New Examples of Oriented Matroids with Disconnected Realization Spaces

Abstract

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller point-sets than the known examples with this property. Moreover, we construct one on 13 points whose realization space is a connected but non-irreducible semialgebraic variety.

1 Oriented Matroids and Matrices

Throughout this section, we fix positive integers r and n.

Let X=(x ₁,…,x _n )∈ℝ^rn be a real (r,n) matrix of rank r, and E={1,…,n} be the set of labels of the columns of X. For such a matrix X, a map $\mathcal{X}_{X}$ can be defined as

$$\mathcal{X}_X:E^r\rightarrow \{-1,0,+1\},\quad \mathcal{X}_X(i_1, \ldots ,i_r):=\mathrm{sgn} \det (x_{i_1},\ldots ,x_{i_r}). $$

The map $\mathcal{X}_{X}$ is called the chirotope of X. The chirotope $\mathcal{X}_{X}$ encodes the information regarding the combinatorial type, which is called the oriented matroid of X. In this case, the oriented matroid determined by $\mathcal{X}_{X}$ is of rank r on E.

We note some properties which the chirotope $\mathcal{X}_{X}$ of a matrix X satisfies.

• 1. $\mathcal{X}_{X}$ is not identically zero.

• 2. $\mathcal{X}_{X}$ is alternating, i.e. $\mathcal{X}_{X}(i_{\sigma(1)},\ldots,i_{\sigma(r)}) =\mathrm{sgn} (\sigma) \mathcal{X}_{X}(i_{1},\ldots,i_{r})$ for all i ₁,…,i _r ∈E and all permutations σ.

• 3. For all i ₁,…,i _r ,j ₁,…,j _r ∈E such that $\mathcal{X}_{X} (j_{k},i_{2},\ldots ,i_{r})\cdot \mathcal{X}_{X} (j_{1},\ldots, j_{k-1},i_{1}, j_{k+1},\ldots, j_{r})\geq 0$ for k=1,…,r, we have $\mathcal{X}_{X}(i_{1},\ldots,i_{r})\cdot \mathcal{X}_{X}(j_{1},\ldots,j_{r}) \geq 0$.

The third property follows from the identity

Generally, an oriented matroid of rank r on E (n points) is defined by a map χ:E ^r→{−1,0,+1}, which satisfies the above three properties ([1]). The map χ is also called the chirotope of an oriented matroid. We use the notation $\mathcal {M}(E,\chi )$ for an oriented matroid which is on the set E and is defined by the chirotope χ.

An oriented matroid $\mathcal {M}(E, \chi )$ is called realizable or constructible, if there exists a matrix X such that $\chi=\mathcal{X}_{X}$. Not all oriented matroids are realizable, but we do not consider the non-realizable case in this paper.

Definition 1.1

A realization of an oriented matroid $\mathcal {M}=\mathcal {M}(E, \chi )$ is a matrix X such that $\mathcal{X}_{X}=\chi$ or $\mathcal{X}_{X}=-\chi$.

Two realizations X,X′ of $\mathcal {M}$ are called linearly equivalent, if there exists a linear transformation A∈GL(r,ℝ) such that X′=AX. Here we have the equation $\mathcal{X}_{X'}=\mathrm{sgn} (\det A)\cdot \mathcal{X}_{X}$.

Definition 1.2

The realization space $\mathcal {R}(\mathcal {M})$ of an oriented matroid $\mathcal {M}$ is the set of all linearly equivalent classes of realizations of $\mathcal {M}$, in the quotient topology induced from ℝ^rn.

Our motivation is as follows: In 1956, Ringel asked whether the realization spaces $\mathcal {R}(\mathcal {M})$ are necessarily connected [6]. It is known that every oriented matroid on less than nine points has a contractible realization space. In 1988, Mnëv showed that $\mathcal {R}(\mathcal {M})$ can be homotopy equivalent to an arbitrary semialgebraic variety [3]. His result implies that they can have arbitrary complicated topological types. In particular, there exist oriented matroids with disconnected realization spaces. Suvorov and Richter-Gebert constructed such examples of oriented matroids of rank 3 on 14 points, in 1988 and in 1996, respectively [5, 7]. However, it is unknown which is the smallest number of points on which oriented matroids can have disconnected realization spaces. See [1] for more historical comments.

