# On the combinatorial structure of 0/1-matrices representing nonobtuse simplices

## Abstract

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 of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in In can be represented by 0/1-matrices *P* of size *n× n* whose Gramians *G* = *P*^{⊤}*P* 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 *P*^{−⊤} of *P* is doubly stochastic and has the same support as *P*. In fact, *P* has a fully indecomposable doubly stochastic pattern. The negative part *C* of *P*^{−⊤} is strictly row-substochastic and its support is complementary to that of *D*, showing that *P*^{−⊤} = *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 *Ŝ* in In having F as a facet. We call *Ŝ* the acute neighbor of *S* at *F*.

If *P* represents a 0/1-simplex that is merely nonobtuse, the inverse of *G* = PΤP is only weakly diagonally dominant and has nonpositive off-diagonal entries. These matrices play an important role in finite element approximation of elliptic and parabolic problems, since they guarantee discrete maximum and comparison principles. Consequently, *P*^{−⊤} can have entries equal to zero. We show that its positive part *D* is still doubly stochastic, but its support may be strictly contained in the support of *P*. This allows *P* to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors *Ŝ* at each of its facets.

In this paper, we study nonobtuse 0/1-simplices *S* having a partly decomposable matrix representation *P*. We prove that if *S* has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of *S* will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all *n*-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to *n*, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.

## Keywords

acute simplex nonobtuse simplex orthogonal simplex 0/1-matrix doubly stochastic matrix fully indecomposable matrix partly decomposable matrix## MSC 2010

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