Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

• Autorzy: Jan Brandts , Apo Cihangir
• Języki publikacji: EN

Abstrakty

EN

The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

Słowa kluczowe

EN

Acute simplex; 0/1-matrix; Hadamard conjecture; hyperoctahedral group; cycle index; Pólya enumeration theorem; Kepler’s tree of fractions; strictly ultrametric matrix

Kategorie tematyczne

• 05A99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2017
• Tom: 5
• Numer: 1
• Strony: 158-201

Daty

wydano

2017-08-28

otrzymano

2017-04-12

zaakceptowano

2017-08-21

online

2017-09-20

Twórcy

autor

Jan Brandts

• ,

autor

Apo Cihangir

• ,

Identyfikatory

DOI

10.1515/spma-2017-0014