# Cubes and adjoints of cross-polytopes

J. Lawrence 111Jim Lawrence, Department of Mathematical Sciences. George Mason
University. Fairfax, VA 22030, U.S.A. E-mail<EMAIL_ADDRESS>I.P. Silva 222
Ilda P. Silva, Departamento de Matemática, Faculdade de Ciencias da
Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, PORTUGAL. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We describe a bijection between oriented cubes and adjoints of cross-
polytopes. This correspondence is used to prove that the real affine cube is,
up to reorientation in the same class, the unique oriented cube that is
realizable. Moreover, its underlying matroid is, up to isomorphism, the unique
combinatorial cube that is realizable over $\mathbb{R}$.

Keywords. $\pm 1$-vectors, representability over the reals, matroids, oriented
matroids, combinatorial cubes, cross-polytopes.

Math. Subsject Classification. 05B35, 52B40, 52C40, 52C07.

##  1 Introduction

We consider $C^{n}:=\\{-1,1\\}^{n}$. The real cube is the oriented matroid of
the affine dependencies of $C^{n}$ over $\mathbb{R}$ and will be denoted ${\bf
Q_{n}}:=Aff(C^{n})$.

Oriented cubes were introduced in [10] as a combinatorial approach, via
oriented matroids, to linear/affine dependencies of $(0,1)-$vectors over the
reals. Oriented cubes are oriented matroids over $C^{n}$ that contain as
signed circuits the smallest signed circuits of the real cube - the rectangles
- and as signed cocircuits the smallest signed cocircuits of the real cube,
the signed cocircuits complementary to the facets and to the skew facets.

So far the real cube is the unique known example of an oriented cube. The
unicity of the real cube would mean a purely combinatorial characterization of
the linear and affine dependencies over the reals of the $\pm 1$ vectors.

We point out that results of E. Mayhew et al. [8], see also [4], prove that
characterizing real representability/orientability of matroids in terms of
excluded minors would be esssentially useless.

The cross-polytope, $\bf O_{n}$, is the oriented matroid of the affine
dependencies of the set
$E=\\{e_{1},\ldots,e_{n}\\}\cup\\{-e_{1},\ldots,-e_{n}\\}$ of points of
$\mathbb{R}^{n}$, where $e_{i}:=$i-th vector of the canonical basis of
$\mathbb{R}^{n}$. The cross-polytope is a polytope obtained by a sequence of
series extensions of a point with $n$-elements. It is a graphic matroid. In
this paper we describe the relationship between combinatorial cubes and
adjoints of the cross-polytope. The main results are: Theorem 3.1 to 3.3.

Roughly speaking, Theorem 3.1 identifies oriented cubes and adjoints of the
cross-polytope. Theorems 3.2. and 3.3 establish that the real cube is, up to
isomorphism, the unique cube whose underlying matroid is representable over
$\mathbb{R}$ and whose orientation is representable over $\mathbb{R}$.

The results are organized in two sections. The next section, section 2, is
devoted to general properties of the adjoints and the extension lattice of the
cross-polytope. The main result is Theorem 2.1 which yelds that, up to
isomorphism $\bf O_{n}$ has a unique representable adjoint. A concise
description of this canonical adjoint of the cross-polytope is given in
Proposition 2.1. Section 3 is used in the proof of the main Theorems.

We assume that the reader is acquainted with matroids and oriented matroids.
We suggest [9] as a reference for matroid theory and [2] for oriented
matroids.

We shall be using oriented matroid standard definitions and notation of [2].

Notation. Given a set $E$, signed subsets or sign vectors,
$X\in\\{+,-,0\\}^{E}$, of $E$ will be denoted as ordered pairs
$X=(X^{+},X^{-})$ where $X^{+}:=\\{e\in E:X(e)=+\\}$, $X^{-}:=\\{e\in
E:X(e)=-\\}$ and $X^{0}:=\\{e\in E:X(e)=0\\}$. The support of $X$ is
$\underline{X}=X^{+}\cup X^{-}$ and $X$ is said to be a signed set
complementary to $X^{0}$.

For every $A\subseteq E$, $X(A)$ denotes the restriction of $X$ to $A$.

Given an oriented matroid ${\cal M}={\cal M}(E)$, its underlying matroid will
be denoted $\underline{\cal M}$.

Given $E\subset\mathbb{R}^{n}$ , ${\cal L}in(E)$ and ${\cal A}ff(E)$, denote,
respectively, the oriented matroids of the linear and of the affine
dependencies of $E$ over $\mathbb{R}$.

##  2 cross-polytopes and their adjoints

###  2.1 The oriented matroid $\bf O_{n}$

The n-cross-polytope, denoted $\bf O_{n}$, is the oriented matroid of affine
dependencies of the set of $2n$-points of the real affine space defined by
$E:=\\{e_{1},\ldots,\ e_{n}\\}\cup\\{-e_{1},\ldots,\ -e_{n}\\}$ where
$e_{i}:=$i-th vector of the canonical basis of $\mathbb{R}^{n}$.

In order to describe explicitly this oriented matroid $\bf O_{n}$ we need some
notation.

We denote by $[n]$ the set $[n]:=\\{1,\ldots,n\\}$ and by
$[n]^{\prime}:=\\{1^{\prime},\ldots,n^{\prime}\\}$ a copy of $[n]$. Subsets of
$[n]$ will be denoted $A,B,\ldots$ and subsets $[n]^{\prime}$ by
$A^{\prime},B^{\prime},\ldots$.

For every subset $A\subseteq[n]$, the set $A^{\prime}:=\\{i^{\prime}:i\in
A\\}$ is the corresponding subset of $[n]^{\prime}$. For every subset
$A^{\prime}\subseteq[n]^{\prime}$, the set $A:=\\{i:i^{\prime}\in
A^{\prime}\\}$ is the corresponding subset of $[n]$.

###### Definition 2.1

The n-cross-polytope is the oriented matroid ${\bf
O_{n}}=O_{n}(([n]\cup[n]^{\prime}),{\cal C})$ whose family of signed circuits
${\cal C}$ is defined as:

${\cal C}=\\{\pm R_{ij}=(\\{i,j\\},\\{i^{\prime},j^{\prime}\\})\ 1<i<j<n\\}.$
$None$

$\bf O_{n}$ is an oriented matroid of rank $n+1$. Its hyperplanes are the
subsets of $[n]\cup[n]^{\prime}$ of the following two types:

type 1) the LV-facets $H_{A}=A\cup([n]\setminus A)^{\prime},\ \
A\subseteq[n]$.

type 2) the interior hyperplanes $H_{ii^{\prime}}=([n]\setminus
i)\cup([n]^{\prime}\setminus i^{\prime}),\ i=1,\ldots n$.

The family $\cal D$ of signed cocircuits of ${\bf O_{n}}$ are the following
signed subsets on $[n]\cup[n]^{\prime}$, the complements of the two types of
hyperplanes:

type 1) the positive set $X_{A^{\prime}}=(([n]\setminus A)\cup
A^{\prime},\emptyset)$ and its opposite, for every $A\subset[n]$.

type 2) $Y_{i}=(\\{i\\},\\{i^{\prime}\\})$ and its opposite, for every
$i\in[n]$.

${\cal D}:=\\{\pm Y_{i}\\}_{i\in[n]}\cup\\{\pm
X_{A^{\prime}}\\}_{A\subseteq[n]}$ $None$

Actually, the oriented matroid ${\bf O_{n}}$ is a graphic matroid and
therefore it has a unique class of orientations [3].

###  2.2 The canonical adjoint $\tilde{\bf O}_{\bf n}$ of $\bf O_{n}$

Let ${\cal M}=(E,{\cal D})$, be an oriented matroid and $\cal D$ its family of
signed cocircuits. The adjoints and the extension lattice of $\cal M$ are
usually described in terms of signed vectors over a subset ${\cal
D}_{1}\subseteq{\cal D}$ satisfying the condition: ${\cal D}={\cal
D}_{1}\uplus-{\cal D}_{1}$ (see [2] for the general definitions).

We describe and consider adjoints and the extension lattice of the cross-
polytope with respect to the family ${\cal D}_{1}$ defined by:

${\cal D}_{1}:=\\{Y_{i}\\}_{i\in[n]}\cup\\{X_{A^{\prime}}\\}_{A\subseteq[n]}$

An (oriented) adjoint of ${\bf O_{n}}$, with respect to ${\cal D}_{1}$, is an
oriented matroid $\tilde{\cal M}=\tilde{\cal M}({\cal D}_{1})$, over the set
${\cal D}_{1}$, satisfying the following two conditions:

(1) $r(\tilde{\cal M}({\cal D}_{1}))=r({\bf O_{n}})=n+1.$

(2) The family $\tilde{\cal D}$ of signed cocircuits of $\tilde{\cal M}$
contains the following $2n$ signed subsets of ${\cal D}_{1}$:

$\tilde{\cal X}[i]=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]\setminus i}\cup
Y_{i},\emptyset),\ \forall\ i\in[n]$ and

$\tilde{\cal X}[i^{\prime}]=(\\{X_{A^{\prime}\cup
i^{\prime}}\\}_{A\subseteq[n]\setminus i},Y_{i}),\ \forall\
i^{\prime}\in[n]^{\prime}$.

