Antipodal Point Arrangements on Spheres and Classification of Normal Systems
AAntipodal Point Arrangements on Spheres,Classification of Normal Systems and Hy-perplane Arrangements over Ordered Fields
Author: C.P. Anil Kumar*
Abstract.
For any positive integer k , we classify the antipodal point ar-rangements (Definitions [2.1,3.1,3.2,5.1,5.2]) on the sphere PF k + F over anordered field F (Definition 1.1), up to an isomorphism, by associating afinite complete set of cycle invariants. The classification for dimension k = 2 is done in Theorem 3.6 and the classification for dimension k > Mathematics Subject Classification (2010).
Primary: 52C35 Secondary:51H10,14P10,14P25,12J15.
Keywords.
Ordered Fields, Point Arrangements, Hyperplane Arrange-ments.
1. Introduction
The main motivation to write this article, arises during the characterizationof isomorphism classes of hyperplane arrangements by associating a finitecomplete set of invariants. This characterization is done in Sections [7,8] over *The author is supported by a research grant and facilities provided by Center for studyof Science, Technology and Policy (CSTEP), Bengaluru, INDIA for this research work. a r X i v : . [ m a t h . C O ] N ov C.P. Anil Kumaran ordered field. Normal systems arise as coarse invariants during classifica-tion of hyperplane arrangements. Before we restate the relevant Problem 1.5regarding classification of normal systems we need a few definitions.
Definition 1.1.
A totally ordered field ( F , ≤ ) satisfying the following two properties • P1: If x, y, z ∈ F then x ≤ y ⇒ x + z ≤ y + z . • P2: If x, y ∈ F then x ≥ , y ≥ ⇒ xy ≥ R is an ordered field with the induced ordering from the field ofreals. Definition 1.2 (Normal System).
Let F be an ordered field and N = { L , L , . . . , L n } be a finite set of linespassing through the origin in F m . Let U = {± v , ± v , . . . , ± v n } be a set ofantipodal pairs of F -vectors on these lines. We say N forms a normal systemif the set B = { v , v , . . . , v n } of F -vectors is maximally linearly independent, that is, any subset of B ofcardinality at most m is linearly independent. Note 1.3.
By an F - vector we mean a vector with coordinates in the field F . Definition 1.4 (Convex Positive Bijection and Isomorphic Normal Systems).
Let F be an ordered field and N = { L , L , . . . , L n } , N = { M , M , . . . , M n } be two finite sets of lines passing through the origin in F m both of them havethe same cardinality n which form normal systems. Let U = {± v , ± v , . . . , ± v n } , U = {± w , ± w , . . . , ± w n } be two sets of antipodal pairs of F -vectors on these lines in N , N respec-tively. We say a bijection δ : U −→ U is a convex positive bijection if δ ( − u ) = − δ ( u ) , u ∈ U and for any base B = { u , u , . . . , u m } ⊂ U and a vector u ∈ U we have u = m (cid:88) i =1 a i u i with a i > , ≤ i ≤ m, if and only if ,δ ( u ) = m (cid:88) i =1 b i δ ( u i ) with b i > , ≤ i ≤ m. We say two normal systems are isomorphic if there exists a convex posi-tive bijection between their corresponding sets of antipodal pairs of normal F -vectors.Now we mention the relevant open problem regarding classification of normalsystems.ntipodal Point Arrangements on Spheres 3 Problem 1.5 (Classification of Normal Systems and Finding Representativesin Each Isomorphism Class).
Classify and enumerate the Normal Systems up to an isomorphism by asso-ciating invariants which can be used to easily construct a family of normalsystems representing each isomorphism class for every positive integer cardi-nality n of the normal system. Here in this article we classify normal systems over an ordered field F up toan isomorphism by associating a finite complete set of cycle invariants andthereby classify the hyperplane arrangements also using one more invariantthe concurrency arrangement sign function. The enumeration problem of thenumber of isomorphism classes of normal systems and the problem of rep-resenting their isomorphism classes by a well defined list of representativesstill remain open (refer to Question 9.1). The exact statement of first mainTheorem Ω about classification of normal systems cannot be stated here asit requires more definitions and concepts which at present are not motivatedand developed. Hence we defer the statement to its appropriate Section 5 ofthe article. Now we mention a few more definitions in order to state secondmain Theorem Σ of the article. Definition 1.6 (A Hyperplane Arrangement).
Let F be a field and m, n be positive integers. We say a set( H mn ) F = { H , H , . . . , H n } of n hyperplanes in F m forms a hyperplane arrangement if • Condition 1: For 1 ≤ r ≤ m, ≤ i < i < . . . < i r ≤ n we have dim F ( H i ∩ H i ∩ . . . ∩ H i r ) = m − r (as an affine space) . • Condition 2: For r > m, ≤ i < i < . . . < i r ≤ n we have H i ∩ H i ∩ . . . ∩ H i r = ∅ . By a hyperplane arrangement, we always mean in general position, (that is,with Conditions 1 , Definition 1.7 (A Bounded/An Unbounded Region).
Let F be an ordered field and m, n be positive integers. Let( H mn ) F = { H , H , . . . , H n } be a hyperplane arrangement where an equation for H i is given by m (cid:88) j =1 a ij x i = b i , with a ij , b i ∈ F , ≤ j ≤ m, ≤ i ≤ n. Then a polyhedral region is defined to be a set of solutions for any choice of n inequalities as follows. { ( x , x , . . . , x m ) ∈ F m | m (cid:88) j =1 a ij x i ≤ , ≥ b i , ≤ i ≤ n } . C.P. Anil KumarA region R is unbounded if there exist v, u ∈ R such that v + t ( u − v ) ∈ R either for all t ≥ t ≤
0. Otherwise R is said to be bounded. Note 1.8.
There are n choices of inequalities for the regions and however onlya few of the regions are actually non-empty as given by the following theoremwhose proof is well known in the literature on hyperplane arrangements. In this article, from now on, a polyhedral region means a non-empty polyhe-dral region.
Theorem 1.9.
Let F be an ordered field and n, m be positive integers. Let ( H mn ) F be a hyperplane arrangement. Then there are • m (cid:80) i =0 (cid:0) ni (cid:1) polyhedral regions, • (cid:0) n − m (cid:1) bounded polyhedral regions and • m − (cid:80) i =0 (cid:0) ni (cid:1) + (cid:0) n − m − (cid:1) unbounded polyhedral regions. Definition 1.10 (Isomorphism Between Two Hyperplane Arrangements).
Let F be an ordered field and( H mn ) F = { H , H , . . . , H n } , ( H mn ) F = { H , H , . . . , H n } be two hyperplane arrangements in F m . We say a map φ : ( H mn ) F −→ ( H mn ) F is an isomorphism between these two hyperplane arrangements if φ is a bi-jection between the sets ( H mn ) F , ( H mn ) F , in particular on the subscripts, andgiven 1 ≤ i < i < . . . < i m − ≤ n with lines L = H i ∩ H i ∩ . . . ∩ H i m − , M = H φ ( i ) ∩ H φ ( i ) ∩ . . . ∩ H φ ( i m − ) , the order of vertices of intersection on the lines L, M agree via the bijectioninduced by φ again on the sets of subscripts of cardinality m (correspondingto the vertices on L ) containing { i , i , . . . , i m − } and (corresponding to thevertices on M ) containing { φ ( i ) , φ ( i ) , . . . , φ ( i m − ) } . There are four possi-bilities of pairs of orders and any one pairing of orders out of the four pairsmust agree via the map induced by φ .We say φ is an isomorphism which is identity on subscripts if in particular φ ( H i ) = H i , ≤ i ≤ n , that is, φ preserves subscripts. Note 1.11.
If there is an isomorphism between two hyperplane arrangements ( H mn ) F i , i = 1 , then there exists a piecewise linear bijection of F m to F m which takes one arrangement to another using suitable triangulation of poly-hedralities. For obtaining a piecewise linear isomorphism extension from ver-tices to the one dimensional skeleton of the arrangements, further subdivisionis not needed. Definition 1.12 (Normal System Associated to a Hyperplane Arrangement).
Let F be an ordered field. Let ( H mn ) F = { H i : m (cid:80) j =1 a ij x j = b i , ≤ i ≤ n } . Thenntipodal Point Arrangements on Spheres 5the normal system N associated to the hyperplane arrangement is given by N = { L i = { t ( a i , a i , . . . , a im ) ∈ F m | t ∈ F } | ≤ i ≤ n } and a set of antipodal pairs of normal F -vectors is given by U = {± v , . . . , ± v n } where 0 (cid:54) = v i ∈ L i , ≤ i ≤ n . For example we can choose by default U = {± ( a i , a i , . . . , a im ) ∈ F m | ≤ i ≤ n } . Definition 1.13 (Hyperplane arrangement given by a normal system).
Let F be an ordered field and N = { L , L , . . . , L n } be a normal system in F m .Let U = {± ( a i , a i , . . . , a im ) | ( a i , a i , . . . , a im ) ∈ L i , ≤ i ≤ n } be a set ofantipodal pairs of F -vectors of the normal system N . We fix the coefficientmatrix [ a ij ] ≤ i ≤ n, ≤ j ≤ m ∈ M n × m ( F ). Let ( H mn ) F = { H , H , . . . , H n } be anyhyperplane arrangement with the normal system N . When we write equationsfor the hyperplane H i , we use the fixed coefficient matrix as H i : m (cid:88) j =1 a ij x j = b i for some b i ∈ F . We say that the hyperplane arrangement ( H mn ) F is given by the normal system N .The normal representation theorem is stated as follows which is proved (referto main Theorem B ) in C. P. Anil Kumar [2] . Theorem 1.14.