One of the main results of this paper is the following.

Theorem 1.3

There exist oriented matroids of rank 3 on 13 points whose realization spaces are disconnected.

Let d and p be positive integers. The solution of a finite number of polynomial equations and polynomial strict inequalities with integer coefficients on ℝ^d is called an elementary semialgebraic set.

Let f ₁,…,f _p ∈ℤ[v ₁,…,v _d ] be polynomial functions on ℝ^d, and V⊂ℝ^d be an elementary semialgebraic set. For a p-tuple ε=(ε ₁,…,ε _p )∈{−,0,+}^p, let

$$V_{\varepsilon}:=\bigl\{ v\in V\mid \mathrm{sgn} \bigl( f_i(v) \bigr)=\varepsilon_i \ \mbox{for}\ i=1,\ldots, p \bigr\} $$

denote the corresponding subset of V. The collection of the elementary semialgebraic sets $(V_{\varepsilon})_{\varepsilon\in \{-,0,+\}^{p}}$ is called a partition of V.

In the case r=3, a triple (i,j,k)∈E ³ is called a basis of χ if χ(i,j,k)≠0. Let B=(i,j,k) be a basis of χ such that χ(B)=+1. The realization space of an oriented matroid $\mathcal {M}= \mathcal {M}(E,\chi)$ of rank 3 can be given by an elementary semialgebraic set

$$\mathcal {R}(\mathcal {M},B):=\bigl\{ X \in \mathbb {R}^{3n}\mid x_i=e_1,\ x_j=e_2,\ x_k=e_3,\ \mathcal{X}_X=\chi \bigr\}, $$

where e ₁,e ₂,e ₃ are the fundamental vectors of ℝ³. For another choice of basis B′ of χ, we have a rational isomorphism between $\mathcal {R}(\mathcal {M},B)$ and $\mathcal {R}(\mathcal {M},B^{\prime})$. Therefore, realization spaces of oriented matroids are semialgebraic varieties.

The universal partition theorem states that, for every partition $(V_{\varepsilon})_{\varepsilon \in \{-,0,+\}^{p}}$ of ℝ^d, there exists a family of oriented matroids $(\mathcal {M}^{\varepsilon})_{\varepsilon \in \{-,0,+\}^{p}}$ such that the collection of their realization spaces with a common basis $(\mathcal {R}(\mathcal {M}^{\varepsilon},B))_{\varepsilon \in \{-,0,+\}^{p}}$ is stably equivalent to the family $( V_{\varepsilon})_{\varepsilon \in \{-,0,+\}^{p}}$. See [2] or [4] for universal partition theorems.

We construct three oriented matroids $\mathcal {M}^{\varepsilon}$ with ε∈{−,0,+} of rank 3 on 13 points, whose chirotopes differ by a sign on a certain triple. These oriented matroids present a partial oriented matroid with the sign of a single base non-fixed, whose realization space is partitioned by fixing the sign of this base. The two spaces $\mathcal {R}(\mathcal {M}^{-})$ and $\mathcal {R}(\mathcal {M}^{+})$ are disconnected, and $\mathcal {R}(\mathcal {M}^{0})$, which is a wall between the two, is connected but non-irreducible. So we also have the following.

Theorem 1.4

There exists an oriented matroid of rank 3 on 13 points whose realization space is connected but non-irreducible.

Remark 1.5

An oriented matroid $\mathcal {M}(E,\chi)$ is called uniform if it satisfies χ(i ₁,…,i _r )≠0 for all i ₁<⋯<i _r ∈E. Suvorov’s example on 14 points is uniform, and the examples which we construct are non-uniform. It is still unknown whether there exists a uniform oriented matroid on less than 14 points with a disconnected realization space.

2 Construction of the Examples

Throughout this section, we set E={1,…,13}.

Let X(s,t,u) be a real (3,13) matrix with three parameters s,t,u∈ℝ given by

This is a consequence of the computation of the following construction sequence. Both operations “∨” and “∧” can be computed in terms of the standard cross product “×” in ℝ³. The whole construction depends only on the choice of the three parameters s,t,u∈ℝ. We have

We set $X_{0}=X(\frac{1}{2},\frac{1}{2},\frac{1}{3})$. The chirotope χ ^ε is the alternating map such that

where ε∈{−,0,+}.