Remark. In what follows we shall denote by $\tilde{Y}$, $\tilde{H}_{i}$,
$\tilde{H}_{i^{\prime}}$ the following subsets of ${\cal D}_{1}$:

$\tilde{Y}:=\\{Y_{1},\ldots,Y_{n}\\}$, $\tilde{H}_{i}:={\cal
X}[i]^{0}=\\{X_{A^{\prime}+i^{\prime}}\\}_{A\subseteq[n]\setminus
i}\cup(\tilde{Y}\setminus Y_{i})$ and $\tilde{H}_{i^{\prime}}:={\cal
X}[i^{\prime}]^{0}=\\{X_{A^{\prime}}\\}_{A\subseteq[n]\setminus
i}\cup(\tilde{Y}\setminus Y_{i})$.

By definition of adjoint $\tilde{H}_{i}$ and $\tilde{H}_{i^{\prime}}$ are
hyperplanes of every adjoint. Actually (Proposition 2.2.2) $\tilde{Y}$ is also
a hyperplane of every adjoint of $\bf O_{n}$.

The compositions of signed cocircuits $\pm\tilde{X}[e],\
e\in[n]\cup[n^{\prime}]$ are the principal covectors of an(any) adjoint of
$\bf O_{n}$.

There are oriented matroids with no adjoint, however every realizable oriented
matroid has at least one realizable adjoint.

Notice that in general, different realizations may produce non-isomorphic
realizable adjoints.

We use the canonical realization of the cross-polytope to define a canonical
adjoint of the cross-polytope. We then prove, Theorem 2.1, that, up to
isomorphism, this is the unique realizable adjoint of $\bf O_{n}$.

###### Definition 2.2

( Canonical representation of $\bf O_{n}$ and canonical adjoint $\tilde{\bf
O}_{\bf n}$)

Let $E_{n}=(e_{1},\ldots,e_{n})$ be the canonical basis of $\mathbb{R}^{n}$
and consider the subsets $V_{n}:=\\{(e_{i},1)\\}_{i\in[n]}\ \ and\ \
V^{\prime}_{n}:=\\{(-e_{i},1)\\}_{i\in[n]}$ of $\mathbb{R}^{n+1}$. Clearly

${\bf O_{n}}:=Aff(E_{n}\cup-E_{n})\simeq Lin(V_{n}\cup V^{\prime}_{n})$

The two types of hyperplanes of the oriented matroid $Lin(V\cup V^{\prime})$
are defined analytically by:

1) For every $i=1,\ldots,n$ , $H_{ii^{\prime}}:(e_{i},0)\cdot(x,x_{n+1})=0$.

2) For every $A\subseteq[n]$, denoting by $v_{A}$ the vector of
$\mathbb{R}^{n}$ defined by $v_{A}(i)=-1\ for\ i\in A$ and $v_{A}(i)=1\ for\
i\in[n]\setminus A$,

$H_{A}:(v_{A},1)\cdot(x,x_{n+1})=0$.

Consider
$\tilde{V}:=\\{(e_{i},0)\\}_{i\in[n]}\cup\\{(v_{A},1)\\}_{A\subseteq[n]}\subseteq\mathbb{R}^{n+1}$.
Let $f:\tilde{V}\longrightarrow{\cal D}_{1}$ be the bijection defined by
$f((e_{i},0))=Y_{i},\ \forall i\in[n]$ and $f((v_{A},0))=X_{A^{\prime}},\
\forall A^{\prime}\subseteq[n]$. The canonical adjoint $\tilde{\bf
O_{n}}=\tilde{\bf O}_{\bf n}({\cal D}_{1})$, of $\bf O_{n}$ with respect to
${\cal D}_{1}$, is the oriented matroid over ${\cal D}_{1}$ obtained as image
of $Lin(\tilde{V})$ by the correspondence $f$.

The next Proposition gives a useful characterizations of the canonical adjoint
that will be used latter.

###### Proposition 2.1

(Affine representations of $\tilde{\bf O}_{\bf n}$ of ${\bf O_{n}}$) Let
$\tilde{\bf O}$ be the canonical adjoint of the cross-polytope $\bf O_{n}$
with respect to ${\cal D}_{1}$.

Let $B=(y_{1},\ldots,y_{n},b)$ be an affine basis of $\mathbb{R}^{n}$ with a
distinguished vertex $b$. For every subset $A\subseteq[n]$ denote by
$b_{A}:=\frac{1}{|A|+1}(b+\sum_{i\in A}y_{i})$ the barycenter of the face
$F_{A}:=conv(b\cup\\{y_{i}\\}_{i\in A})$ of the simplex $conv(B)$.

1\. The canonical adjoint $\tilde{\bf O_{n}}$ of $\bf O_{n}$ is isomorphic,
via the correspondence $f:{\cal D}_{1}\longrightarrow\mathbb{R}^{n}$ defined
by $f(Y_{i})=y_{i}$, $f(X_{A^{\prime}})=b_{[n]\setminus A}$, to the acyclic
oriented matroid $Aff(\\{y_{i}\\}_{i\in[n]}\cup\\{b_{A}\\}_{A\subseteq[n]})$.

2\. For every $c\in int(conv(B))$ there is a unique affine representation
$f_{c}:{\cal D}_{1}\longrightarrow\mathbb{R}^{n}$ of $\tilde{\bf O_{n}}$
satisfying the following condition:

$f_{c}(y_{i})=y_{i},\ \ f_{c}(X_{[n]^{\prime}})=b\ \ and\ \
f_{c}(X_{\emptyset})=c.$

The representation $f_{c}$ is extended to every $X_{A^{\prime}}$,
$A\subseteq[n]$, defining $f_{c}(X_{A\textasciiacute})=S_{A}\cap
F_{[n]\setminus A}$, where $F_{[n]\setminus
A}:=conv(\\{y_{i}\\}_{i\in[n]\setminus A}\cup b)$, and $S_{A}$ is the affine
space of $\mathbb{R}^{n}$ defined by $S_{A}:=aff(\\{y_{i}\\}_{i\in A}\cup c)$.

We will denote by $B(Y_{1},\ldots,Y_{n},X_{[n]})$ any affine representation of
$\tilde{\bf O_{n}}$ defined by an affine basis $B=(y_{1},\ldots y_{n},b)$ and
any choice of the interior point $c\in int(conv(B))$ that represents
$X_{\emptyset}$

Proof. Consider the canonical representation of $\tilde{\bf O_{n}}\simeq
Lin(\tilde{V})$ where
$\tilde{V}:=\\{(e_{i},0)\\}_{i\in[n]}\cup\\{(v_{A},1)\\}_{A\subseteq[n]}\subseteq\mathbb{R}^{n+1}$.

1) To obtain a represention of $\tilde{\bf O}_{\bf n}$ described in 1),
consider the hyperplane $\tilde{H}$ of $\mathbb{R}^{n+1}$ defined by:

${H}:x_{1}+\ldots+x_{n}+(n+2)x_{n+1}=2.$

For every $v\in\tilde{V}$ let $v^{\prime}:=\lambda^{+}v\cap\tilde{H}$ be the
intersection of the positive halfline $\lambda^{+}v,\
\lambda^{+}\in\mathbb{R}^{+}$ with the hyperplane ${H}$. Consider

$\tilde{V}^{\prime}:=\\{v^{\prime}\in{H}:\ v\in{\cal V}\\}.$

By definition of $\tilde{V}^{\prime}$, $\tilde{\bf O}_{\bf
n}:=Lin(\tilde{V})\simeq Aff(\tilde{V}^{\prime})$.

We leave it to the reader to verify that the points
$y_{i}=(2e_{i},0)=\lambda^{+}(e_{i},0)\cap{H}$ and
$b=v^{\prime}_{[n]}:=(v_{[n]},1)=(-1_{[n]},1)$ form an affine basis of the
hyperplane $(H)$ and that for every $A\subset[n]$, the halfline
$\lambda^{+}(v_{A},1)$ intersects the hyperplane ${H}$ in the point
$b_{A}=\frac{1}{n-|A|+1}(b+\sum_{i\in[n]\setminus A}y_{i})$, yelding the
result.

2) To obtain a representation of $\tilde{\bf O_{n}}$ with $c=\beta
b+{\sum_{i=1}}^{n}\alpha_{i}y_{i}$ with $\beta,\alpha_{i}>0$ and
$\beta+{\sum_{i=1}}^{n}\alpha_{i}=1$, consider the hyperplane

$H_{c}:\ \alpha_{1}x_{1}+\ldots+\alpha_{n}x_{n}+(\beta+1)x_{n+1}=2\beta$

As before it is straightforward to prove that for every $v\in{\cal V}$ the
positive halfline $\lambda^{+}v$ intersects $H_{c}$. Taking
$B=(y_{1},\ldots,y_{n},b)$ with $y_{i}=\frac{2\beta}{\alpha_{i}}e_{i},\
i=1,\ldots,n$ , $b=(v_{[n]},0)$ and $c=\beta
b+{\sum_{i=1}}^{n}\alpha_{i}y_{i}=\beta(v_{\emptyset},1)$. We conclude from 1)
that the point $b_{c_{A}}:=\lambda^{+}(v_{A},1)\cap H_{c}$ that satisfies
$b_{c_{A}}:=\lambda^{+}(v_{A},1)\cap H_{c}=S_{A}\cap F_{[n]\setminus A}$.