Let F be an ordered field. Let ( H mn ) F , ( H mn ) F be two hyper-plane arrangements with sets U , U as antipodal pairs of normal F - vectors.Then there exists an isomorphism between the hyperplane arrangements upto translation of any hyperplane if and only if there exists a convex positivebijection between U , U .Restating the theorem, we have that, if two hyperplane arrangements ( H mn ) F , ( ˜ H mn ) F are isomorphic then their associated normal systems N and ˜ N are iso-morphic. Conversely, if we have two hyperplane arrangements ( H mn ) F , ( ˜ H mn ) F ,whose associated normal systems N and ˜ N are isomorphic, then, there existtranslates of each of the hyperplanes in the hyperplane arrangement ( H mn ) F ,giving rise to a translated hyperplane arrangement ( H mn ) F , such that, thisarrangement and ( ˜ H mn ) F are isomorphic. Note 1.15.
In C. P. Anil Kumar [2] it has been proved that a normal system isa coarse invariant for a hyperplane arrangement (refer to Normal Represen-tation Theorem 1.14). Also we have observed in the same article in the case ofdimension two that any two normal systems of the same cardinality are iso-morphic. However in dimensions more than two, there exist non-isomorphicnormal systems which therefore excludes the possibility of any two hyperplanearrangements with these two normal systems being isomorphic.
We need some more definitions before we state the second main theorem. C.P. Anil Kumar
Definition 1.16 (Concurrency Arrangement Sign Function).
Let F be an ordered field and( H mn ) F = { H i : m (cid:88) j =1 a ij x j = c i , ≤ i ≤ n } , be a hyperplane arrangement in F m . Let N = { L i = { λ ( a i , a i , . . . , a im ) | λ ∈ F } , ≤ i ≤ n } be the associated normal system with U = {± ( a i , a i , . . . , a im ) ∈ F m | ≤ i ≤ n } be a set of antipodal pairs of vectors on the lines of the normal system. Let P n = {± [ a i : a i : . . . : a im ] | ≤ i ≤ n } ⊂ PF ( m − F be the associated antipodal point arrangement on the ( m −
1) -dimensionalsphere. For every 1 ≤ i < i < . . . < i m +1 ≤ n , consider the hyperplane M { i ,i ,...,i m +1 } passing through the origin in F n in the variables y , y , . . . , y n whose equation is given by M { i ,i ,...,i m +1 } ( y , y , . . . , y n ) =Det a i a i · · · a i ( m − a i m y i a i a i · · · a i ( m − a i m y i ... ... . . . ... ... ... a i m − a i m − · · · a i m − ( m − a i m − m y i m − a i m a i m · · · a i m ( m − a i m m y i m a i m +1 a i m +1 · · · a i m +1 ( m − a i m +1 m y i m +1 = 0Then the associated concurrency arrangement of hyperplanes passing throughthe origin in F n is given by( C n ( nm +1 )) F = { M { i ,i ,...,i m +1 } | ≤ i < i < . . . < i m +1 ≤ n } . For this concurrency arrangement ( C n ( nm +1 )) F associated to the hyperplanearrangement ( H mn ) F the sign function is defined as: S : ( C n ( nm +1 )) F −→ {± } ,M { i ,i ,...,i m +1 } −→ sign ( M { i ,i ,...,i m +1 } ( c , c , . . . , c n )) Definition 1.17 (Concurrency Arrangement Sign Function Induced by a Con-vex Positive Bijection).
Let F be an ordered field and( H mn ) F k = { H k : m (cid:88) j =1 a kij x j = c ki , ≤ i ≤ n } , k = 1 , n hyperplanes in the space F m . Let U k = {± ( v ki = ( a ki , a ki , . . . , a kim )) ∈ F m | ≤ i ≤ n } , k = 1 , N k = { L ki = { λ ( a ki , a ki , . . . , a kim ) | λ ∈ F } , ≤ i ≤ n } , k = 1 , δ : U −→ U be a convex positive bijection. Then this isomorphism δ givesrise to a concurrency arrangement of the second hyperplane arrangement forthe choice of normals δ ( v ) , δ ( v ) , . . . , δ ( v n )of the hyperplanes given by (cid:0) ( C n ( nm +1 )) F (cid:1) δ = { M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } | ≤ i < i < . . . < i m +1 ≤ n } where M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } ( y π (1) , y π (2) , . . . , y π ( n ) ) =Det ←− δ ( v i ) −→ y π ( i ) ←− δ ( v i ) −→ y π ( i ) ... ... ... ... ←− δ ( v i m − ) −→ y π ( i m − ) ←− δ ( v i m ) −→ y π ( i m ) ←− δ ( v i m +1 ) −→ y π ( i m +1 ) = 0and a sign function for the corresponding choice of constants d i ∈ {± c i } , ≤ i ≤ n with d π ( i ) = c π ( i ) if δ ( v i ) = v π ( i ) and d π ( i ) = − c π ( i ) if δ ( v i ) = − v π ( i ) for a permutation π ∈ S n again induced by δ on the subscripts. We definethe induced concurrency arrangement sign function of the second hyperplanearrangement by S δ : (cid:0) ( C n ( nm +1 )) F (cid:1) δ −→ {± } with M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } −→ sign ( M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } ( d π (1) , d π (2) , . . . , d π ( n ) )) . Now we state the second main theorem.
Theorem Σ (Second Main Theorem). Let F be an ordered field. Let( H mn ) F k = { H ki : m (cid:88) j =1 a kij x j = c ki , ≤ i ≤ n } , k = 1 , n hyperplanes in the space F m . With thenotations in Definitions [1.16, 1.17], we have, ( H mn ) F k , k = 1 , δ : U −→ U a convex positive bijection and2. The concurrency arrangement sign maps S : ( C n ( nm +1 )) F −→ {± } , S δ : (cid:0) ( C n ( nm +1 )) F (cid:1) δ −→ {± } are such that for all M { i ,i ,...,i m +1 } ∈ ( C n ( nm +1 )) F ,(a) either S (cid:0) M { i ,i ,...,i m +1 } (cid:1) = S δ (cid:0) M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } (cid:1) , C.P. Anil Kumar(b) or S (cid:0) M { i ,i ,...,i m +1 } (cid:1) = − S δ (cid:0) M δ { π ( i ) ,π ( i ) ,...,π ( i m +1 ) } (cid:1) . With relevance to antipodal point arrangements (refer to Definition 5.1) ornormal systems, the theory of matroids is a well studied subject. Matroidsare combinatorial abstractions of vector configurations and hyperplane ar-rangements. E. Katz [4] gives a survey of this theory aimed at algebraicgeometers. Here in this article we study specific kind of antipodal pairs ofvectors arranged on spheres, vector configurations, which are associated tonormal systems that arise from hyperplane arrangements and classify themcombinatorially. The method of associating cycle invariants as a combinato-rial model to point arrangements in the plane has already been explored byauthors J.E.Goodman and R.Pollack [3]. Also the slope problem mentionedin chapter 10, page 60 in M. Aigner and G. M. Ziegler [1], Proofs from THEBOOK, explains a similar method.Section 2 defines a k -dimensional sphere PF k + F over an ordered field F as ageneralization of the k -dimensional sphere S k ⊂ R k +1 . Section 3 is devotedto the classification of antipodal point arrangements on PF F in two dimen-sions. Theorem 3.6 states the classification theorem in the case of dimensiontwo. Section 4 revisits the two non-isomorphic examples of normal systemsin dimension three that are mentioned in C. P. Anil Kumar [2] and computesthe combinatorial invariants. Sections [5,6] are devoted to classification ofantipodal point arrangements on PF k + F in higher dimensions for k >
2. The-orem Ω in Section 6 states the classification theorem in higher dimensions.In Sections [7,8] we combinatorially classify hyperplane arrangements overan ordered field by proving Theorem Σ using associated normal system andthe concurrency arrangement sign function. In final Section 9 we pose openQuestion 9.1 about normal systems.
2. The Analogue of Spheres over Ordered Fields
Over the field R of reals the k -dimensional sphere S k is defined as S k = { ( x , x , . . . , x k +1 ) ∈ R k +1 | k +1 (cid:88) i =1 x i = 1 } . We also observe that every line L ⊂ R k +1 passing through the origin meetsthe sphere in two distinct points on either side of the origin. With this ob-servation we define the analogue of the sphere over an ordered field F .ntipodal Point Arrangements on Spheres 9 Definition 2.1.
Let F be an ordered field and k ∈ N . The k -dimensionalsphere in F k +1 is defined as PF k + F = { [( x , x , . . . , x k +1 )] | [( x , x , . . . , x k +1 )] = { λ ( x , x , . . . , x k +1 ) | λ ∈ F + , thatis, λ > } , (cid:54) = ( x , x , . . . , x k +1 ) ∈ F k +1 } Now we define linear independence.
Definition 2.2.
Let F be an ordered field and l, k ∈ N , ≤ l ≤ k + 1. We saypoints [( x i , x i , . . . , x ik +1 )] ∈ PF k + F , ≤ i ≤ l ≤ k + 1are linearly independent if Rank [ x ij ] l × ( k +1) = l (maximum possible rank) . Otherwise they are said to be linearly dependent. This does not depend onthe choice of the representatives.Here we define what we mean by a positive combination.
Definition 2.3 (Definition of Positive Combination).
