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摘要翻译：
在1994年的一篇开创性论文中，Lusztig通过引入任意（广义的，部分的）标志簇G/P的全非负部分(G/P)_{\geq0}来推广全正性理论。他把这个空间称为“非凡的多面体子空间”，并推测分解成细胞，这随后被第一作者证明。随后，第二作者通过猜想这个单元分解空间是一个与闭球同胚的正则CW复形，提出了一个具体的猜想，即这个单元分解空间是一个多面体的次优空间。本文利用离散Morse理论证明了该猜想的同伦等价性。我们明确地证明了胞的边界与球面同伦，胞的闭包是可收缩的。后半部分推广了Lusztig的一个结果:(G/P)_{\geq0}--上维单元的闭包--是可收缩的。对于除射影空间外的所有G/P空间，我们关于胞体边界的结果，即使是顶维胞体(G/P)_{>0}的边界与球面同伦的特例，也是新的。
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英文标题：
《Discrete Morse theory for totally non-negative flag varieties》
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作者：
Konstanze Rietsch and Lauren Williams
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最新提交年份：
2010
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分类信息：
一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Representation Theory 表象理论
分类描述：Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
代数和群的线性表示，李理论，结合代数，多重线性代数
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英文摘要：
In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a "remarkable polyhedral subspace", and conjectured a decomposition into cells, which was subsequently proven by the first author. Subsequently the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's that (G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere, is new for all G/P other than projective space.
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PDF链接：
https://arxiv.org/pdf/0810.4314
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