###  2.3 Extension lattice and properties of the adjoints of ${\bf O_{n}}$

A single element extension of a matroid ${\cal M}={\cal M}(E)$ is a matroid
${\cal M}\cup p$, over the set $E\cup p$, with the same rank as $\cal M$ such
that $({\cal M}\cup p)\setminus p={\cal M}$.

Single element extensions of an oriented matroid are described by
localizations in ${\cal M}$. If $\cal D$ is the family of signed cocircuits of
$\cal M$ and ${\cal D}_{1}\subset\cal D$ is such that ${\cal D}={\cal
D}_{1}\uplus-{\cal D}_{1}$, a localization in $\cal M$ with respect to ${\cal
D}_{1}$ is a signed subset $\tilde{\cal E}=(\tilde{\cal E}^{+},\tilde{\cal
E}^{-})\in\\{+,-,0\\}^{{\cal D}_{1}}$ that defines the position of the new
point $p$ with respect to the hyperplanes of $\cal M$ in the following way
(for further details see [2]) :

For every $X=(X^{+},X^{-})\in\tilde{\cal E}^{+}$, $(X^{+}\cup p,X^{-})$ is a
signed cocircuit of ${\cal M}\cup p$.

For every $X=(X^{+},X^{-})\in\tilde{\cal E}^{-}$, $(X^{+},X^{-}\cup p)$ is a
signed cocircuit of ${\cal M}\cup p$.

For every $X=(X^{+},X^{-})\in\tilde{\cal E}^{0}$, $(X^{+},X^{-})$ is a signed
cocircuit of ${\cal M}\cup p$.

A sign vector $\tilde{\cal E}=(\tilde{\cal E}^{+},\tilde{\cal
E}^{-})\in\\{+,-,0\\}^{{\cal D}_{1}}$ defines a localization in $\cal M$ if it
defines an extension in every contraction ${\cal M}/L$ by a hyperline $L$.

We recall that given an oriented matroid ${\cal M}=(E,{\cal D})$, $\cal D$
being its family of signed cocircuits, the contraction of $\cal M$ by $L$ is
the (representable) oriented matroid of rank 2, ${\cal M}/L=(E\setminus
L,{\cal D}/L)$, with family of signed cocircuits ${\cal
D}/L:=\\{X=(X^{+},X^{-})\in{\cal D}:\ L\subset X^{0}\\}$. A sign vector
$\tilde{\cal E}=(\tilde{\cal E}^{+},\tilde{\cal E}^{-})\in\\{+,-,0\\}^{{\cal
D}_{1}}$ defines a localization in $\cal M$ if and only if it satisfies the
following condition:

(Loc) For every hyperline $L$ of $\cal M$ the restriction $\tilde{\cal
E}/L=(\tilde{\cal E}/L^{+},\tilde{\cal E}/L^{-})$ of $\cal E$ to ${\cal D}/L$
is orthogonal to every signed triplet of cocircuits of ${\cal D}/L$ of the
following forms: $(\\{X,Y\\},Z)$ such that $X^{+}\cap Y^{+}\subseteq Z^{+}$
and $(\\{X,Y,Z\\},\emptyset)$ such that $X^{+}\cap Y^{+}\subseteq Z^{-}$.

The extension lattice  $\tilde{\cal E}({\cal M})$ of a matroid $\cal M$ is the
poset of localizations with respect to ${\cal D}_{1}$ ordered by the usual
order $\tilde{\cal E}_{1}=(\tilde{\cal E}_{1}^{+},\tilde{\cal
E}_{1}^{-})\preceq\tilde{\cal E}_{2}=(\tilde{\cal E}_{2}^{+},\tilde{\cal
E}_{2}^{-})$ iff $\tilde{\cal E}_{1}^{+}\subseteq\tilde{\cal E}_{2}^{+}$ and
$\tilde{\cal E}_{1}^{-}\subseteq\tilde{\cal E}_{2}^{-}$, endowed with an
element $\hat{1}$.

The next Theorem A of (Theorem 7.5.8 [2] page 311) will be used in the sequel
to obtain properties of the adjoints of the cross-polytope.

Theorem A ( Bachem and Kern [1])

Consider an oriented matroid ${\cal M}=(E,{\cal D})$, $\cal D$ its family of
signed cocicuits. Let ${\cal D}_{1}\subset{\cal D}$ be such that ${\cal
D}={\cal D}_{1}\cup-{\cal D}_{1}$.

If $\tilde{\cal M}=({\cal D}_{1},\tilde{\cal L})$ is an adjoint of $\cal M$,
with family of signed covectors $\tilde{\cal L}$ then every signed covector
$\tilde{V}=(\tilde{V}^{+},\tilde{V}^{-})\in\tilde{\cal L}$ is a localization
in $\mathcal{M}$.

###### Proposition 2.2

(Properties of every adjoint of ${\bf O_{n}}$)

Consider the cross-polytope ${\bf O_{n}}$ and let ${\cal D}_{1}$ be the
subfamily of its signed cocircuits defined by

${\cal D}_{1}:=\\{Y_{i}\\}_{i\in[n]}\cup\\{X_{A^{\prime}}\\}_{A\subseteq[n]}.$

Let $\tilde{\cal M}=\tilde{\cal M}({\cal D}_{1})$ be an adjoint of ${\bf
O_{n}}$ with respect to ${\cal D}_{1}$. Then,

  1. 1.

For every $i\in[n]$ and every $A\subseteq[n]\setminus i$ the signed set
$\tilde{C}(i,A)=(\\{Y_{i},X_{A^{\prime}\cup i^{\prime}}\\},X_{A^{\prime}})$ is
a signed circuit of $\tilde{\cal M}$.

  2. 2.

$\tilde{\cal M}$ is an acyclic oriented matroid whose LV-face lattice is
isomorphic to the face lattice of a simplex with vertices
$Y_{1},\ldots,Y_{n},X_{[n]^{\prime}}$ and whose facets are the hyperplanes
$\tilde{Y}=\\{Y_{1},\ldots,Y_{n}\\}$ and $\tilde{\cal H}_{i}:=X[i]^{0}$ for
every $i\in[n]$.

  3. 3.

Let $\tilde{X}[0]$ denote the set of positive signed cocircuits of
$\tilde{\cal M}$ whose complement is the facet $\tilde{Y}$.
$\tilde{X}[0]:=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]},\emptyset)$ defines a
single element extension ${\bf O_{n}}\cup 0$ of ${\bf O_{n}}$. The oriented
matroids ${\bf O_{n}}\cup 0$ and ${\bf O_{n}}$ have exactly the same number of
hyperplanes and of signed cocircuits.

  4. 4.

For every $i\in[n]$, ${\bf O_{n}}/i\simeq{\bf O_{n}}/i^{\prime}\simeq{\bf
O_{n-1}}\cup 0$ where ${\bf O_{n-1}}\cup 0$ is the extension of $\bf O_{n-1}$
defined in 3).

Moreover,

$\tilde{\cal M}(\tilde{H}_{i})$ is an adjoint of ${\bf O_{n}}/i$ with respect
to ${\cal D}_{1}/i:=\\{X=(X^{+},X^{-})\in{\cal D}_{1}:\ i\in X^{0}\\}$ and

$\tilde{\cal M}(\tilde{H}_{i^{\prime}})$ is an adjoint of ${\bf
O_{n}}/i^{\prime}$ with respect to ${\cal
D}_{1}/i^{\prime}:=\\{X=(X^{+},X^{-})\in{\cal D}_{1}:\ i^{\prime}\in X^{0}\\}$
.

Proof. Let $\tilde{\cal M}=\tilde{\cal M}({\cal D}_{1})$ be an adjoint of the
cross-polytope ${\bf O_{n}}$.

1) The existence of the rank preserving embedding of the poset, $L^{op}$, the
poset of flats of ${\bf O_{n}}$ with the reverse order, into the lattice of
flats of $\tilde{\cal M}$, implies that for every $i\in[n]$ and every
$A\subseteq[n]\setminus i$ the set
$C=\\{Y_{i},X_{A^{\prime}+i},X_{A^{\prime}}\\}$ is a flat and a circuit of
rank 2 of $\tilde{\cal M}$.

On the other hand, by Theorem A every covector of $\tilde{\cal M}$ is a
localization in ${\bf O_{n}}$ and, by definition of localization with respect
to ${\cal D}_{1}$, must be orthogonal to the signature:
$\tilde{C}(i,A)=(\\{Y_{i},X_{A^{\prime}+i}\\},X_{A^{\prime}})$ of $C$,
implying that $\tilde{C}(i,A)$ must be one of the two opposite signed circuits
of $\cal M$ with support $C$.

For every $i\in[n]$ and every $A\subseteq[n]\setminus i$ the three signed
cocircuits $Y_{i},X_{A^{\prime}},\ X_{A^{\prime}+i^{\prime}}$ are the signed
cocircuits of the contraction ${\bf O_{n}}/L$ of $\bf O_{n}$ by the hyperline
$L=H_{ii^{\prime}}\cap H_{A}\cap H_{A+i}=A\cup([n]\setminus(A+i))^{\prime}$,
proving 1).

2) By definition of adjoint for every $i\in[n]$,
$\tilde{X}[i]:=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]\setminus
i}\cup(\\{Y_{i}\\},\emptyset)$ is a positive signed cocircuit of $\tilde{\cal
M}$ and consequently $\tilde{H}_{i}:=\tilde{X}[i]^{0}$ is a LV-facet of
$\tilde{\cal M}$.