Let F be an orderedfield and k ∈ N . Let [( x i , x i , . . . , x ik +1 )] ∈ PF k + F , ≤ i ≤ k + 1 be linearlyindependent. We say [( x k +21 , x k +22 , . . . , x k +2 k +1 )] ∈ PF k + F is a positive combination of[( x i , x i , . . . , x ik +1 )] ∈ PF k + F , ≤ i ≤ k + 1if there exist λ i > , ≤ i ≤ k + 1 such that( x k +21 , x k +22 , . . . , x k +2 k +1 ) = k +1 (cid:88) i =1 λ i ( x i , x i , . . . , x ik +1 ) . Again here sign of λ i does not depend on the choice of representatives.Now we define antipodes on a sphere PF k F . Definition 2.4 (Definition of Antipodes).
Let F be an ordered field and k ∈ N .We say points[( x , x , . . . , x k , x k +1 )] , [( y , y , . . . , y k , y k +1 )] ∈ PF k F are antipodes if there exists λ < , λ ∈ F such that ( x , x , . . . , x k +1 ) = λ ( y , y , . . . , y k +1 ). We denote − [( x , x , . . . , x k +1 )] = [( y , y , . . . , y k +1 )] . Now we mention the definition of a sphere algebraic set.0 C.P. Anil Kumar
Definition 2.5 (Sphere Algebraic Set).
Let F be an ordered field and n ≥ f ∈ F [ X , X , . . . , X n , X n +1 ] let V n ( f ) = { ( x , x , . . . , x n , x n +1 ) ∈ F n +1 | f ( x , x , . . . , x n , x n +1 ) = 0 } We say V n ( f ) is a sphere algebraic set if every line L = { λ ( x , x , . . . , x n , x n +1 ) | λ ∈ F } for a point ( x , x , . . . , x n , x n +1 ) ∈ ( F n +1 ) ∗ intersects V n ( f ) in twopoints λ ( x , x , . . . , x n , x n +1 ) , λ ( x , x , . . . , x n , x n +1 )for some λ , λ ∈ F with λ λ < F . Theorem 2.6.
Let F be an ordered field. For n ≥ , let f n − [ x , x , . . . , x n − ] = f ( n − , [ x , x , . . . , x n − ] + f ( n − , [ x , x , . . . , x n − ]+ f ( n − , [ x , x , . . . , x n − ] be a polynomial of degree , with homogeneous decomposition into f ( n − , , f ( n − , , (cid:54) = f ( n − , . Suppose f n − has no solutions in F n − and the homo-geneous polynomial equation f ( n − , [ x , x , . . . , x n − ] = 0 has only origin in F n − as a solution. Let f n, ( x , x , . . . , x n ) = x n f n − ( x x n , x x n , . . . , x n − x n ) be its homogenization. Suppose in addition the range of f n, ( x , x , . . . , x n ) for ( x , x , . . . , x n ) ∈ ( F n ) ∗ is contained in the subset of squares in F . Thenthe zero set V ( f n, − defined by the equation f n, = 1 is a sphere algebraicset and f n, − is an irreducible polynomial.Proof. The proof is immediate. We only prove that f n, − n ≥
2. Suppose it is reducible then we have f n, − g ( x , x , . . . , x n ) h ( x , x , . . . , x n )where both g, h are linear polynomials in x , x , . . . , x n with nonzero constantcoefficients. Over ordered fields not every ray passing through the originintersects one of these two zero sets g = 0 , h = 0 which are affine hyperplanes.Hence we arrive at a contradiction. (cid:4) Example 2.7. • Let F = Q ∩ R . • Let F = K ⊂ R where K is the smallest extension of Q which is positivelyquadratically closed (refer to Definition 2.8 and Theorem 2.9).For n ≥ , let f n, ( x , x , . . . , x n ) = x + x + . . . + x n , let f n, ≡ , f n, ≡ . Some examples where such polynomials exist are those ordered fields whichare positively quadratically closed.ntipodal Point Arrangements on Spheres 11
We begin with the definition of a positively quadratically closed field.
Definition 2.8 (Positively Quadratically Closed).
Let F be an ordered field.We say F is positively quadratically closed if for every a ∈ F , a > b ∈ F such that a = b .Now we prove the following theorem. Theorem 2.9.
Let F be an ordered field. Then there exists a positively quadrat-ically closed ordered field F P QC containing F which extends the total orderon F .Proof. Consider the field F to be the compositum of quadratic extensionsobtained by adjoining square roots of positive elements of F . F = (cid:95) d> F [ √ d ] . Then we can extend the total order on F to F as follows. The extension isdone step by step. Consider F [ √ d ] with d >
0. Then we define a + b √ d > • if a > , b > • if a > , b < a > b d or • if a < , b > b d > a or • if a = 0 , b > • if a < , b = 0.We define a + b √ d = 0 if and only if a = 0 , b = 0. Otherwise in all othercases, we define a + b √ d < a + b √ d > c + e √ d if ( a − c ) + ( b − e ) √ d >
0. This gives a totalorder on F [ √ d ] extending the total order on F . Having extended the totalorder to F [ √ d ] we prove that the total order is extendable to the field F . ByZorn’s lemma there exists a maximal field K ⊆ F for which the total orderis extendable. Now if F (cid:54) = K then there exists d ∈ F such that √ d / ∈ K . Sowe can extend the total order to the field K [ √ d ] which is a contradiction tomaximality. Hence we have F = K .Now we construct a field F i inductively which is the compositum of the squareroots of all positive elements of F i − for i >
1. Then consider the field givenby F P QC = (cid:91) i ≥ F i This field is positively quadratically closed ordered field containing F andwhich extends the total order on F . (cid:4) Note 2.10. If F is a positively quadratically closed ordered field then every linepassing through the origin in the affine space A n F has a pair of antipodal unitvectors, that is, the sphere S n − F is complete with respect to the lines passing through the origin or equivalently every such line intersects the sphere in apair of antipodal unit vectors.
3. Antipodal Point Arrangements on the
Sphere PF F Now we define antipodal point arrangements on the 2 -sphere PF F . Definition 3.1 (Antipodal Point Arrangement on the
Sphere PF F ). Let F be an ordered field. We say a set P n = {± P , ± P , . . . , ± P n } ⊂ PF F ofpoints is a point arrangement on the sphere if three points of P n are linearlydependent (refer to Definition 2.2) then some two of them are antipodal. Definition 3.2 (Isomorphism Between two Antipodal Point Arrangements onthe
Sphere PF F ). Let F be an ordered field. Two point arrangements P n = {± P , ± P , . . . , ± P n } , Q m = {± Q , ± Q , . . . , ± Q m } ⊂ PF F are isomorphic if n = m and there is a bijection φ : P n −→ Q n between thetwo sets such that the following occurs. • φ ( − A ) = − φ ( A ) for all A ∈ P n . • for any A, B, C, D ∈ P n if D is a positive combination of A, B, C if andonly if φ ( D ) is a positive combination of φ ( A ) , φ ( B ) , φ ( C ). PF F We begin this section with the standard arrangement.
Let F be anordered field. The arrangement S ⊂ PF F consists of four antipodal pairs ofpoints given by S = { x = [(1 , , , x = [( − , , , y = [(0 , , , − y = [(0 , − , ,z = [(0 , , , − z = [(0 , − , , P (cid:0) a point in I - Octant (cid:1) , − P (cid:0) antipode point of P in V II - Octant (cid:1) } ⊂ PF F There are twenty four symbols that we associate to this standard arrange-ment. Before we actually describe these symbols we mention four importantaspects.1. A symbol is of the form a −→ ( b, c, d )2. We say that it is compatible or associated to an antipodal point ar-rangement if a, b, c, d represent elements of the arrangement in PF F such that a is a positive combination of b, c, d .ntipodal Point Arrangements on Spheres 133. If we give an anticlockwise local orientation to the plane a ⊥ with thedirection ray a ∈ PF F representing the thumb then ignoring signs theline cycle is clockwise oriented and is given by( bcd ) , and not by ( bdc ) . For example to get the symbol P −→ ( y, x, z ) for S refer to the firstoctant view in Figure 1. Here the line cycle ( yxz ) is obtained by movingclockwise around P .4. The ordered triple ( b, c, d ) where a is a positive combination of b, c, d has negative determinant.The associated symbols for the standard arrangement S are given by P −→ ( y, x, z ) , P −→ ( x, z, y ) , P −→ ( z, y, x ) ,x −→ ( − y, P, − z ) , x −→ ( P, − z, − y ) , x −→ ( − z, − y, P ) ,z −→ ( − x, P, − y ) , z −→ ( P, − y, − x ) , z −→ ( − y, − x, P ) ,y −→ ( − z, P, − x ) , y −→ ( − x, − z, P ) , y −→ ( P, − x, − z ) , − x −→ ( − P, y, z ) , − x −→ ( z, − P, y ) , − x −→ ( y, z, − P ) , − z −→ ( − P, x, y ) , − z −→ ( x, y, − P ) , − z −→ ( y, − P, x ) , − y −→ ( − P, z, x ) , − y −→ ( x, − P, z ) , − y −→ ( z, x, − P ) , − P −→ ( − x, − y, − z ) , − P −→ ( − z, − x, − y ) , − P −→ ( − y, − z, − x ) . In the above symbols the triples are all negatively (clockwise) oriented, thatis, given that P is in the first octant the triples have determinant negative. Here we explore the symmetry involved in the above set of 24 compatiblesymbols. We state the following theorem on the action of the symmetry group S on the set of symbols and describe the transitive orbits. Theorem 3.3.
