with a certain stratification of G/P defined by Lusztig. We also verify the same closure relations for these strata.
The totally nonnegative part of a partial flag variety G/P has been shown by the first author to be a union of semi-algebraic cells. Moreover she showed that the closure of a cell is the union of smaller cells. In this note we provide glueing maps for each of the cells to prove that the totally nonnegative part of G/P is a CW complex. This generalizes a result of Postnikov, Speyer and the second author for Grassmannians.
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。
On Positivity for the Peterson Variety
本文探讨旗流形 $G/B$ 及其子变形上同调环中的几何正性，阐述了 Schubert 变形基的非负结构常数，并解释了该正性现象如何扩展至特定的幂零 Hessenberg 变形（Peterson 变形），同时提出了向其他 Hessenberg 变形扩展的开放问题。
In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties.
Patch ideals encode neighbourhoods of a variety in GL_n/B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen-Macaulay and Gorenstein. Consequently, we combinatorially describe the singular locus of the Peterson variety; give an explicit equivariant K-theory localization formula; and extend some results of [B. Kostant '96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties. We conjecture that the projectivized tangent cones are Cohen-Macaulay, and that their h-polynomials are nonnegative and up…
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…
We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
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 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.
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 …
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…
We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\C)$ relying in particular on the positivity of the structure constants, which are enumerative Gromov--Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterso…
We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and…
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。
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。
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。
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。
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, …
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。
Theta Positivity in Lagrangian Grassmannian
本文研究拉格朗日格拉斯曼流形的 theta 非负部分，证明其具有轨道分解且同胚于闭球，并阐明其与其他正结构及广义 Plücker 坐标非负性的关系。
Total positivity and symmetric spaces
通过取Lusztig全正部分在对称空间中的Hausdorff闭包定义全正性，引入双Bruhat胞腔并给出显式正参数化，证明胞腔分解和闭包关系。
Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let $\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$ inside $\mathbb{R}^n$, where each $V_k$ has dimension $k$. Lusztig (1994, 1998) introduced the totally positive part $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$ as the subset of partial flags which can be represented by a totally positive $n\times n$ matrix, and defined the totally nonnegative part $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ as the closure of $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$. On the other …
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