The circuits $\tilde{C}(i,A)$ of $\tilde{\cal M}$ imply that if $\tilde{\cal
M}$ is acyclic than it is a simplex with vertices $Y_{1},\ldots.Y_{n}$ and
$X_{[n]^{\prime}}$. We claim that $\tilde{Y}:=\\{Y_{1},\ldots Y_{n}\\}$ is a
facet of $\tilde{\cal M}$.

To prove the claim we use the axioms for maximal vectors (topes) of an
oriented matroid [11] to conclude that the signed set
$\tilde{X}[0]:=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]},\emptyset)$,
complementary of $\tilde{Y}$, is a positive cocircuit of $\tilde{\cal M}$.

Let $\tilde{\cal T}$ denote the family of maximal covectors (topes) of the
adjoint $\tilde{\cal M}$ of $\bf O_{n}$, consider the family $\tilde{\cal P}$
of principal maximal covectors defined by: $\tilde{\cal
P}:=\\{\tilde{X}[e_{1}]\circ...\circ\tilde{X}[e_{n}]:\
e_{i}\in\\{i,i^{\prime}\\},\ \forall i\in[n]\\}$. Taking the restriction of
(all the vectors of) $\tilde{\cal P}$ to $\tilde{Y}$ we conclude:

$\tilde{\cal P}(\tilde{Y})=\\{+,-\\}^{\tilde{Y}}$ $None$

This equality implies ([11] ) that $\tilde{Y}$ is an independent set of
$\tilde{\cal M}$.

On the other hand, the restriction to ${\cal D}_{1}\setminus\tilde{Y}$ of
every vector of $\tilde{\cal P}$ is the same vector:

$\tilde{X}[0]:=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]},\emptyset)$ $None$

From $(1)$ and $(2)$ we conclude that $\tilde{X}[0]\circ\tilde{\cal
T}\subseteq\tilde{\cal T}$ implying that $\tilde{X}[0]$ is a positive covector
of $\tilde{\cal M}$ with $\tilde{Y}$ as the complementary flat. Since
$\tilde{Y}$ is independent $r(\tilde{Y})=r(\tilde{\cal M})-1$ implying
$\tilde{Y}$ is actually a facet of $\tilde{\cal M}$.

3) We concluded in 2) that the positive sign vector $\tilde{X}[0]$ is a
cocircuit of $\tilde{\cal M}$ and therefore, by Theorem A, a localization in
$\bf O_{n}$, with respect to ${\cal D}_{1}$. The single element extension
${\bf O_{n}}\cup 0$ of ${\bf O_{n}}$ obtained by localizing the new element
$0$ according to $\tilde{X}[0]$ contains as signed cocircuits the signed-
vectors of the following set:

$\overline{\cal D}_{1}:=\\{X_{A^{\prime}}\cup 0=(A^{\prime}\cup([n]\setminus
A)\cup
0,\emptyset)\\}_{A\subseteq[n]}\cup\\{Y_{i}=(i,i^{\prime})\\}_{i\in[n]}.$

We leave to the reader the verification that for every hyperpline $L$ of $\bf
O_{n}$ the hyperplane of ${\bf O_{n}}\cup 0$ spanned by $L\cup 0$ always
contains a hyperplane of $\bf O_{n}$, implying that the family $\overline{\cal
D}$ is actually the family of signed cocircuits of the extension ${\bf
O_{n}}\cup 0$, proving 3).

4) For every $i\in[n]$, the isomorphisms ${\bf O_{n}}/i\simeq{\bf
O_{n}}/i^{\prime}\simeq{\bf O_{n-1}}\cup 0$ follow directly from the
definition of the cross-polytope $\bf O_{n}$, the details are left to the
reader.

The fact that $\tilde{\cal M}(\tilde{H}_{i})$ is an adjoint of ${\bf O_{n}}/i$
with respect to ${\cal D}_{1}/i$ is a consequence of the hypothesis that
$\tilde{\cal M}$ is an adjoint of ${\bf O_{n}}$ with respect to ${\cal D}_{1}$
and the fact that the family of signed cocircuits of ${\bf O_{n}}/i$ is ${\cal
D}/i:=\\{X=(X^{+},X^{-})\in{\cal D}:\ i\in X^{0}\\}={\cal D}_{1}/i\cup-{\cal
D}_{1}/i=\tilde{H}_{i}\cup-\tilde{H}_{i}$.

The case $\tilde{\cal M}(\tilde{H}_{i^{\prime}})$ is similiar.

###### Theorem 2.1

The canonical oriented adjoint $\tilde{\bf O}_{\bf n}$ of the cross-polytope
$\bf O_{n}$ described in Proposition 2.1 is, up to isomorphism, the unique
realizable oriented adjoint, with respect to ${\cal D}_{1}$, of ${\bf O_{n}}$.

Proof. The proof is by induction on $n$. For $n=2$ the result is trivially
true. Assume that the result is true for $k<n$ and consider a realizable
adjoint $\tilde{\cal M}=\tilde{\cal M}({\cal D}_{1})$ of $\bf O_{n}$. By
definition ${\cal D}_{1}=\tilde{H}_{n}\cup\tilde{H}_{n^{\prime}}\cup Y_{n}$
and by induction hypothesis, and Proposition 2.2-3) and 4), the restriction of
$\cal M$ to each one of the hyperplanes $\tilde{H}_{n}=\\{X_{A^{\prime}\cup
n^{\prime}}\\}_{A\subseteq[n-1]}\cup(\tilde{Y}\setminus Y_{n})$ and
$\tilde{H}_{n^{\prime}}=\\{X_{A^{\prime}}\\}_{A\subseteq[n-1]}\cup(\tilde{Y}\setminus
Y_{n})$ is isomorphic to the canonical adjoint of $\bf O_{n-1}$.

By Proposition 2.2.1) and 2) $\cal M$ is acyclic with vertices $Y_{1},\ldots
Y_{n}$ and $X_{[n]^{\prime}}$. Let $f:{\cal
D}_{1}\longrightarrow\mathbb{R}^{n}$ be an affine representation of $\cal M$.
Consider $f(Y_{i})=y_{i}$, $f(X_{[n]^{\prime}})=b$ and $f(X_{\emptyset})=c$.
$f(\tilde{H}_{n})=B(Y_{1},\ldots,Y_{n-1},X_{[n]^{\prime}})$ is contained in
the affine hyperplane $H=aff(y_{1}\ldots,y_{n-1},b)$ of $\mathbb{R}^{n}$. The
point $f(X_{n})\in relintconv(y_{1},\ldots,y_{n-1},b)$ and
$B=(y_{1},\ldots,y_{n},b)$ is an affine basis of $\mathbb{R}^{n}$.

From proposition 2.2.1 we know that $c$ must be a point in the relative
interior of the line segment $y_{n}f(X_{n})$, therefore a point in the
interior of $conv(B)$ and $f(\tilde{H}_{n^{\prime}}$ is contained in the
affine hyperplane $H^{\prime}=aff(y_{1},\ldots,y_{n},c)$. Then, for each
$i\in[n]$ the circuit $\tilde{C}(i,\emptyset)$ defines the point $f(X_{i})$,
as the intersection point $f(X_{i})=aff(y_{i},c)\cap
conv(\\{y_{k}\\}_{k\in[n]\setminus i}\cup b)=aff(y_{i},c)\cap
conv(f(\tilde{H}_{i}))$. By Proposition 2.1 $f(\tilde{H}_{i})$ is completely
determined by the affine base $B\setminus y_{i}$ and by $c_{i}$, yelding that
for every $A\subset[n]$ $f_{(}X_{A^{\prime}})=F_{[n]\setminus A}\cap S_{A}$
and $\tilde{\cal M}=B(Y_{1},\ldots,Y_{n},X_{[n]^{\prime}})=\tilde{\bf O}_{\bf
n}$.

##  3 Cubes, adjoints and extension spaces of the cross-polytope

We start by recalling the definitions of cubes and of oriented cubes from
[10]:

Notation. We consider $C^{n}:=\\{-1,1\\}^{[n]}$.

For every $A\subseteq[n]$, $v_{A}$ denotes the vector of $C^{n}$ defined by:
$v_{A}(i)=-1\ for\ i\in A$ and $v_{A}(i)=1\ for\ i\notin A$. It is sometimes
useful to write $v_{A}=(-1_{A},1_{[n]\setminus A})$.

Given $B\subseteq[n]$ and $v\in C^{n}$, ${}_{-B}v$ denotes the vector of
$C^{n}$ whose coordinates are the opposite of those in $v$ in the entries
indexed by $B$, i. e. ${}_{-B}v(i)=-v(i)\ for\ i\in B$ and ${}_{-B}v(i)=v(i)\
for\ i\notin B$.

Notice that ${}_{-B}v_{A}=v_{A\Delta B}$ where $A\Delta B:=A\setminus B\cup
B\setminus A$.

Given a subset $V\subseteq C^{n}$ we denote by ${}_{-A}V$ the subset of
$C^{n}$ defined by: ${}_{-A}V:=\\{_{-A}v\in C^{n}:\ v\in V\\}$.

The real cube is the oriented matroid ${\bf Q_{n}}=Aff(C^{n})$ of the affine
dependencies of $C^{n}$ over the reals.