The group S acts on the set S = { p −→ ( q, r, s ) | p, q, r, s ∈ {± a, ± b, ± c, ± d } such that {± p } ∩ {± q } = {± p } ∩ {± r } = {± p } ∩ {± s } = {± q } ∩ {± r } = {± q } ∩ {± s } = {± r } ∩ {± s } = ∅} of all symbols with the action given by p −→ ( q, r, s ) Apply (12) to get − p −→ ( − r, − q, − s ) p −→ ( q, r, s ) Apply (23) to get r −→ ( − q, p, − s ) p −→ ( q, r, s ) Apply (34) to get s −→ ( − q, − r, p ) p −→ ( q, r, s ) Apply (14) to get − p −→ ( − s, − r, − q ) . • The set S has elements. Then each transitive orbit of an elementunder the action of S contains elements. There are orbits. • There are orbits (192 elements satisfying property 4) that arise ascompatible symbols associated to concrete four antipodal point arrange-ments. • Each transitive orbit is the set of all compatible symbols correspondingto one fixed four antipodal pairs of points of the point arrangement onthe sphere PF F provided one of the symbols in the orbit is compatible. • Moreover the action of S on the set S is free.Proof. We have S ) = 4 ∗ ∗ = 384. We observe that the action iscompatible with the relations(12)(23)(12) = (23)(12)(23) , (23)(34)(23) = (34)(23)(34) , (34)(14)(34) = (14)(34)(14) , (12)(34) = (34)(12) , (14)(23) = (23)(14) , (12) = (23) = (34) = (41) = identity. So we have an action of the symmetric group S on the set S of symbols.The set of compatible symbols, as a transitive orbit, obtained by the actionof S on the compatible symbol P −→ ( y, x, z ) is precisely the above given 24compatible symbols of the standard arrangement in Section 3.1.1. Similarlyfor every transitive orbit if one of the symbols is compatible then all theremaining 23 symbols of the orbit are compatible. The rest of the proof ofthe theorem is immediate. (cid:4) Let F be an ordered field. Here we associate line cycles to the points of the standardarrangement S . Later we use this as a local dictionary for an antipodalarrangement on PF F to characterize the arrangement up to an isomorphism.Consider the standard four antipodal point arrangement S given by P = x = [(1 , , , − P = − x = [( − , , ,P = y = [(0 , , , − P = − y = [(0 , − , ,P = z = [(0 , , , − P = − z = [(0 , − , ,P = P a point in I - Octant, − P = − P its antipode point in V II - Octant in PF F The compatible 24 symbols (an S transitive orbit) gives rise to the followingdictionary of line cycles using subscripts { , , , } and symbols { + , −} ateach point of the arrangement with ( x, y, z ) denoting a positively orientedbasis of the arrangement. τ +4 = (213) at P , τ − = (231) at − P ,τ +3 = (142) at P , τ − = (124) at − P ,τ +2 = (341) at P , τ − = (314) at − P ,τ +1 = (243) at P , τ − = (234) at − P . Now we prove a theorem that given the dictionary of line cycles there is aunique way to recover back the 24 compatible symbols, an S orbit of thearrangement, which is compatible with the standard arrangement.We state the theorem as follows.ntipodal Point Arrangements on Spheres 15 Theorem 3.4.
Let F be an ordered field. Let P = {± P , ± P , ± P , ± P } ⊂ PF F be any four antipodal point arrangement on the sphere. Suppose the linecycles are given by τ +4 = (213) at P , τ − = (231) at − P ,τ +3 = (142) at P , τ − = (124) at − P ,τ +2 = (341) at P , τ − = (314) at − P ,τ +1 = (243) at P , τ − = (234) at − P . Then the map δ : P −→ S given by δ : P −→ x, − P −→ − x, P −→ y, − P −→ − y,P −→ z, − P −→ − z, P −→ P, − P −→ − P is an isomorphism, that is, it is a convex positive bijection. Also − δ is anisomorphism. The S invariant set of compatible symbols are given by P −→ ( P , P , P ) , P −→ ( P , P , P ) , P −→ ( P , P , P ) ,P −→ ( − P , P , − P ) , P −→ ( P , − P , − P ) , P −→ ( − P , − P , P ) ,P −→ ( − P , P , − P ) , P −→ ( P , − P , − P ) , P −→ ( − P , − P , P ) ,P −→ ( − P , P , − P ) , P −→ ( − P , − P , P ) , P −→ ( P , − P , − P ) , − P −→ ( − P , P , P ) , − P −→ ( P , − P , P ) , − P −→ ( P , P , − P ) , − P −→ ( − P , P , P ) , − P −→ ( P , P , − P ) , − P −→ ( P , − P , P ) , − P −→ ( − P , P , P ) , − P −→ ( P , − P , P ) , − P −→ ( P , P , − P ) , − P −→ ( − P , − P , − P ) , − P −→ ( − P , − P , − P ) , − P −→ ( − P , − P , − P ) . Proof.
Let us denote { P = x, − P = − x, P = y, − P = − y,P = z, − P = − z, P = P, − P = − P } . The octant views are given in Figure 1 based on the point P lying in variousoctants with respect to a positively oriented system ( x, y, z ).However first we show that ( P , P , P ) is positively oriented, that is, itsdeterminant is positive and the symbol P −→ ( P , P , P ) is compatible. Apriori we do not know the orientation of ( P , P , P ) and the compatibilitysigns of the symbols.Consider the following choices. ( ± P , ± P , ± P ). Out of these( P , P , P ) , ( − P , − P , P ) , ( P , − P , − P ) , ( − P , P , − P )have the same sign of the determinant and the remaining( − P , − P , − P ) , ( P , P , − P ) , ( − P , P , P ) , ( P , − P , P )have the same sign of the determinant.If the second set of determinants are positive then we argue as follows us-ing Theorem 3.3. Suppose P −→ ( − P , − P , − P ) is compatible then we6 C.P. Anil Kumar Figure 1.
Four Point Arrangements on the Sphere S have − P −→ ( P , P , P ) is compatible. Hence τ − = (243) which is in-valid. Suppose P −→ ( P , P , − P ) is compatible then we have − P −→ ( − P , P , − P ) is compatible. Hence τ − = (142) which is invalid. Suppose P −→ ( P , − P , P ) is compatible then we have − P −→ ( − P , P , − P ) iscompatible. Hence τ − = (243) which is invalid. Suppose P −→ ( − P , P , P )is compatible then we have − P −→ ( − P , P , − P ) is compatible. Hence τ − = (341) which is invalid.If the first set of determinants are positive then we argue as follows usingTheorem 3.3. We have P −→ ( − P , − P , P ) , P −→ ( − P , P , − P ) , P −→ ( P , − P , − P ) also give invalid line cycles. Hence we conclude that ( P , P , P )is positively oriented and the symbol P −→ ( P , P , P ) is compatible.This proves the theorem. We also note that the over all total flip given by ν : P −→ S , ν : P −→ − x, − P −→ x, P −→ − y, − P −→ y,P −→ − z, P −→ z, P −→ − P, − P −→ P is also an isomorphism, that is, a convex positive bijection. Using these linecycles we can write down all the S invariant set of 24 compatible symbols. (cid:4) There are other isomorphisms from P to S as well and below we describeall of them via the automorphism group Aut( S ). Here we compute the automorphism group of the standard antipodal pointarrangement.
Theorem 3.5.
Let F be an ordered field. Let S ⊂ PF F be the standardarrangement. Then Aut( S ) = S ⊕ ( Z / Z ) . ntipodal Point Arrangements on Spheres 17 Proof.
We have the twenty four compatible symbols of the standard arrange-ment given in Section 3.1.1. If p −→ ( q, r, s ) is one such compatible symbolthen the map δ : S −→ S , δ : P −→ p, − P −→ − p, y −→ q, − y −→ − q,x −→ r, − x −→ − r, z −→ s, − z −→ − s is an automorphism. We also have if φ is an automorphism then − φ is also anautomorphism and moreover either φ ( P ) −→ ( φ ( y ) , φ ( x ) , φ ( z )) or − φ ( P ) −→ ( − φ ( y ) , − φ ( x ) , − φ ( z )) gives rise to a compatible symbol and the other oneis not a compatible symbol. Hence we getAut( S ) = S ⊕ ( Z / Z ) . This proves the theorem. (cid:4) PF F Now we prove an isomorphism theorem for antipodal point arrangements onthe sphere PF F which can be generalized to higher dimensions in Theorem Ω. Let P n = {± P , ± P , . . . , ± P n } ⊂ PF F be an antipodal point arrangement.We introduce an equivalence relation ∼ on the symmetric group S n as follows.Let g, h ∈ S n then g ∼ h if g = h or g = h − . This is an equivalencerelation with reflexive, symmetric and transitive properties. The equivalenceclasses being [ τ, τ − ]. Any element of order at most two is an equivalence classcontaining just one element. Remaining equivalence classes has two elements.The antipode map a : PF F −→ PF F (reflection about the origin in threedimensions) has a negative determinant. The line cycles τ + i for P i , τ − i for − P i associated to a pair of antipodes in P n are mutually inverses of each other asthere is a reflection about the origin is involved. So we actually obtain( n −
1) - cycles with τ − i = ( τ + i ) − ∈ S n − ( { , , . . . , i − , i +1 , . . . , n } ) , ≤ i ≤ n. Now we consider the local scenario by restricting to just four antipodal pairs.The restriction map | local A and inverse map ( ∗ ) − commutes. We observethat For any four subset A ⊂ { , , . . . , n } , ( τ − i ) | localA = (( τ + i ) − ) | localA = (( τ + i ) | localA ) − , i ∈ A Now we statethe theorem as follows.