The oriented matroid which we will study is $\mathcal {M}^{\varepsilon}:=\mathcal {M}(E,\chi^{\varepsilon})$.

Remark 2.1

We can replace X ₀ with $X(\frac{1}{2},\frac{1}{2},u^{\prime})$ where u′ is chosen from $\mathbb {R}\backslash\{-1,0,\allowbreak \frac{1}{2},1,\frac{3}{2},2,3\}$. We will study the case $0<u^{\prime}<\frac{1}{2}$. If we choose u′ otherwise, we can get other oriented matroids with disconnected realization spaces.

In the construction sequence, we need no assumption on the collinearity of x ₉,x ₁₂,x ₁₃. Hence every realization of $\mathcal {M}^{\varepsilon}$ is linearly equivalent to a matrix X(s,t,u) for certain s,t,u, up to multiplication on each column with positive scalar.

Moreover, we have the rational isomorphism

$$\mathcal {R}^\ast \bigl(\chi^\varepsilon \bigr)\times (0,\infty )^{12}\cong \mathcal {R}\bigl(\mathcal {M}^\varepsilon\bigr), $$

where $\mathcal {R}^{\ast}(\chi^{\varepsilon}):= \{ (s,t,u)\in \mathbb {R}^{3}\mid \mathcal{X}_{X(s,t,u)} =\chi^{\varepsilon}\}$. Thus we have only to prove that the set $\mathcal {R}^{\ast}(\chi^{\varepsilon})$ is disconnected (resp. non-irreducible) to show that the realization space $\mathcal {R}(\mathcal {M}^{\varepsilon})$ is disconnected (resp. non-irreducible).

The equation $\mathcal{X}_{X(s,t,u)}=\chi^{\varepsilon}$ means that

$$ \mathrm{sgn} \det(x_i,x_j,x_k)=\chi^\varepsilon (i,j,k),\quad \mbox{for all}\ (i,j,k)\in E^3. $$

(1)

We write some of them which give the equations on the parameters s,t,u. Note that for all (i,j,k)∈E ³({i,j,k}≠{9,12,13}), the sign is given by

$$\chi^\varepsilon (i,j,k)=\mathrm{sgn} \det (x_i,x_j,x_k)|_{s=t=1/2,u=1/3}. $$

From the equation sgndet(x ₂,x ₃,x ₅)=sgn(s)=sgn(1/2)=+1, we get s>0. Similarly, we get det(x ₂,x ₅,x ₄)=1−s>0; therefore,

$$ 0<s<1. $$

(2)

From the equations det(x ₁,x ₇,x ₃)=t>0,det(x ₁,x ₄,x ₇)=1−t>0, we get

$$ 0<t<1. $$

(3)

Moreover, we have the inequalities

(4)

(5)

(6)

(7)

(8)

From the equation det(x ₉,x ₁₂,x ₁₃)=u(1−2s)(1−2t+tu−su), we get

$$ \mathrm{sgn} \bigl(u(1-2s)(1-2t+tu-su)\bigr)=\varepsilon. $$

(9)

Conversely, if we have Eqs. (2)–(9), then we get (1).

We can interpret a (3,13) matrix as the set of vectors {x ₁,…,x ₁₃}⊂ℝ³. After we normalize the last coordinate for x _i  (i∈E∖{1,2,9}), we can visualize the matrix on the affine plane {(x,y,1)∈ℝ³}≅ℝ². Figure 1 shows the affine image of X ₀. See Figs. 2, 3 for realizations of $\mathcal {M}^{\varepsilon}$.

Fig. 1

Column vectors of X ₀

Fig. 2

Realization of $\mathcal {M}^{-}$ (on the left) and that of $\mathcal {M}^{+}$ (on the right)

Fig. 3

Realizations of $\mathcal {M}^{0}$

Proof of Theorem 1.3

We prove that $\mathcal {R}^{\ast}(\chi^{-})$ and $\mathcal {R}^{\ast}(\chi^{+})$ are disconnected. From Eqs. (2)–(9), we obtain

$$\mathcal {R}^\ast \bigl(\chi^- \bigr)= \left\{ (s,t,u)\in \mathbb {R}^3 \,\, \begin{array}{|c} 0<s<1,\,0<u<t<1-u, \\ (1-t)^2-su>0,\ t^2-(1-s)u>0, \\ (1-2s)(1-2t+tu-su)<0 \end{array} \right\}, $$

$$\mathcal {R}^\ast \bigl(\chi^+ \bigr)= \left\{ (s,t,u)\in \mathbb {R}^3 \,\, \begin{array}{|c} 0<s<1,\ 0<u<t<1-u,\\ (1-t)^2-su>0,\ t^2-(1-s)u>0,\\ (1-2s)(1-2t+tu-su)>0 \end{array} \right\}. $$

First, we show that $\mathcal {R}^{\ast}(\chi^{-} )$ is disconnected; more precisely, that it consists of two connected components. We do this by proving the next proposition.