Notice that the circuits with smallest number of elements of the real cube are
the signed rectangles $R(v;A,B)=(\\{v,_{-(A\uplus
B)}v\\},\\{_{-A}v,_{-B}v\\}),\ with\ A\cap B=\emptyset$. We denote by $\cal R$
the family of all the signed rectangles of $\bf Q_{n}$.

A facet of the real cube is a hyperplane $H_{i+}:=\\{v\in C^{n}:\ v(i)=1\\}$
or $H_{i-}:=\\{v\in C^{n}:\ v(i)=-1\\}$ and a skew-facet is a hyperplane
$H_{ij+}:=\\{v\in C^{n}:\ v(j)=v(i)\\}$ or $H_{ij-}:=\\{v\in C^{n}:\
v(j)=-v(i)\\}$. We denote by $\cal F$ the signed cocircuits of the real cube
complementary to the facets and by $\overline{\cal F}$ the signed cocircuits
complementary of the facets and to the skew-facets. $\overline{\cal F}$ is the
collection of signed cocircuits of $\bf Q_{n}$ with smallest cardinality.

###### Definition 3.1

( Cubes and Oriented Cubes [10])

A cube over $C^{n}$ is a matroid $M=M(C^{n})$ satisfying the following two
conditions:

C1) every (unsigned) rectangle of $C^{n}$ is a circuit of $M$

C2) every facet and skew facet of $C^{n}$ is an (unsigned) cocircuit and a
hyperplane of $C^{n}$.

Cubes may or may not be orientable. Orientable cubes have a canonical
orientation ${\cal M}(C^{n})$ that satisfies the following equivalent
conditions:

C1) all the signed rectangles of $\bf Q_{n}$ are circuits of $\cal M$.

C2) all the signed cocircuits of $\bf Q_{n}$ that are facets and skew-facets
are signed cocircuits of $\cal M$.

An oriented cube is a cube isomorphic to an oriented cube over $C^{n}$, ${\cal
M}(C^{n})$, canonically oriented.

Remark. Note that to conclude that an oriented matroid ${\cal M}(C^{n})$ is an
oriented cube one only has to verify the following conditions:

C1) every signed rectangle is a circuit.

C’2) the facets are hyperplanes of $\cal M$.

###### Proposition 3.1

Let $\tilde{\cal M}=\tilde{\cal M}({\cal D}_{1})$ be an adjoint of the cross-
polytope ${\bf O_{n}}$ with respect to ${\cal
D}_{1}:=\\{Y_{i}\\}_{i\in[n]}\cup\\{X_{A^{\prime}}\\}_{A\subseteq[n]}$. Let
${\cal D}_{1}^{+}$ be the subset of ${\cal D}_{1}$ defined by ${\cal
D}_{1}^{+}:=\\{X_{A^{\prime}}\\}_{A\subseteq[n]}$ of ${\cal D}_{1}$.

The restriction $\tilde{\cal M}({\cal D}_{1}^{+})$ of $\tilde{\cal M}$ to
${\cal D}_{1}^{+}$ is an oriented cube.

Proof. Let $\tilde{\cal M}^{\prime}:=\tilde{\cal M}({\cal D}_{1}^{+})$ be the
restriction of an adjoint of ${\bf O_{n}}$ to ${\cal D}_{1}^{+}$. We identify
${\cal D}_{1}^{+}$ with $C^{n}$ via the bijection that to every
$X_{A^{\prime}},\ A\subseteq[n]$ assigns the vector $v_{A}\in C^{n}$ and prove
that ${\cal M}({\cal D}_{1}^{+})$ satisfies the conditions $C1)$ and
$C^{\prime}2)$ of the above Remark.

C’2) is a direct consequence of the definitions of adjoint and of restriction.
Notice that for every $i\in[n]$ the restriction of the signed cocircuits
$\tilde{X}[i]$ and $\tilde{X}[i^{\prime}]$ of $\tilde{\cal M}$ to ${\cal
D}_{1}^{+}$ are the positive cocircuits of $\tilde{\cal M}^{\prime}$:

$\tilde{X}[i]({\cal D}_{1}^{+})=(\\{X_{A^{\prime}}\\}_{A\subseteq[n]\setminus
i},\emptyset)$ and $\tilde{X}[i]({\cal D}_{1}^{+})=(\\{X_{A^{\prime}\cup
i^{\prime}}\\}_{A\subseteq[n]\setminus i},\emptyset)$

whose complementary sets in ${\cal D}_{1}^{+}$, are the LV-facets of
$\tilde{\cal M}^{\prime}$ :

$H_{i^{+}}:=\tilde{H_{i}}({\cal D}_{1}^{+})=\tilde{X}[i]^{0}({\cal
D}_{1}^{+})$ and $H_{i-}:=\tilde{H}_{i^{\prime}}({\cal
D}_{1}^{+})=\tilde{X}[i]^{0}({\cal D}_{1}^{+})$.

The lattice of faces of $\tilde{\cal M}^{\prime}$ is therefore isomorphic to
the lattice of faces of $\bf Q_{n}$.

C1) We prove that every signed rectangle $(\\{{X},_{-(A\uplus
B)}{X}\\},\\{_{-A}{X},_{-B}{X}\\})$ is a circuit of $\tilde{\cal M}^{\prime}$,
where for every $X\in{\cal D}_{1}^{+}$, $X=X_{C^{\prime}}$ for some
$C\subset[n]$, and every $D\subset[n]$, ${}_{-D}X=X_{(C\Delta D)^{\prime}}$.
The proof is by induction on $n$.

For $n=2$ it is trivial, in this case $\tilde{\cal M}$ must be the oriented
matroid represented in Figure 1 and clearly $\tilde{\cal M}^{\prime}$ is a
rectangle, a cube of rank $3$.

Figure 1:  $\tilde{O}_{2}$, the unique adjoint of $\bf O_{2}$

Assume that C1) is true for every adjoint $\tilde{\cal M}$ of $\bf O_{n-1}$.

Consider an adjoint $\tilde{\cal M}$ of $\bf O_{n}$, the restrictions
$\tilde{\cal M}(\tilde{H}_{n})$ and $\tilde{\cal M}(\tilde{H}_{n^{\prime}})$
are isomorphic to adjoints of $\bf O_{n-1}$ (Proposition 2.2.4)), therefore
all the rectangles contained in the facets $\tilde{H}_{n+}$ and of
$\tilde{H}_{n-}$ of $\tilde{\cal M}^{\prime}$ are rectangles of $\tilde{\cal
M}^{\prime}$ and we are left with proving that the signed rectangles of the
form $R(X;A,B\uplus n)=(\\{X,_{-(A\uplus B\uplus n)}X\\},\\{_{-A}X,_{-(B\cup
n)}X\\})$, with $X\in{\cal D}_{1}^{+}$, $A\uplus B\subseteq[n-1]$ are signed
circuits of $\tilde{\cal M}^{\prime}$. This will be done by induction on the
cardinality of $B$ assuming, without loss of generality, that $n\in X^{+}$,
i.e. $X$ belongs to the facet $\tilde{H}_{n+}$ of $\tilde{\cal M}^{\prime}$

When $|B|=0$, every rectangle $R(X;A,n)=(\\{X,_{(A\cup
n}X\\},\\{_{-A}X,_{-n}X\\})$ with $A\subset[n-1]$ is a circuit of $\tilde{\cal
M}^{\prime}$.

In fact, by Proposition 2.2.1), since $n\in X^{+}$ the signed circuits
$C=(\\{Y_{n},_{-n}X\\},X)$ and $D=(\\{Y_{n},_{-(A\cup n)}X\\},_{-A}X)$ are
circuits of $\tilde{\cal M}$. Eliminating $Y_{n}$ between $-C$ and $D$ we
conclude that $R(X;A,n)$ is a circuit of $\tilde{\cal M}^{\prime}$.

Assume that $R(X;A,B\cup n)$ is a circuit of $\tilde{\cal M}^{\prime}$ for
$|B|=k$ and consider a rectangle $R_{B}:=R(X;A,B\cup n)$ with $|B|=k+1$.

Consider $B_{b}=B\setminus b$, for some $b\in B$. The rectangle
$R(X;A,B_{b}\cup n)=(\\{X,_{-(A\cup B_{b}\cup n)}X\\},\\{_{-A}X,_{-(B_{b}\cup
n)}X\\})$ is a circuit of $\tilde{\cal M}$.

Consider the rectangle $S=(\\{_{-(B_{b}\cup n)}X,_{-(A\cup B\cup
n)}X\\},\\{_{-(B\cup n)}X\\},_{-(A\cup B_{b}\cup n)}X\\})$ contained in the
hyperplane $H_{n^{\prime}}$ of $\tilde{\cal M}$ and therefore a rectangle of
$\tilde{\cal M}^{\prime}$. Eliminating ${}_{-(B_{b}\cup n)}X$ between
$R(X;A,B_{b}\cup n)$ and $S$ we conclude that $R_{B}$ must be a signed
rectangle of $\tilde{\cal M}^{\prime}$, proving the Lemma.

The next Theorem describes completely the relation between oriented cubes and
adjoints of the cross-polytope.