Theorem 3.6.
Let F be an ordered field. The following two assertions holdtrue.1. The line cycles of the antipodal pairs of points of a point arrangement P n ⊂ PF F determines the collection of local S - invariant set of com-patible symbols for every four subset of antipodal pairs of points in P n .
2. Let P n = {± P , ± P , . . . , ± P n } , P n = {± P , ± P , . . . , ± P n } be twopoint arrangements. Let ( τ + i ) j be the line cycle associated to P ji and ( τ − i ) j be the line cycle associated to − P ji for j = 1 , , ≤ i ≤ n . Thereexists a convex positive bijection (an isomorphism) δ : P n −→ P n . ifand only if there exist • a permutation π ∈ S n and • a sign vector µ = ( µ (1) , µ (2) , . . . , µ ( n )) ∈ ( Z / Z ) n = {± } n with the property that(a) either ( τ [ µ ( i ) ∗ (+)] π ( i ) ) = π ( τ + i ) π − , ( τ [ µ ( i ) ∗ ( − )] π ( i ) ) = π ( τ − i ) π − , ≤ i ≤ n (b) or an overall total flip (here we can choose − µ in place of µ ) ( τ [ µ ( i ) ∗ (+)] π ( i ) ) = [ π ( τ + i ) π − ] − = π ( τ − i ) π − , ( τ [ µ ( i ) ∗ ( − )] π ( i ) ) = [ π ( τ − i ) π − ] − = π ( τ + i ) π − , ≤ i ≤ n where • [ µ ( i ) ∗ (+)] = + , [ µ ( i ) ∗ ( − )] = − if µ ( i ) = + . • [ µ ( i ) ∗ (+)] = − , [ µ ( i ) ∗ ( − )] = + if µ ( i ) = − .Proof. We prove the second assertion first. Suppose δ : P n −→ P n is anisomorphism. Then the permutation π and the signed vector µ are definedby the equation δ ( P i ) = µ ( i ) P π ( i ) , δ ( − P i ) = − µ ( i ) P π ( i ) , ≤ i ≤ n. Now we this definition of π, µ the property 2a is satisfied. If we choose for µ the following definition δ ( P i ) = − µ ( i ) P π ( i ) , δ ( − P i ) = µ ( i ) P π ( i ) , ≤ i ≤ n then π, µ satisfies the property 2b. This proves one way implication.Now we prove the other way implication where we are given the permutation π and the signed vector µ and changing µ to − µ if necessary we assumethat the property 2a holds. First we localize to any two corresponding fourantipodal point arrangements {± P i , ± P j , ± P k , ± P l } , {± P π ( i ) , ± P π ( j ) , ± P π ( k ) , ± P π ( l ) } . Since property 2a holds and the restriction map and the inverse map com-mutes with respect to localization there is an isomorphic way to identify thesetwo arrangements using local line cycles via the local chart as the standardarrangement S using Theorem 3.4. Using this chart we conclude that locallythere exists an isomorphism of the four antipodal point arrangements givenby δ : P i −→ µ ( i ) P π ( i ) , − P i −→ − µ ( i ) P π ( i ) , P j −→ µ ( j ) P π ( j ) , − P j −→ − µ ( j ) P π ( j ) P k −→ µ ( k ) P π ( k ) , − P k −→ − µ ( k ) P π ( k ) , P l −→ µ ( l ) P π ( l ) , − P l −→ − µ ( l ) P π ( l ) ntipodal Point Arrangements on Spheres 19These local isomorphisms patch up and extend uniquely to an isomorphismdefined as δ ( P i ) = µ ( i ) P π ( i ) , δ ( − P i ) = − µ ( i ) P π ( i ) , ≤ i ≤ n. We also observe that − δ : P n −→ P n is an isomorphism. This proves theisomorphism theorem in two dimensions.Now we prove the first assertion. The local cycles of four antipodal sub-arrangements determine the S invariant set of 24 compatible symbols usingTheorem 3.4. Hence the first assertion follows and we can write down all thecompatible symbols of the given arrangement. (cid:4)
4. Examples of two Non-isomorphic Normal Systems in ThreeDimensions over Rationals: Revisited
Consider the normal systems whose associated sets of antipodal vectors aregiven by U = {± u i | ≤ i ≤ } , U = {± v i | ≤ i ≤ } with U ∩U = {± u , ± u , ± u , ± u , ± u } = {± v , ± v , ± v , ± v , ± v } where u = (1 , ,
0) = v , u = (0 , ,
0) = v , u = (0 , ,
1) = v ,u = (cid:0) , , (cid:1) = v , u = (cid:0) , , (cid:1) = v , u = (cid:0) , , (cid:1) , v = (cid:0) , , (cid:1) . Here below we find out line cycles of each point with respect to the givennotation.We have proved that these two are non-isomorphic normal systems by asso-ciating graphs of compatible pairs mentioned in C. P. Anil Kumar [2]. Forexample, from the 15 equations below for U the vertex {− u , u } has degreeone and is only compatible with { u , − u } (equation (5) in the first set).From the 15 equations below for U we observe that there is no vertex of de-gree one as we observe that if a vertex of the associated graph of compatiblepairs has a positive degree then the degree is at least two.Now we mention the following (cid:0) (cid:1) = 15 equations for U .1. 3 u = u + 2 u + 2 u = (1 , , u = u + 4 u + 8 u = (1 , , u = 6 u + 6 u + 7 u = (6 , , u = 3 u + 4 u + 9 u = (4 , , u + 21 u = 2 u + 22 u = (12 , , u = 41 u + 20 u + 63 u = (48 , , u + 9 u = 4 u + 6 u = (2 , , u = 3 u + u + 9 u = (6 , , u + 9 u = 10 u + 22 u = (12 , , u = 2 u + 6 u + 3 u = (1 , , u = 6 u + 5 u + 11 u = (6 , , u = 18 u + 41 u + 11 u = (6 , , u = 13 u + 30 u + 9 u = (24 , , u = 26 u + 45 u + 66 u = (41 , , u + 27 u = 27 u + 11 u = (9 , , τ +1 ) = (24653) at u , ( τ − ) = (23564) at − u , ( τ +2 ) = (13546) at u , ( τ − ) = (16453) at − u , ( τ +3 ) = (16452) at u , ( τ − ) = (12546) at − u , ( τ +4 ) = (15326) at u , ( τ − ) = (16235) at − u , ( τ +5 ) = (16432) at u , ( τ − ) = (12346) at − u , ( τ +6 ) = (15324) at u , ( τ − ) = (14235) at − u . Now we mention the following (cid:0) (cid:1) = 15 equations for U .1. 3 v = v + 2 v + 2 v = (1 , , v = v + 4 v + 8 v = (1 , , v = 2 v + 6 v + 9 v = (2 , , v = 3 v + 4 v + 9 v = (4 , , v = 5 v + 6 v + 22 v = (9 , , v = 7 v + 12 v + 81 v = (16 , , v + 9 v = 4 v + 6 v = (2 , , v + 11 v = 3 v + 9 v = (3 , , v + 27 v = 6 v + 22 v = (4 , , v = 2 v + 6 v + 3 v = (1 , , v = 2 v + 5 v + 6 v = (2 , , v = 2 v + 7 v + 11 v = (2 , , v + 44 v = 18 v + 27 v = (9 , , v = 2 v + 21 v + 45 v = (12 , , v + 11 v = 3 v + 9 v = (2 , , τ +1 ) = (24653) at v , ( τ − ) = (23564) at − v , ( τ +2 ) = (13564) at v , ( τ − ) = (14653) at − v , ( τ +3 ) = (14652) at v , ( τ − ) = (12564) at − v , ( τ +4 ) = (16532) at v , ( τ − ) = (12356) at − v , ( τ +5 ) = (14632) at v , ( τ − ) = (12364) at − v , ( τ +6 ) = (14532) at v , ( τ − ) = (12354) at − v .
5. Antipodal Point Arrangements on Higher DimensionalSpheres and Classification of Normal Systems
Here we mainly associate combinatorial invariants to antipodal point arrange-ments to classify them and hence classify the normal systems up to an isomor-phism. These combinatorial invariants turn out to be oriented cycles of pointsntipodal Point Arrangements on Spheres 21of the orthogonally projected arrangements along small sub-arrangements.We begin with the required definitions.
Definition 5.1 (Antipodal Point Arrangement on the k - Sphere PF k + F ). Let F be an ordered field. We say a set P n = {± P , ± P , . . . , ± P n } ⊂ PF k + F of points is a point arrangement on the sphere if for any 1 ≤ i < i < . . . ⊥ . We have row rank of T is k .Since row - rank ( T ) = col - rank ( T ) , Rank + N ullity = n we have dim (Ker( T )) = n − k . Define on F m with m > < v = ( x , x , . . . , x m ) , w = ( y , y , . . . , y m ) > F m = m (cid:88) i =1 x i y i . This is a symmetric bilinear form with the property that • < v, v > F m ≥ v ∈ F m . • < v, v > F m = 0 ⇐⇒ v = 0.2 C.P. Anil KumarThen for w ∈ F n , w ∈ F k < T w , w > F k = w t T w = w t T t w = < w , T t w > F n . Now we observe that if w ∈ Ker( T ) ⇐⇒ < w , T t w > F n = 0 for all w ∈ F k .So we conclude thatKer( T ) ⊥ = Ran( T t ) = Span < v i : 1 ≤ i ≤ k > . So we conclude that Ker( T ) (cid:77) Ran( T t ) = F n . We define the orthogonal projections as
P, Q : F n −→ F n such that P | Ker( T ) = 0 , P | Ran( Tt ) = Id, Q | Ker( T ) = Id, Q | Ran( Tt ) = 0 . These projections satisfy the following relations. I = P + Q, P = P, Q = Q, P t = P, Q t = Q, that is < P w , w > F n = < w , P w > F n , < Qw , w > F n = < w , Qw > F n for w , w ∈ F n . This proves the existence of orthogonal projections. We begin with a definition.
