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# Class Note for MATH 796 at KU

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Date Created: 02/06/15

Wednesday 326 Oriented Matroids Last time Let A H1l r r Hn be a hyperplane arrangement in Rail Let 1 r r r Zn be af ne linear forms such that H E 6 Rd l 0 for all ii For c 01 Won 6 70 let gt 0 if c F EERdl zi5lt0 if c 7 430 if 570 If F a 0 then it is called a face of A and c CF is the corresponding covectori y A faces of A LQAQA yQA U 0 big face lattice of A ordered byFSF ifFQF Consider the linear forms 139 that were used in representing each face by a covectorr Specifying i is equivalent to specifying a normal vector 17139 to the hyperplane H with i 17139 L As we know the vectors 17 represent a matroid whose lattice of ats is precisely LAl Scaling 17139 equivalently 1 by a nonzero constant A E R has no effect on the matroid represented by the ails but what does it do to the covectors If A gt 0 then nothing happens but if A lt 0 then we have to switch and 7 signs in the ith position of every covectorr So in order to gure out the covectors we need not just the normal vectors 17139 but an orientation for each one Example Lets go back to the two arrangements considered at the start Their regions are labeled by the following covectors Now you should object that the oriented normal vectors are the same in each case Yes but this couldn7t happen if the arrangements were central because two vector subspaces of the same space cannot possibly be parallel In fact if A is a central arrangement then the oriented normals determine yA uniquelyl Proposition The covectors of A are preserved under the operation of negation changing all 7s to 77s and vice versa if and only if A is central In fact the maximal covectors that can be negated are exactly those that correspond to bounded regionsl Example 1 Consider the central arrangement A whose hyperplanes are the zero sets of the linear forms 11y 217y 3172 1yzl The corresponding normal vectors are V 1711 1 l 174 where 171 1710 172 110 173 101 174 0171 The projectivization projA looks like this 7 7 77 74 77 Each region F that borders the equator has a polar opposite 7F such that C7F 70Fl The regions with covectors 7 7 7 and 77 do not border the equator ie they are bounded in proj Since they do not border the equator neither do their opposites in A so those opposites do not occur in MMM 1n the gure of Example 1 consider the point p 2 3 4 That three lines intersect at 17 means that there is a linear dependence among the corresponding normal vectors 172 i 173 174 07 or on the level of linear forms 1 127131401 Of course knowing which subsets of V are linearly dependent is equivalent to knowing the matroid M represented by Vi lndeed 172173174 is a circuit of Ml However 1 tells us more than that there exists no 139 E R3 such that 21 gt 0 31 lt 0 and 41 gt 0 That is A has no covector of the form 7 for any 96 E 7 We say that 07 is the corresponding oriented Circuiti For c E 70 write c7z39 l Q7 c77z39 l c177 De nition Let n be a positive integer A Circuit system for an oriented matroid is a collection 5 of ntuples c E 70 satisfying the following properties 1 000 ltgi 2 If c E if then 7c E if 3 If cc E if and c f c then it is not the case that both 0 C c and c C CL 4 If cc E if and c f c and there is some i with Ci and c 7 then there exists d E 9 with di 0 and for allja i d CCUC F andd Cc UcLi Again the idea is to record not just the linearly dependent subsets of a set 1 i i i Zn of linear forms but also the sign patterns of the corresponding linear dependences or syzygieslli Condition 1 says that the empty set is linearly independent Condition 2 says that multiplying any syzygy by 71 gives a syzygyi Condition 3 as in the de nition of the circuit system of an unoriented matroid must hold if we want circuits to record syzygies with minimal supporti Condition 4 is the oriented version of circuit exchanger Suppose that we have two syzygies n n 7ij 2ij 07 j1 j1 with 71 gt 0 and 75 lt 0 for some ii Multiplying by positive scalars if necessary hence not changing the sign patterns we may assume that 71 77 Then i6ij 0 j1 where Sj 7139 7 In particular 6139 0 and Sj is positive respi negative if and only if at least one of 717 is positive respi negative 0 The set 0 Uc la 6 9 forms a circuit system for an ordinary matroidi 0 Just as every graph gives rise to a matroid any loopless directed graph gives rise to an oriented matroid homework probleml As in the unoriented setting the circuits of an oriented matroid represent minimal obstructions to being a covectori That is for every real hyperplane arrangement A we can construct a circuit system if for an oriented matroid such that if k is a covector of A and c is a circuit then it is not the case that 16 2 0 and K 2 cm More generally we can construct an oriented matroid from any real pseudosphere arrangement ie a col lection of homotopy d 7 lspheres embedded in R such that the intersection of the closures of the spheres in any subcollection is connected or empty Here is an example of a pseudocircle arrangement in R2 394 ln fact7 the Topological Representation Theorem of Folkman and Lawrence 1978 says that every oriented matroid can be realized by such a pseudosphere arrangementi However7 there exist lots of oriented matroids that cannot be realized as hyperplane arrangements A Example Pappus7 Theorem from Euclidean geometry says the following Let a 25 a 12 c be distinct points in R2 such that a 25 and a 12 c are collineari Then the three points z W O m 7 yiac ac 2Wn are collineari o If we perturb the green line a little bit so that it meets z and y but not 2 we obtain a pseudoline arrangement whose oriented matroid M cannot be realized by means of a line arrangementi o Pappus7 Theorem can be proven using analytic geometry The equations that say that 17y72 are collinear work over any eld Therefore7 unorienting M produces a matroid that is not representable over any eld

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