###### Theorem 3.1

Consider $\tilde{\cal O}_{n}$ the class of all oriented adjoints of the cross-
polytope $\bf O_{n}$ with respect to ${\cal
D}_{1}:=\\{Y_{i}\\}_{i\in[n]}\cup\\{X_{A^{\prime}}\\}_{A\subseteq[n]}$ and
$\tilde{\cal C}_{n}$ the class of all oriented cubes over ${\cal
D}_{1}^{+}:=\\{X_{A^{\prime}}\\}_{A\subseteq[n]}$.

The correspondence $f:\tilde{\cal O}_{n}\longrightarrow\tilde{\cal C}_{n}$
defined by $f(\tilde{\cal M})=\tilde{\cal M}({\cal D}_{1}^{+})$, that assigns
to every oriented adjoint $\tilde{\cal M}\in\tilde{\cal O}_{n}$ its
restriction to ${\cal D}_{1}^{+}$, is a bijection.

The proof of Theorem 3.1 requires proving that every oriented cube can be
extended, in a unique way, to an oriented matroid isomorphic to an adjoint of
the cross-polytope. In order to do so we need some preliminary lemmas.

###### Lemma 3.1

(A property of flats and covectors of cubes)

Consider an oriented cube ${\cal M}={\cal M}(C^{n})$ and a flat $F$ of $\cal
M$.

For every $i\in[n]$ let ${}_{-i}F:=\\{_{-i}v:\ v\in F\\}$. Then,

1) Either $F\cap\ _{-i}F=\emptyset$ or $F=\ _{-i}F$.

2) Let $V=(V^{+},V^{-})$ be a vector of $\cal M$ such that $V^{0}=F$.

(i) if $F\cap\ _{-i}F=\emptyset$, define $F+:=F\cap H_{i^{+}}$ and $F-:=F\cap
H_{i^{-}}$, the following holds: either ${}_{-i}(F+)\subseteq V^{+}$ and
${}_{-i}(F-)\subseteq V^{-}$ or ${}_{-i}(F+)\subseteq V^{-}$ and
${}_{-i}(F-)\subseteq V^{+}$.

(ii) if $F=_{-i}F$ then $V(H_{i^{-}})=_{-i}V(H_{i^{+}})$.

Proof. 1) We prove that if $F\cap_{-i}F\not=\emptyset$ then $F=_{-i}F$: assume
that there is some element $v\in F\cap\ _{-i}F$, i.e. $v,_{-i}v\in F$. If for
some $A\subseteq[n]\setminus i$, ${}_{-A}v\in F$, also ${}_{-(A\cup i)}v$ must
belong to $F$ since the rectangle $(\\{v,_{-(A\cup
i)}v\\},\\{_{-i}v,_{-A}v\\})$ is a circuit of $\cal M$. 1) is proved.

2) Assume that $V=(V^{+},V^{-})$ is a covector of $\cal M$ complementary to
the flat $F$. In both cases we use orthogonality between covectors and the
rectangles of the cube $\cal M$ to obtain the result:

(i) if $F\cap\ _{-i}F=\emptyset$, then for every $v,v^{\prime}\in F+$
orthogonality of $V$ with the rectangle
$(\\{v,_{-i}v^{\prime}\\},\\{_{-i}v,v^{\prime}\\})$ implies that
${}_{-i}v,_{-i}v^{\prime}\in_{-i}(F+)$ must have the same sign in $V$. For
every $v\in F+$ and $w\in F-,\ w=_{-(A\cup i)}v$ for some $\
A\subseteq[n]\setminus i$, orthogonality of $V$ with the rectangle
$(\\{v,w\\},\\{_{-i}v,_{-i}w\\})$ implies that ${}_{-i}v\in_{-i}(F+)$ and
${}_{-i}w\in_{-i}(F-)$ must have different signs in $V$. The result then,
follows.

###### Definition 3.2

Flats parallel and anti-parallel to $i$

Let $F$ be a flat of an oriented cube ${\cal M}={\cal M}(C^{n})$.

The flat $F$ is parallel to $i$ if it satisfies the condition: $F=_{-i}F$.

The flat $F$ is anti-parallel to $i$ if it satisfies the condition:
$F\cap_{-i}F=\emptyset$.

###### Lemma 3.2

(Extensions by the point at infinity of a class of parallel edges of a cube)
Consider an oriented cube ${\cal M}=(C^{n},{\cal D})$, ${\cal D}$ its family
of signed cocircuits. For every $i\in[n]$ define

${\cal Y}_{i}:=\\{Y=(Y^{+},Y^{-})\in{\cal D}:Y^{0}$ is a hyperplane parallel
to $i$ $\\}$

${\cal X}_{i}:=\\{X=(X^{+},X^{-})\in{\cal D}:X^{0}=(F+)\uplus(F-)$ is a
hyperplane anti-parallel to $i$ and $\ {}_{-i}(F-)\subseteq X^{+}\
and_{-i}(F+)\subseteq X^{-}\\}$, where $(F+):=X^{0}\cap H_{i^{+}}$ and
$(F-):=X^{0}\cap H_{i^{-}}$

Then

1\. $({\cal X}_{i},{\cal Y}_{i})$ is a localization in $\cal M$ defining a
single element extension ${\cal M}\cup y_{i}$ of $\cal M$.

2\. $\cal M$ and its single element extension ${\cal M}\cup y_{i}$ have the
same number of hyperplanes and signed cocircuits.

The new element $y_{i}$ will be called the point at infinity of direction $i$.

Proof. From the definitions of ${\cal X}_{i}$ and ${\cal Y}_{i}$ it is clear
that ${\cal D}={\cal X}_{i}\uplus-{\cal X}_{i}\uplus{\cal Y}_{i}$.

For 1. we have to prove that for every hyperpline $L$ of ${\cal M}$, $({\cal
X}_{i}\uplus{\cal Y}_{i})/L$ is a localization in the rank 2 realizable
oriented matroid ${\cal M}/L$.

Let $L$ be a hyperline of ${\cal M}$.

Case 1) If $L$ is parallel to $i$ then, by Lemma 3.1, every hyperplane
containing $L$ is also parallel to $i$ and in this case $({\cal
X}_{i}\uplus{\cal Y}_{i})/L={\cal Y}_{i}/L={\cal D}/L$, and $y_{i}$ is a loop
in ${\cal M}/L$.

Case 2) If $L$ is anti-parallel to $i$ then $L\cap\ _{-i}L=\emptyset$.
Consider $v\in L$ and the hyperplane $H=cl_{\cal M}(L\cup_{-i}v),\ v\in L$.
Since for every $w\in L\setminus v$, the set $\\{v,w,_{-i}v,_{-i}w\\}$ is a
circuit of $\cal M$ we conclude that $H=L\cup\ _{-i}L$ and therefore $H$ is
the unique hyperplane parallel to $i$ that contains $L$. Moreover, for every
hyperplane anti-parallel to $i$, containing $L$, its complementary signed
cocircuit $X=(X^{+},X^{-})$ in ${\cal X}_{i}$ is the signed cocircuit that
satisfies the condition ${}_{-i}(L-)\subseteq X^{+}$ and ${}_{-i}(L+)\subseteq
X^{-}$, where $(L+):=L\cap H_{i^{+}}$ and $(L-):=L\cap H_{i^{-}}$. The
localization of the new point $y_{i}$ in ${\cal M}/L$ is represented in the
next figure, clearly a localization in ${\cal M}/L$.

Figure 2:  Localization of $y_{i}$ in ${\cal M}/L$

Now consider 2. By definition of the extension ${\cal M}\cup y_{i}$, for every
$X=(X^{+},X^{-})\in{\cal X}_{i}$ the signed set $X+{y_{i}}=(X^{+}\cup
y_{i},X^{-})$ is a signed cocircuit of ${\cal M}\cup y_{i}$ and its complement
of $(X+{y_{i}})^{0}=X^{0}$ is a hyperplane of both $\cal M$ and ${\cal M}\cup
y_{i}$. For every pair of opposite signed cocircuits $\pm
Y=(Y^{+},Y^{-})\in{\cal Y}_{i}$, its complement, $Y^{0}$, in $C^{n}$ is a
hyperplane of $\cal M$ and in $C^{n}\cup y_{i}$, $Y^{0}\cup y_{i}$, is a
hyperplane of ${\cal M}\cup y_{i}$.

We claim that ${\cal M}\cup y_{i}$ has no other hyperplanes. Any other
hyperplane of ${\cal M}\cup y_{i}$ would be of the form $L\cup y_{i}$ for some
hyperline $L$ of ${\cal M}$. There is no such hyperplane if $L$ is parallel to
$i$, because in this case $y_{i}\in L$. On the other hand if $L$ is anti-
parallel to $i$ the unique hyperplane of ${\cal M}\cup y_{i}$ that contains
$y_{i}$ is the hyperplane $H=L\cup_{-i}L\cup y_{i}$ and $H\setminus y_{i}$ is
an hyperplane of $\cal M$. The result is proved.

Remark. Notice that by definition of the localization $({\cal X}_{i},{\cal
Y}_{i})$ the extension ${\cal M}\cup y_{i}$ contains as hyperplanes all the
sets $H_{j^{+}}\cup y_{i}$, $H_{j^{-}}\cup y_{i}$ for $j\not=i$ implying that
for every $v\in C^{n}$ the set $\\{v,_{-i}v,y_{i}\\}$ is a circuit and a flat
of ${\cal M}\cup y_{i}$. Moreover, supposing that $v\in H_{i^{+}}$ the
signature of this circuit must be $\pm C$ where
$C=(\\{_{-i}v,y_{i}\\},\\{v\\})$ since for every hyperline $L$ intersection of
a facet $H_{j^{\epsilon}}$ with $H_{i}^{+}$ , $y_{i}$ is placed in every
hyperplane containing $L$, opposite to ${}_{-i}L=H_{j^{\epsilon}}\cap
H_{i^{-}}$.