Proposition 2.2

Proof

There are two cases:

Note that

(10)

(11)

(12)

(⊂) For the case 1−2s>0 and 1−2t+tu−su<0, the inequality 2t−1>0 follows from Eq. (10). Since we have 0<s<1/2<t<1, we get

(13)

For the other case 1−2s<0, similarly, we get 1−2t>0 from Eq. (10). Since we have 0<t<1/2<s<1, we get

(14)

(⊃) For the component 0<s<1/2<t<1, the inequalities 1−2t+tu−su<0, (1−t)²−su>0, 1−t−u>0 follow from (13). Thus we get t ²−(1−s)u>0 from Eq. (11). The inequality u<t holds because t>1/2 and u<1−t.

For the other component 0<t<1/2<s<1, similarly, we get the inequalities 1−2t+tu−su>0,t ²−(1−s)u>0,t−u>0 from (14), and (1−t)²−su>0 from Eq. (12). Last, we get u<1−t from t<1/2 and u<t. □

For the set $\mathcal {R}^{\ast}(\chi^{+})$, we have the following proposition.

Proposition 2.3

The proof is similar to that of Proposition 2.2 and is omitted.

Proof of Theorem 1.4

We show that $\mathcal {R}^{\ast}(\chi^{0})$ consists of two irreducible components whose intersection is not empty. From Eqs. (2)–(9), we get

$$\mathcal {R}^\ast \bigl(\chi^0\bigr) = \left\{ (s,t,u)\in \mathbb {R}^3\, \begin{array}{|c} 0<s<1,\,0<u<t<1-u, \\ (1-t)^2-su>0,\ t^2-(1-s)u>0,\\ (1-2s)(1-2t+tu-su)=0 \end{array}\right\}. $$

Here we have the decomposition

The intersection of the two irreducible components is the set

$$\biggl\{(s,t,u)\in \mathbb {R}^3 \bigg| s=t=\frac{1}{2}, 0<u<\frac{1}{2}\biggr\}\cong \biggl\{ X \biggl(\frac{1}{2},\frac{1}{2},u\biggr)\bigg| 0<u<\frac{1}{2}\biggr\}. $$

The proof is also similar to that of Proposition 2.2 and is omitted. □

Figure 3 shows two realizations of $\mathcal {M}^{0}$. On the left, it shows the affine image of $X(\frac{1}{2},\frac{3}{8},\frac{1}{4})$, on the irreducible component 1−2s=0. On the right, it shows the image of $X( \frac{3}{4},\frac{11}{24},\frac{2}{7})$, on the other component 1−2t+tu−su=0. These images can be deformed continuously to each other via $X(\frac{1}{2},\frac{1}{2},u)$ $(0<u<\frac{1}{2})$.

We set

$$\mathcal {R}^\ast :=\left\{(s,t,u)\in \mathbb {R}^3 \left| \begin{array}{c} 0<s<1,\,0<u<t<1-u,\\ (1-t)^2-su>0,\,t^2-(1-s)u>0 \end{array} \right. \right\}. $$

The set $\mathcal {R}^{\ast}\times (0,\infty)^{12}$ is rationally isomorphic to a realization space of a partial oriented matroid with the sign χ(9,12,13) non-fixed. The collection of semialgebraic sets $(\mathcal {R}^{\ast}(\chi^{\varepsilon}))_{\varepsilon \in\{-,0,+\} }$ is a partition of $\mathcal {R}^{\ast}$. Figure 4 illustrates this partition in 3-space.

Fig. 4

$\mathcal {R}^{\ast}$ (on the top) and its partition $(\mathcal {R}^{\ast}(\chi^{\varepsilon}))_{\varepsilon \in \{-,0,+\}}$