Definition 5.3 (Orthogonally Projected Antipodal Point Arrangements).
Let F be an ordered field. Let P n = {± P , . . . , ± P n } be an antipodal point ar-rangement in the k -dimensional sphere PF k + F . Let A = {± P i , ± P i , . . . , ± P i r }⊂ P n be an antipodal point sub-arrangement with 1 ≤ r ≤ k −
2. We canorthogonally project using Q A the sub-arrangement P n \A to the space or-thogonal to the space spanned by vectors of A to obtain an antipodal pointarrangement P A n − r = { P A j = Q A ( P j ) , − P A j = Q A ( − P j ) | ≤ j ≤ n, j (cid:54) = i l , ≤ l ≤ r } in the k − r dimensional sphere PF ( k − r )+ F .Now we prove a theorem on the signs. Theorem 5.4 (Sign of the Combination does not change after Projection).
Let F be an ordered field. Let P n = {± P , . . . , ± P n } be an antipodal pointarrangement in the k - dimensional sphere PF k + F . Let A = {± P n } . Let P A n − denote the projected arrangement. Suppose P i = [( x i , x i , . . . , x ik , x ik +1 )] , − P i = − [( x i , x i , . . . , x ik , x ik +1 )] , ≤ i ≤ n. Suppose we have ( x j , x j , . . . , x jk , x jk +1 ) = k (cid:88) l =1 λ l ( x i l , x i l , . . . , x i l k , x i l k +1 )+ λ n ( x n , x n , . . . , x nk , x nk +1 )ntipodal Point Arrangements on Spheres 23 for some j / ∈ { i , i , . . . , i k , n } , λ l , λ n ∈ F ∗ . Suppose we have P A j = Q A ( x j , x j , . . . , x jk , x jk +1 ) = k (cid:88) l =1 β l P A i l = k (cid:88) l =1 β l Q A ( x i l , x i l , . . . , x i l k , x i l k +1 ) then sign ( λ l ) = sign ( β l ) , ≤ l ≤ k. Proof.
This theorem is immediate. (cid:4)
Now we prove an isomorphism theorem about signs for antipodal point ar-rangements on spheres PF k + F . Theorem 5.5 (An isomorphism theorem).
Let F be an ordered field. Let P jn = {± P j , ± P j , . . . , ± P jn } be two antipodalpoint arrangements in PF k + F for j = 1 , . Let ± P ji , ± P ji , . . . , ± P ji k +1 , ± P jl be k +2 antipodal pairs of points of the arrangement. Let A = { i , i , . . . , i k +1 } .With respect to the set A let P jl = m (cid:88) r =1 ( λ li r ) jA P ji r , j = 1 , . Suppose we have sign (( λ li r ) A ) = sign (( λ li r ) A ) for any such choice.Then the map δ : P i −→ P i , δ : − P i −→ − P i , ≤ i ≤ n is an isomorphism of antipodal point arrangements P jn .Proof. This theorem is immediate and δ is a convex positive bijection. (cid:4) Let F be an ordered field. Let P n = {± P , ± P , . . . , ± P n } be an antipodalpoint arrangement in PF k + F . Let A = {± P j , ± P j , . . . , ± P j k − } ⊂ P n be a subset of cardinality k −
2. Then consider the projected arrangement P A n − k +2 ⊂ PF F . These arrangements on the two dimensional spheres giverise to clockwise oriented line cycles at each point of P A n − k +2 denoted asfollows. ( τ + j ) A ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at P A j , ( τ − j ) A ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at − P A j both of which are ( n − k + 1) -cycles which are mutual inverses of each otherfor 1 ≤ j ≤ n, j (cid:54) = j l , ≤ l ≤ k −
6. The First Main Theorem
Now we prove the following isomorphism theorem, the first main theorem forthe line cycle invariants.
Theorem Ω (First Main Theorem). Let F be an ordered field. Let P n = {± P , ± P , . . . , ± P n } be an antipodalpoint arrangement on PF k + F . The line cycle invariants of antipodal pairs P j , − P j , ≤ j ≤ n given by mutually inverse cycles( τ + j ) A ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at P A j , ( τ − j ) A ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at − P A j after projection along the small sub-arrangement A = {± P j , ± P j , . . . , ± P j k − } ⊂ P n \{± P j } given by P A n − k +2 = {± P A i | i ∈ { , , . . . , n }\{ j , j , . . . , j k − }} for all such possible choices of A determines the antipodal point arrangementup to an isomorphism, that is, for l = 1 , P ln = {± P l , ± P l , . . . , ± P ln } ⊂ PF k + F be two antipodal point arrangements with the line cycle invariants of antipo-dal pairs P lj , − P lj , ≤ j ≤ n given by mutually inverse cycles( τ + j ) A l ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at ( P lj ) A , ( τ − j ) A l ∈ S n − k +1 (cid:0) { , , . . . , n }\{ j , j , . . . , j k − , j } (cid:1) at − ( P lj ) A after projection along the small sub-arrangement A = {± P lj , ± P lj , . . . , ± P lj k − } ⊂ P ln \{± P lj } , then they are isomorphic if and only if there exist1. a permutation π ∈ S n and2. a sign vector µ ∈ ( Z / Z ) n = {±} n such that for all invariant line cycles either( τ [ µ ( j ) ∗ (+)] π ( j ) ) π ( A )2 = π ( τ + j ) A π − , which is equivalent to ( τ [ µ ( j ) ∗ ( − )] π ( j ) ) π ( A )2 = π ( τ − j ) A π − holds or with a total flip the following holds. (Also we could flip the sign of µ ) ( τ [ µ ( j ) ∗ ( − )] π ( j ) ) π ( A )2 = π ( τ + j ) A π − , which is equivalent to ( τ [ µ ( j ) ∗ (+)] π ( j ) ) π ( A )2 = π ( τ − j ) A π − wherentipodal Point Arrangements on Spheres 251. π ( A ) = {± P π ( j ) , ± P π ( j ) , . . . , ± P π ( j k − ) } .2. • [ µ ( j ) ∗ (+)] = + , [ µ ( j ) ∗ ( − )] = − if µ ( j ) = +. • [ µ ( j ) ∗ (+)] = − , [ µ ( j ) ∗ ( − )] = + if µ ( j ) = − . Proof.
To determine the arrangement up to an isomorphism we do the fol-lowing. Let P i , P i , . . . , P i k +1 , P l be k + 2 points of the arrangement. With respect to A define the coefficients λ li j by letting P l = k +1 (cid:88) j =1 λ li j P i j . To determine the arrangement up to an isomorphism using Theorem 5.5we need to determine the signs of λ li j , ≤ j ≤ m using the combinatorialinvariants.Now when k = 2 we know that the line cycles give rise to the compatibleset of S -invariant set of 24 symbols locally for all the sub-arrangementsusing Theorem 3.6. These determine the signs and hence the antipodal pointarrangement on PF F is determined up to an isomorphism.Now we consider a general value of k . Now using various orthogonal projec-tions along subsets of A and repeated application of Theorem 5.4 we canrecover the signs of the coefficients λ li j from the combinatorial line cycle in-variants. Now we use Theorem 5.5.The rest of Theorem Ω also follows. (cid:4)
7. Concurrency Arrangement Sign Function and Classificationof Hyperplane Arrangements
We have already done the classification of a normal system which gives acoarse invariant for a hyperplane arrangement. In this section, with one moreinvariant which is the concurrency arrangement sign function, we completelyclassify the hyperplane arrangements up to an isomorphism over an orderedfield.
In this section we define what an orientation of simplex is and prove an usefullemma and a proposition.
Definition 7.1 (Vertex Orientation).
Let F be an ordered field. Let P i = ( x i , . . . , x im ) ⊂ F m +1 , ≤ i ≤ m + 16 C.P. Anil Kumarform the vertices of a simplex ∆ m P P . . . P m +1 . We say that the vertexorientation [ P P . . . P m +1 ] of the simplex ∆ m P P . . . P m +1 is positive ifDet x x · · · x m x x · · · x m ... ... . . . ... ...1 x m x m · · · x mm x ( m +1)1 x ( m +1)2 · · · x ( m +1) m is positive. Here we have [ P P . . . P m +1 ] = [ P σ (1) P σ (2) . . . P σ ( m +1) ] for anyeven permutation σ ∈ S m +1 the symmetric group on ( m + 1) -letters.Now we prove an useful lemma. Lemma 7.2.