Proof of Theorem 3.1. In Proposition 3.2 we proved that the restriction to
${\cal D}_{1}^{+}$ of any adjoint of ${\bf O_{n}}$, with respect to ${\cal
D}_{1}$ is an oriented cube.

In order to prove the theorem we consider an oriented cube, ${\cal M}={\cal
M}(C^{n})$ and prove that it can be extended, in a unique way, to an oriented
matroid isomorphic to an adjoint of the cross-polytope.

By Lemma 3.2 ${\cal M}$, admits a single element extension by a new point
$y_{i}$, the point at infinity in direction $i$.

To conclude the proof of the Theorem we have to prove that the $n$ single-
element extensions $y_{1},\ldots,y_{n}$ of the cube $\cal M$ are compatible,
resulting in a unique oriented matroid ${\cal
M}(C^{n}\cup\\{y_{1},\ldots,y_{n}\\})$ that is isomorphic to an adjoint of
${\bf O_{n}}$.

Since no single-element extension by a point $y_{i}$ creates new hyperplanes
(Lemma 3.2), we can perform successively, in any order, the $n$ elementary
extensions by elements $y_{i}$. Note, that the description of the
localizations defining the extension of the oriented matroid ${\cal
M}(C^{n}\cup\\{y_{i_{1}},y_{i_{2}},\ldots,y_{i_{k}}\\}),\ k<n$ by
$y_{i_{k+1}}$ is exactly the same localization that describes the single
element extension of ${\cal M}(C^{n})$ by $y_{i_{k+1}}$.

When the $n-th$ extension by an element $y_{i_{n}}$ is completed, independenly
of the order, we obtain the same matroid ${\cal
M}(C^{n}\cup\\{y_{1},\ldots,y_{n}\\})$.

Consider the bijection
$f:C^{n}\cup\\{y_{1},\ldots,y_{n}\\}\longrightarrow{\cal D}_{1}$ defined for
every $i\in[n]$ by $f(y_{i})=Y_{i}=(\\{i\\},\\{i^{\prime}\\})$, and for every
$A\subseteq[n]$ by $f(v_{A})=X_{A^{\prime}}$. By construction, the image
$f({\cal M}(C^{n}\cup\\{y_{1},\ldots,y_{n}\\}))$ of the extended matroid
${\cal M}(C^{n}\cup\\{y_{1},\ldots,y_{n}\\})$ is an adjoint of the cross-
polytope ${\bf O_{n}}$ with respect to ${\cal D}_{1}$.

###### Theorem 3.2

If ${\cal M}(C^{n})$ is a realizable oriented cube then ${\cal
M}(C^{n})\simeq{\bf Q_{n}}$

To prove the theorem we prove that every realization ${\cal A}ff(W)$ of $\cal
M$ can be extended to a realizable adjoint of the cross-polytope $\bf O_{n}$.
Then Theorem 2.1 ensures that ${\cal M}\simeq{\bf Q_{n}}$.

We use the next two lemmas (actually from lemma 3.4 we only need (i)).

###### Lemma 3.3

( in Theorem 1 of [10] ) Let ${\cal M}={\cal M}(C^{n})$ be an oriented cube.
Then for every $i\in[n]$ the reorientation of ${\cal M}$ obtained by reversing
signs on a facet $H_{i^{-}}$ is an oriented cube ${}_{-H_{i^{-}}}{\cal
M}=_{-H_{i^{-}}}{\cal M}(C^{n})$ whose facets are the facets $H_{i-}$,
$H_{i+}$ and the skew-facets $H_{ij+},H_{ij-}$, $j\in[n]\setminus i$, of $\cal
M$ .

Proof. The proof is straightforward from the definitions and is left to the
reader.

###### Lemma 3.4

Let ${\cal M}={\cal M}(C^{n})$ be a realizable oriented cube of rank $n+1$.
Let $W\subset\mathbb{R}^{n}$ be a realization of $\cal M$ such that ${\cal
M}\simeq{\mathcal{A}}ff(W)$ via the bijection $\sigma:C^{n}\longrightarrow W$,
$\sigma(v_{A})=w_{A}$ for every $A\subseteq[n]$. Then the following two
conditions are satisfied for $n\geq 2$:

(i) All the line segments $w_{A}w_{[n]\setminus A}$ with $A\subseteq[n]$ meet
in the same point $O\in\mathbb{R}^{n}$. The point $O$ is a point in the
interior of the polytope $conv(W)$ and will be called the center of (the
realization) $W$.

(ii) For $n\geq 3$, let $O_{i^{+}}$, $O_{i^{-}}\in\mathbb{R}$ denote the
centers of the facets $H_{i^{+}},H_{i^{-}}$ of $W$, let ${S}:=\\{O_{i^{+}}$,
$O_{i^{-}}:\ i\in[n]\\}$. Then the oriented matroid $\mathcal{A}ff(S)$ is a
cross-polytope - a polar of the the polytope $conv(W)$.

Notation. To simplify notation we use the same notation $H_{i^{+}}$, $H_{i-}$,
$H_{ij+}$, $H_{ij-}$ for the facets and skew-facets of ${\cal M}$ and of its
realization. By an opposite pair of points of the realization $W$ is meant any
pair of points the form $w_{A}$, $w_{[n]\setminus A}$.

Proof. (i) Let $w_{A}$, $w_{B}$ and $w_{C}$, $A,B,C\subseteq[n-1]$, be three
distinct points of the facet $H_{n^{+}}$ of ${\cal M}_{W}:={\cal A}ff(W)$. The
opposite points $w_{[n]\setminus A}$, $w_{[n]\setminus B}$ and
$w_{[n]\setminus C}$ are points of the facet $H_{n^{-}}$. The signed rectangle
$C(A,B)=(\\{w_{A},w_{[n]\setminus A}\\},\\{w_{B},w_{[n]\setminus B}\\})$ is a
circuit of ${\mathcal{M}}_{W}$ and therefore the four points are the vertices
of a quadrilateral whose diagonals $w_{A}w_{[n]\setminus A}$ and
$w_{B}w_{[n]\setminus B}$ intersect in a point $O\in
relint(w_{A}w_{[n]\setminus A})\cap relint(w_{B}w_{[n]\setminus B})$.

Similarly, considering the circuits $C(A,C)=(\\{w_{A},w_{[n]\setminus
A}\\},\\{w_{C},w_{[n]\setminus C}\\})$ and $C(B,C)=(\\{w_{B},w_{[n]\setminus
B}\\},\\{w_{C},w_{[n]\setminus C}\\})$ we conclude that the diagonal
$w_{C}w_{[n]\setminus C}$ of both quadrilaterals must intersect the other
diagonal. Let $O_{A}$ be the intersection point of the diagonals of $C(A,C)$
and $O_{B}$ the intersection point of the diagonals of $C(B,C)$ (Figure 3)

Figure 3:

To conclude the proof of (i) we claim that : $O_{A}=O_{B}=O.$

In fact, if $O_{A}\not=O_{B}$ then the line segment $w_{C}w_{[n]\setminus C}$
would be contained in the affine plane $\pi:=aff(C(A,B))$. On the other hand,
since no cube contains three colinear points , the plane $\pi$ would be the
affine plane spanned by the points $w_{A},w_{B},w_{C}$. But, since $n\geq 3$,
this affine plane is contained in the facet $H_{n^{+}}$ and therefore it
cannot contain points of $H_{n^{-}}$, otherwise $rank({\cal M})<n+1$. But this
contradicts the assumption that $C(A,B)\subset\pi$. We conclude then that
$O_{A}=O_{B}$ and, consequently, it must be $O_{A}=O_{B}=O$.

By definition $O$ is in the relative interior of every line segment connecting
opposite vertices of $W$. It is clearly a point of the convex polytope
$conv(W)$ which is in none of its facets, therefore $O\in int(conv(W))$.

(ii) We prove that $O\in relint(O_{n^{+}}O_{n^{-}})$, where $O_{n^{+}}$,
$O_{n^{-}}$ are the centers of the facets $H_{n+}$, $H_{n-}$ of ${\cal
M}_{W}$. The other cases are similar.

Let $w_{A},w_{B}$ be two distinct points of the facet $H_{n^{+}}$ such that
$B\not=A,[n-1]\setminus A$. This is possible because $n\geq 3$.

Consider the circuits $C=(\\{w_{A}w_{[n-1]\setminus
A}\\},\\{w_{B}w_{[n-1]\setminus B}\\})$ contained in $H_{n^{+}}$ and
$C^{\prime}=(\\{w_{A}w_{[n]\setminus A}\\},\\{w_{B}w_{[n]\setminus B}\\})$
contained in $H_{n^{-}}$. By definition
$O_{n^{+}}:=relint(w_{A}w_{[n-1]\setminus A})\cap
relint(w_{B}w_{[n-1]\setminus B})$ and $O_{n^{-}}:=relint(w_{A\cup
n}w_{[n]\setminus A})\cap relint(w_{B\cup n}w_{[n]\setminus B})$.

