New Examples of Oriented Matroids with Disconnected Realization Spaces
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New Examples of Oriented Matroids with Disconnected Realization Spaces Yasuyuki Tsukamoto
Received: 3 February 2012 / Revised: 15 July 2012 / Accepted: 31 August 2012 / Published online: 12 October 2012 © Springer Science+Business Media, LLC 2012
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. Keywords Oriented matroids · Realization space
1 Oriented Matroids and Matrices Throughout this section, we fix positive integers r and n. Let X = (x1 , . . . , xn ) ∈ Rrn 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 XX can be defined as XX : E r → {−1, 0, +1},
XX (i1 , . . . , ir ) := sgn det(xi1 , . . . , xir ).
The map XX is called the chirotope of X. The chirotope XX encodes the information regarding the combinatorial type, which is called the oriented matroid of X. In this case, the oriented matroid determined by XX is of rank r on E. We note some properties which the chirotope XX of a matrix X satisfies. 1. XX is not identically zero. 2. XX is alternating, i.e. XX (iσ (1) , . . . , iσ (r) ) = sgn(σ )XX (i1 , . . . , ir ) for all i1 , . . . , ir ∈ E and all permutations σ . 3. For all i1 , . . . , ir , j1 , . . . , jr ∈ E such that XX (jk , i2 , . . . , ir ) · XX (j1 , . . . , jk−1 , i1 , jk+1 , . . . , jr ) ≥ 0 for k = 1, . . . , r, we have XX (i1 , . . . , ir ) · XX (j1 , . . . , jr ) ≥ 0. Y. Tsukamoto () Department of Human Coexistence, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan e-mail: [email protected]
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Discrete Comput Geom (2013) 49:287–295
The third property follows from the identity det(x1 , . . . , xr ) · det(y1 , . . . , yr ) r det(yk , x2 , . . . , xr ) · det(y1 , . . . , yk−1 , x1 , yk+1 , . . . , yr ), = k=1
for all x1 , . . . , xr , y1 , . . . , yr ∈ Rr . 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 M(E, χ) for an oriented matroid which is on the set E and is defined by the chirotope χ . An oriented matroid M(E, χ) is called realizable or constructible, if there exists a matrix X such that χ = XX . 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 M = M(E, χ) is a matrix X such that XX = χ or XX = −χ . Two realizations X, X of M are called linearly equivalent, if there exists a linear transformation A ∈ GL(r, R) such that X = AX. Here we have the equation XX = sgn(det A) · XX . Definition 1.2 The realization space R(M) of an oriented matroid M is the set of all linearly equivalent classes of realizations of M, in the quotient topology induced fr
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