Sergey Fomin, Andrei Zelevinsky
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.
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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 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 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 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 …
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。
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.
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。
Haar measure for totally positive split real semisimple quantum groups of classical type
本文针对经典型分裂实半单李群，构造了积分其量子包络代数Borel部分的局部紧量子群，计算了其Haar权与模数据，并将其Drinfeld双视为全正半群函数代数的解析量子化。
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…
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。
Remarks on totally positive flag manifolds
本文研究实数域上分裂半单群的全正部分与全正抛物子群的交，为全正旗流形提供新的注记。
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。
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 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。
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 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.
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, …
We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.
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…
We introduce a new class of "electrical" Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type A electrical Lie group is isomorphic to the symplectic group. The nonnegative part (EL_{2n})_{\geq 0} of the electrical Lie gr…
Regularity of Unipotent Elements in Total Positivity
本文研究分裂连通约化群全非负幺半群中幂幺元的正则性，证明其在某些 Levi 子群中正则，证实 Lusztig 猜想。
On a conjecture concerning totally extremal ideal Perron similarities
本文通过证明完全极值归一化理想 Perron 相似的行在 Hadamard 积下构成群，并进一步证明非奇异矩阵的行在 Hadamard 积下构成群当且仅当它是有限阿贝尔群的特征标表，从而肯定地解决了 Gershnik 等人提出的关于完全极值理想 Perron 相似的猜想。
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