Consider now the signed rectangles of $W$ defined as:

$D=(\\{w_{A},w_{[n]\setminus A}\\},\\{w_{[n-1]\setminus A},w_{A\cup n}\\})$
and $D^{\prime}=(\\{w_{B},w_{[n]\setminus B}\\},\\{w_{[n-1]\setminus
B},w_{B\cup n}\\})$.

Denote by $conv(D),conv(D^{\prime})$ the quadrilaterals of $\mathbb{R}^{n}$
whose vertices are the vertices of the circuits $D$ and $D^{\prime}$. The line
segment $O_{n^{+}}O_{n^{-}}=conv(D)\cap conv(D^{\prime})$ (figure 4).

Figure 4:

On the other hand since, by definition, the center $O$ is the intersection of
the diagonals of the $D$ and $D^{\prime}$, it follows that $O\in
relintO_{n^{+}}O_{n^{-}}$.

In a similar way we conclude that $O\in relintO_{i^{+}}O_{i^{-}}$, for every
$i\in[n]$, implying that $\forall\ i<j\leq n$
$C_{ij}=(\\{O_{i^{+}},O_{i^{-}}\\},\\{O_{j^{+}},O_{j^{-}}\\})$ is a circuit of
the matroid $\mathcal{A}ff(S)$, $S=\\{O_{i^{+}},O_{i^{-}}:i\in[n]\\}$. Since
the points $O_{i^{+}}$ and $O_{i^{-}}$ lie in the relative interior of
opposite facets of the convex polytope $conv(W)$ that has dimension $n$, the
$n$ vectors $O_{i^{-}}O_{i^{+}}$ form a basis of $\mathbb{R}^{n}$implying that
for every $i\in[n]$, $S\setminus\\{O_{i^{+}},O_{i^{-}}\\}$ must be a
hyperplane of $\mathcal{A}ff(S)$. This oriented matroid is an n-cross-
polytope.

Proof of Theorem 3.2

Let ${\cal M}_{W}={\cal A}ff(W)$ be an affine realization in $\mathbb{R}^{n}$
of a (realizable) oriented cube ${\cal M}={\cal M}(C^{n})$ via the
correspondence $v_{A}\longrightarrow w_{A}\in\mathbb{R}^{n},\ \forall
A\subseteq[n]$. By Lemma 3.3 the reorientation of $\cal M$, ${}_{-H_{i-}}{\cal
M}$, obtained by reversing signs on the facet $H_{i-}$ is an oriented cube
whose facets are the 2n-hyperplanes of ${\cal M}$: $H_{i^{+}}$, $H_{i^{-}}$,
$H_{ij+}$, $H_{ij-}$, for every $j\in[n]\setminus i$.

Considering $\mathbb{R}^{n}$ as a subset of the real projective space
$\mathbb{P}_{n}(\mathbb{R})$, representations of the oriented cube
${}_{-H_{i-}}{\cal M}$ can be obtained as images $f_{i}(W)$ of $W$ under
suitable projective transformation
$f_{i}:\mathbb{P}_{n}(\mathbb{R})\longrightarrow\mathbb{P}_{n}(\mathbb{R})$
admissible for $W$. In every such representation $f_{i}(W)$ of
${}_{-H_{i-}}{\cal M}$, the images, $f_{i}(w_{A})$, $f_{i}(w_{A\cup i})$ of
the extremities of the edges $w_{A}w_{A\cup i}$ of $W$ are opposite points of
$f_{i}(W)$. By lemma 3.4 all of the line segments $f_{i}(w_{A})f_{i}(w_{A\cup
i})$ meet in the center $C_{i}$ of $f_{i}(W)$. Therefore all the edges of
${\cal A}ff(W)$ meet in the point
$f_{i}^{-1}(C_{i})\in\mathbb{P}_{n}(\mathbb{R})$. This shows that every
realization ${\cal A}ff(W)$ of $\cal M$ can be extended to a realizable
adjoint of the cross-polytope. By Theorem 2.1 there is a unique realizable
adjoint of the cross-polytope. By Theorem 3.1 ${\cal A}ff(W)$ must be
isomorphic to the real affine cube ${\cal A}ff(C^{n})$.

Remark. Actually a suitable projective transformation $f_{i}$ can be
constructed in the following way, using the notation of Lemma 3.3: identify,
as usual, $\mathbb{R}^{n}$ with
$\mathbb{R}^{n}\times{1}\subseteq\mathbb{R}^{n+1}$ in such a way that the
center $O_{W}$ of $W$ has coordinates $({\bf 0},1)$. Let $\pi_{i}$ be an
affine plane of $\mathbb{R}^{n+1}$ cutting the line segment $O_{W}O_{i^{+}}$
in such a way that it separates the points of the facet $H_{i^{-}}$ and the
origin of $\mathbb{R}^{n}$ from the points of $H_{i^{+}}$. The projection
$f_{i}(W)$ from the origin $O\in\mathbb{R}^{n+1}$ of $W$ in the plane $\pi$ is
a representation of ${}_{-H_{i-}}{\cal M}$. Let $C_{i}$ be the center of
$f_{i}(W)$. The point $f_{i}^{-1}(C_{i})$ of the real projective space
$\mathbb{P}_{n}(\mathbb{R})$ is the meeting point of all the edges
$w_{A}w_{A\cup i}$ of $W$.

The next Theorem 3.3. is a nonoriented version of Theorem 3.2.

###### Theorem 3.3

Let ${\cal M}(C^{n})$ be an oriented cube whose underlying matroid
$\underline{\cal M}(C^{n})$ is realizable. Then $\underline{\cal
M}(C^{n})\simeq\underline{\bf Q_{n}}$.

The proof of this Theorem is an easy consequence of Theorem 3.2 and of the
following Theorem B from [10]:

Theorem B( Corollary 1.3) of [10]) Let $M(C^{n})$ be an oriented cube whose
underlying matroid is realizable. Then every class of orientations of $\cal M$
contains a unique acyclic orientation whose signed rectangles and signed
cocircuits complementary of the facets and skew-facets are signed as in ${\cal
A}ff(C^{n})$.

Proof of theorem 3.3. Let $W\subseteq\mathbb{R}^{n}$ be a realization of the
underlying matroid. By lemma 3.4 we may assume without loss of generality that
${\cal A}ff(W)$ is a realizable oriented cube. Theorem 3.2 then ensures that
${\cal A}ff(W)\simeq{\bf Q_{n}}$ and the result follows.

##  4 Final Remarks

From Theorem 3.2 we know that if an oriented cube other than the real cube
exists, it must be non-realizable. Moreover, from Theorem 3.3, we know that it
may be non-realizable either because it is a non-realizable orientation of the
real affine cube, or because its underlying matroid is a cube that is non-
representable over $\mathbb{R}$.

The following conjectures play a fundamental role in this context:

Conjecture 1. (M. Las Vergnas [7]): The real affine cube $\bf Q_{n}$ has a
unique class of orientations. Equivalently, the canonical adjoint ${\tilde{\bf
O}}_{\bf n}$ of the cross-polytope $\bf O_{n}$ has a unique class of
orientations.

Conjecture 2. (I. P. Silva [10]): The real affine cube $\bf Q_{n}$ is the
unique oriented cube. Equivalently, the canonical adjoint ${\tilde{\bf
O}}_{\bf n}$ of the cross-polytope is the unique adjoint of the cross-polytope
$\bf O_{n}$.

Several constructions and examples of (unoriented) cubes non-representable
over $\mathbb{R}$ are given [10]. All the known examples are non-orientable
matroids. New families of minor-minimal excluded minors for orientability
based on extensions of the cross-polytope are described in [12].

## References

  * [1] A. Bachem, W. Kern, Adjoints of oriented matroids, Combinatorica, 6(1986), 299-308.
  * [2] Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, Encyclopedia of Math. Appl., vol. 46, Cambridge University Press, 1993.
  * [3] R. Bland, M. Las Vergnas, Orientability of matroids, J. Combinatorial Theory, Ser. B, 24 (1978), 94-123.
  * [4] R. Campbell, J. Geelen, On the complex-representable excluded minors for real-representability, Preprint 2018.
  * [5] M. Las Vergnas, Convexity in Oriented Matroids, J. Combinatorial Theory, Ser. B, 29(1980), 231-243.
  * [6] J. Folkman, J. Lawrence, Oriented Matroids, J. Combinatorial Theory, Ser. B, 25 (1978), 199-236.
  * [7] M. Las Vergnas, J.P. Roudneff, I. Salaűn, Regular polytopes, preprint 1989 (unpublished).
  * [8] E.Mayhew, M. Newman, G. Whittle, On excluded minors for real representability, J. Combinatorial Theory, Ser.B, 99(2009), 685-689.
  * [9] J. Oxley,Matroid Theory, 2nd edition, Oxford Science Publications 2014.
  * [10] I.P.F. da Silva, Cubes and orientability, Discrete Mathematics, 308(2008), 3574-3585.
  * [11] I.P.F. da Silva, Axioms for maximal vectors of an oriented matroid, a combinatorial characterization for the regions fo an arrangement of pseudohyperplanes, European J. of Combinatorics, 16(1995), 125-145.
  * [12] I.P.F. da Silva, On minor-minimal non orientable matroids with 2n-elements and rank n, European J. of Combinatorics, 30(2009),1825-1832.