Let F be an ordered field. Let H i : m (cid:88) j =1 a ij x j = c i , with a ij , c i ∈ F , ≤ j ≤ m, ≤ i ≤ m + 1 be ( m + 1) - hyperplanes forming the simplex ∆ m H H . . . H m H m +1 in F m .Let P i = ( x i , . . . , x im ) , ≤ i ≤ m + 1 be the opposite vertices of the simplexwith respect to planes H , H , . . . , H m +1 respectively. Suppose the outwardnormal for each face of the simplex is given by ( a i , a i , . . . , a im ) ∈ F m , ≤ i ≤ m + 1 , that is, m (cid:88) j =1 a ij x ij < c i , ≤ i ≤ m + 1 . The vertex orientation [ P P . . . P m +1 ] is positive if and only if the determi-nant Det a a · · · a m c a a · · · a m c ... ... . . . ... ... a m a m · · · a mm c m a ( m +1)1 a ( m +1)2 · · · a ( m +1) m c m +1 is positive.Proof. First we prove the forward implication ( ⇒ ). Consider Figure 2 whendimension m = 2. We prove this lemma as follows. First let us assumewe have a standard simplex with P the origin, P = (1 , , . . . , , P =(0 , , . . . , , P m +1 = (0 , , . . . , P P P . . . P m +1 ] ofthe simplex ∆ m P P P . . . P m +1 agrees with the outward pointing normal asthe equations for the hyperplanes are given by H : m (cid:88) i =1 x i = 1 , H j +1 : − x j = 0 , ≤ j ≤ m, ntipodal Point Arrangements on Spheres 27 Figure 2.
Dimension m = 2 for a Triangleand the above determinant becomesDet · · · − · · · · · · − · · · − = ( − m ( − m = 1 > x x · · · x m − x m x x · · · x m − x m x x · · · x m − x m ... ... ... . . . ... ...1 x ( m − x ( m − · · · x ( m − m − x ( m − m x m x m · · · x m ( m − x mm x ( m +1)1 x ( m +1)2 · · · x ( m +1)( m − x ( m +1) m > Det x x · · · x m − x m x − x x − x · · · x m − − x m − x m − x m x − x x − x · · · x m − − x m − x m − x m ... ... ... ... ... ...0 x ( m − − x x ( m − − x · · · x ( m − m − − x m − x ( m − m − x m xm − x xm − x · · · xm ( m − − x m − xmm − x m x ( m +1)1 − x x ( m +1)2 − x · · · x ( m +1)( m − − x m − x ( m +1) m − x m > if and only if Det · · · x − x x − x · · · x m − − x m − x m − x m x − x x − x · · · x m − − x m − x m − x m ... ... ... ... ... ...1 x ( m − − x x ( m − − x · · · x ( m − m − − x m − x ( m − m − x m xm − x xm − x · · · xm ( m − − x m − xmm − x m x ( m +1)1 − x x ( m +1)2 − x · · · x ( m +1)( m − − x m − x ( m +1) m − x m > a a · · · a m c a a · · · a m c ... ... . . . ... ... a m a m · · · a mm c m a ( m +1)1 a ( m +1)2 · · · a ( m +1) m c m +1 > a a · · · a m c − m (cid:80) j =1 a j x j a a · · · a m a m a m · · · a mm a ( m +1)1 a ( m +1)2 · · · a ( m +1) m > c − m (cid:88) j =1 a j x j > , for 2 ≤ i ≤ m + 1 , m (cid:88) j =1 a ij ( x ij − x j ) = m (cid:88) j =1 a ij x ij − c i < . Hence the first simplification is done by translating the space F m and assum-ing without loss of generality that the vertex P of the simplex∆ m P P . . . P m P m +1 is the origin.So let H : m (cid:88) j =1 a j x j = c , for 2 ≤ i ≤ m + 1 , H i : m (cid:88) j =1 a ij x j = 0 , where a ij , c ∈ F , ≤ j ≤ m, ≤ i ≤ m + 1with P = (0 , , . . . , , P i = ( x i , x i , . . . , x im ) , ≤ i ≤ m + 1and we have 0 < c for 2 ≤ i ≤ m + 1 , m (cid:80) j =1 a ij x ij < · · · x x · · · x m − x m x x · · · x m − x m ... ... ... . . . ... ...1 x ( m − x ( m − · · · x ( m − m − x ( m − m x m x m · · · x m ( m − x mm x ( m +1)1 x ( m +1)2 · · · x ( m +1)( m − x ( m +1) m > . Now we make the second simplification as follows. Replace a j by a j c , ≤ j ≤ m. ntipodal Point Arrangements on Spheres 29For 2 ≤ i ≤ m + 1 replace a ij by a ij | m (cid:80) j =1 a ij x ij | . This can be done becauseDet a a · · · a m c a a · · · a m a m a m · · · a mm a ( m +1)1 a ( m +1)2 · · · a ( m +1) m > a c a c · · · a m c a | m (cid:80) j =1 a j x j | a | m (cid:80) j =1 a j x j | · · · a m | m (cid:80) j =1 a j x j | a m | m (cid:80) j =1 a mj x mj | a m | m (cid:80) j =1 a mj x mj | · · · a mm | m (cid:80) j =1 a mj x mj | a ( m +1)1 | m (cid:80) j =1 a ( m +1) j x ( m +1) j | a ( m +1)2 | m (cid:80) j =1 a ( m +1) j x ( m +1) j | · · · a ( m +1) m | m (cid:80) j =1 a ( m +1) j x ( m +1) j | > H : m (cid:88) j =1 a j x j = 1 , for 2 ≤ i ≤ m + 1 , H i : m (cid:88) j =1 a ij x j = 0 , where a ij ∈ F , ≤ j ≤ m, ≤ i ≤ m + 1with P = (0 , , . . . , , P i = ( x i , x i , . . . , x im ) , ≤ i ≤ m + 1and we have for 2 ≤ i ≤ m + 1 , m (cid:80) j =1 a ij x ij = − < · · · x x · · · x m − x m x x · · · x m − x m ... ... ... . . . ... ...1 x ( m − x ( m − · · · x ( m − m − x ( m − m x m x m · · · x m ( m − x mm x ( m +1)1 x ( m +1)2 · · · x ( m +1)( m − x ( m +1) m > . Let { e i , ≤ i ≤ m } be the standard column basis for F m . Let f = ( a , a , . . . , a m ) and f i − = ( a i , a i , . . . , a im ) t , y i − = ( x i , x i , . . . , x im ) t , ≤ i ≤ m + 1 . Define a transformation A : F m −→ F m given on the basis by Ae i = y i , ≤ i ≤ m. A ) > ≤ i ≤ m, < f , y i > = 1 ⇒ < A t f , e i > = 1 ⇒ A t f = (1 , , , . . . , t ∈ F m ⇒ f t A = (1 , , , . . . , . We also conclude thatfor 1 ≤ i ≤ m, < f i , y i > = − , < f i , y j > = 0 for j (cid:54) = i, ≤ j ≤ m ⇒ < A t f i , e i > = − , < A t f i , y j > = 0 for j (cid:54) = i, ≤ j ≤ m ⇒ A t f i = − e i ∈ F m ⇒ f ti A = − e ti . Then we have a a · · · a m a a · · · a m ... ... . . . ... a m a m · · · a mm a ( m +1)1 a ( m +1)2 · · · a ( m +1) m A = − I m × m . So Det a a · · · a m a a · · · a m ... ... . . . ... a m a m · · · a mm a ( m +1)1 a ( m +1)2 · · · a ( m +1) m = ( − m Det ( A − ) . SoDet a a · · · a m a a · · · a m a m a m · · · a mm a ( m +1)1 a ( m +1)2 · · · a ( m +1) m = ( − m ( − m Det ( A − ) > . This proves the forward implication and with the last determinant equality,the converse ( ⇐ ) also holds, completing the proof of the lemma. (cid:4) Note 7.3.
1. In Lemma 7.2 the determinant sign is negative if we change any tworows i < j , that is, when the vertex orientation [ P P . . . P i − P j P i +1 . . . P j − P i P j +1 . . . P m +1 ] is negative. Also the determinant sign is negative if we keep the orderof the rows same but change any row vector to its negative vector.2. This lemma is useful in fixing the signs of the sign map coherently (referto [12, 13]) under the isomorphism as mentioned in Theorem Σ . Now we prove another useful proposition.ntipodal Point Arrangements on Spheres 31
Proposition 7.4.
