关于表示非钝角单形的0/1-矩阵的组合结构
On the combinatorial structure of 0/1-matrices representing nonobtuse simplices
摘要：
0/1-单形是单位n-立方体I^n的n+1个仿射独立顶点的凸包。如果它的二面角都不是钝角，则它是非钝角;如果它们都不是直角，则它是锐角。I^n中的锐0/1-单形可以表示为大小为n × n的0/1-矩阵P，其Gramian的逆矩阵是严格对角占优的，且元素为负非对角元素。本文证明了P的转置逆的正部分D是双随机的，且与P具有相同的支撑，P^-T的负部分C是严格行次随机的，且其支撑与D的支撑互补，从而证明了P^-T=D-C没有零元，且有正的行和.因此，对于锐0/1-小面S的每个小面F，在I^n中存在至多一个以F为小面的其他锐0/1-单形T。我们称T为S在F处的急性邻居。如果P表示一个0/1-单纯形，它仅仅是非钝角的，那么P^-T可以有等于零的元素。它的正部分D仍然是双随机的，但它的支集可以严格包含在P的支集中。在理论上，这可能会导致一个非钝角0/1-单形S在它的每个面上有几个非钝角邻居。其次，我们研究了具有部分可分解的矩阵表示P的非钝角0/1-单形S。我们证明了这样的单形也具有至少两个对角块的分块对角矩阵表示，并证明了具有部分可分解的矩阵表示的非钝角单形可以分裂成相互正交的完全不可分解的单形面，其维数之和为n。利用这一见解，我们能够扩展一个邻居定理的急性单纯形一个更大的类nonobtuse单纯形。
英文摘要：
A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I^n can be represented by 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse of P is doubly stochastic and has the same support as P. The negated negative part C of P^-T is strictly row-substochastic and its support is complementary to that of D, showing that P^-T=D-C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex T in I^n having F as a facet. We call T the acute neighbor of S at F. If P represents a 0/1-simplex that is merely nonobtuse, P^-T can have entries equal to zero. Its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors at each of its facets. Next, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that such a simplex also has a block diagonal matrix representation with at least two diagonal blocks, and show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal fully indecomposable simplicial facets whose dimensions add up to n. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.
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