Let F be an ordered field. Let ( H mn ) F k = { H ki : m (cid:88) j =1 a kij x j = c ki , ≤ i ≤ n } , k = 1 , be two hyperplane arrangements of n hyperplanes in the space F m which areisomorphic. For any ≤ i < i < . . . < i m < i m +1 < n consider thesimplices ∆ m H ki H ki . . . H ki m H ki m +1 , k = 1 , . Let the opposite vertex of theplane H ki j in the simplex ∆ m H ki H ki . . . H ki m H ki m +1 be denoted by P ki j = H ki ∩ H ki ∩ . . . H ki j − ∩ H ki j +1 ∩ . . . ∩ H ki m ∩ H ki m +1 , k = 1 , . Then we have • either the vertex orientation [ P i P i . . . P i m P i m +1 ] of the simplex ∆ m H i H i . . . H i m H i m +1 uniformly agrees with the vertex orientation [ P i P i . . . P i m P i m +1 ] of the simplex ∆ m H i H i . . . H i m H i m +1 indepen-dent of the choice ≤ i < i < . . . < i m < i m +1 < n , • or the vertex orientation [ P i P i . . . P i m P i m +1 ] of the simplex ∆ m H i H i . . . H i m H i m +1 uniformly disagrees with the vertex orientation [ P i P i . . . P i m P i m +1 ] of the simplex ∆ m H i H i . . . H i m H i m +1 indepen-dent of the choice ≤ i < i < . . . < i m < i m +1 < n .Proof. If P ki j = ( x ki j , x ki j , . . . , x ki j m ) ∈ F m , ≤ j ≤ m + 1 , k = 1 , P ki P ki . . . P ki m P ki m +1 ] of the simplex∆ m H ki H ki . . . H ki m H ki m +1 is given by sign of the determinantDet x ki x ki · · · x ki ( m − x ki m x ki x ki · · · x ki ( m − x ki m x ki x ki · · · x ki ( m − x ki m ... ... ... . . . ... ...1 x ki m − x ki m − · · · x ki m − ( m − x ki m − m x ki m x ki m · · · x ki m ( m − x ki m m x ki m +1 x ki m +1 · · · x ki m +1 ( m − x ki m +1 m for k = 1 ,
2. Hence it is enough to prove that the sign of ratio the determinantseither always positive or is always negative.First let us assume that i = 1 , i = 2 , . . . , i m = m . For k = 1 , L ki = H k ∩ . . . ∩ H ki − ∩ H ki +1 ∩ . . . ∩ H km be m -lines passing through P k = H k ∩ H k ∩ . . . ∩ H km . Also let us assume that P k is the origin for k = 1 , H km +1 , H kr , m + 2 ≤ r ≤ n be the two planes ofthe arrangements ( H mn ) F k , k = 1 ,
2, respectively. Let P ki = L ki ∩ H km +1 and Q ki = L ki ∩ H kr , ≤ i ≤ m . We conclude that for 1 ≤ i ≤ m the points P k , P ki , Q ki are collinear and lie on the line L ki for k = 1 , P is in between P i , Q i P is in between P i , Q i for 1 ≤ i ≤ m . In the vector notation,if for some, λ ki ∈ F ∗ = F \{ } , −−−→ P k Q ki = λ ki −−−→ P k P ki , k = 1 , λ i λ i > ≤ i ≤ m. Now for k = 1 , P k P k . . . P km P k ] for the simplex∆ m H k H k . . . H km H km +1 for the choice 1 = i < i = 2 < . . . < i m = m . This proves uniformity of agreement or disagreement of vertex orientations ofsimplices containing origin as a vertex in both the arrangements. Now witha similar reasoning the proof of uniformity can be extended to all pairs ofcombinatorially similar simplices 1 ≤ i < i < . . . < i m < i m +1 ≤ n inboth the arrangements ( H mn ) F k , k = 1 , H ki j ofntipodal Point Arrangements on Spheres 33the simplex ∆ m H ki H ki . . . H ki m H ki m +1 at a time. This completes the proof ofthe proposition. (cid:4) Here in this section we prove an extension theorem to obtain a bijectionbetween cones of the concurrency arrangements. We need a definition.
Definition 7.5 (Simplex Polyhedrality).
Let ( H mn ) F = { H , H , . . . , H n } be a hyperplane arrangement of n hyperplanes in an m - dimensional spaceover the ordered field F . We say a set of m + 1 hyperplanes { H i , H i , . . . , H i m +1 | ≤ i < i < . . . < i m < i m +1 ≤ n } give rise to an m - dimensional simplex polyhedrality of the arrangement ifthe equations of these m + 1 hyperplanes gives rise to a bounded polyhedralregion (refer to Definition 1.7) of the arrangement.Now we prove another required theorem before we state the extension theo-rem. Theorem 7.6.
Let F be an ordered field and ( H mn ) F = { H , H , . . . , H n } , ( H mn ) F = { H , H , . . . , H n } be two arrangements which are isomorphic by an iso-morphism which is identity on the subscripts. Let ( C n ( nm +1 )) F , ( C n ( nm +1 )) F be theirassociated concurrency arrangements respectively and let the constant coeffi-cients be given by ( b , b , . . . , b n ) , ( c , c , . . . , c n ) respectively which lie in the interior of two cones of the concurrency arrange-ments ( C n ( nm +1 )) F , ( C n ( nm +1 )) F respectively. Suppose the subscripts ≤ i < i <. . . < i m < i m +1 ≤ n gives rise to an m - dimensional simplex polyhedrality(refer to Definition 7.5) of both the arrangements. Let the constant coeffi-cients ( b , b , . . . , b n ) , ( c , c , . . . , c n ) which lie in the interior of the two conesbe moved to the interior of their adjacent cones passing through single bound-ary hyperplanes (co-dimension one) corresponding to ≤ i < i < . . .
2, on the line (cid:84) i ∈ A H ji , there is a swap of points (cid:92) i ∈ A H ji ∩ H jj m , (cid:92) i ∈ A H ji ∩ H jj m +1 . H mn ) F , ( ˜ H mn ) F are isomorphicby an isomorphism which is also identity on the subscripts. This proves thetheorem. (cid:4) m - dimensional simplex polyhedralities of a hyperplane ar-rangement. Here we mention a note that given a hyperplane arrangementhow many m -dimensional simplex polyhedralities exist in the arrangement. Note 7.7 (Number of Simplex Polyhedralities of a Hyperplane Arrangement).
Let F be an ordered field and ( H mn ) F = { H i : m (cid:80) j =1 a ij x j = c i , ≤ i ≤ n, a ij , c i ∈ F } be a hyperplane arrangement in F m . Let ( C n ( nm +1 )) F be its associated con-currency arrangement in F n . Let C denote the convex cone containing thepoint ( c , c , . . . , c n ) of ( C n ( nm +1 )) F in its interior.The number of simplex polyhedralities of the hyperplane arrangement ( H mn ) F = { H , . . . , H n } is precisely equal to the number of co-dimension oneboundary hyperplanes of F m in the concurrency arrangement of the convexcone C containing ( c , c , . . . , c n ) . For ≤ i < i < . . . < i m < i m +1 ≤ n ,the simplex polyhedrality ∆ m H i H i . . . H i m H i m +1 gives the boundary hyper-plane M { i ,i ,...,i m ,i m +1 } ∈ ( C n ( nm +1 )) F of the convex cone C and vice-versa. The extension theorem is stated as follows.
Theorem 7.8.
Let F be an ordered field. Let ( H mn ) F k = { H ki : m (cid:88) j =1 a kij x j = c ki , ≤ i ≤ n } , k = 1 , be two hyperplane arrangements of n hyperplanes in the space F m . With thenotations in Definitions 1.16, let ( C n ( nm +1 )) F , ( C n ( nm +1 )) F be their associated con-currency arrangements respectively. Let ( c , c , . . . , c n ) ∈ C, ( c , c , . . . , c n ) ∈ D where C, D represent convex cones in the concurrency arrangements ( C n ( nm +1 )) F , ( C n ( nm +1 )) F respectively. Suppose the arrangements ( H mn ) F k , k = 1 , are isomor-phic to each other under the isomorphism which is identity on the subscripts(refer to Definition 1.10). Then there exists a bijection ψ : ( C n ( nm +1 )) F −→ ( C n ( nm +1 )) F such that ψ ( C ) = D and for ˜ C ∈ ( C n ( nm +1 )) F , ˜ D ∈ ( C n ( nm +1 )) F , if ψ ( ˜ C ) = ψ ( ˜ D ) then the ˜ C, ˜ D give rise to hyperplane arrangements which arealso isomorphic under the isomorphism which is identity on subscripts.Proof. We define the bijection as follows. Using Theorem 7.6 we move toadjacent cones to obtain cones which are isomorphic under an isomorphismwhich is identity on subscripts. By any finite sequence of corresponding movesof juxtaposed cones from C to ˜ C in ( C n ( nm +1 )) F and D to ˜ D in ( C n ( nm +1 )) F wentipodal Point Arrangements on Spheres 35obtain an extension with ψ ( ˜ C ) = ˜ D with the required property. We onlyneed to prove that this extension is well defined. For this we consider anysequence of moves of the constant coefficients from C back to C giving riseto a loop through various adjacent cones of the concurrency arrangement( C n ( nm +1 )) F . Then at this sequence we evaluate the signs of the hyperplanes ofthe concurrency arrangements. The set of signs will come back to the originalset of signs while moving and coming back to C . A similar pattern of signchanges occurs while moving from D ∈ ( C n ( nm +1 )) F because C and D have thesame set of boundary hyperplanes. Hence we must come back to D again. Soa loop of moves has to go to a loop of moves. This fact gives a well definedand unique bijection ψ : ( C one n ( nm +1 )) F −→ ( C one n ( nm +1 )) F with the required property. This proves the theorem. (cid:4) We state one more definition.
Definition 7.9 (Infinity Arrangement).
Let F be an ordered field and ( H nm ) F be a hyperplane arrangement. We say( H nm ) F is an infinity arrangement if there exists a permutation σ ∈ S n suchthat the hyperplane H σ ( l ) is a hyperplane at infinity with respect to thearrangement { H σ (1) , H σ (2) , . . . , H σ (1 − } , ≤ l ≤ n.
8. Proof of Second Main Theorem
Now we prove the isomorphism theorem which is the second main Theorem Σof the article.
Proof.
In the concurrency arrangement associated to a hyperplane arrange-ment over the ordered field, the constant coefficients corresponding to pointsin a convex cone C and its opposite cone − C are all isomorphic under iso-morphisms which are identity on subscripts.To prove the forward implication ( ⇒ ) we observe the following in a sequenceof steps.1. Since the given hyperplane arrangements are isomorphic there exists aconvex positive bijection δ : U −→ U .2. Assume without loss of generality that δ induces trivial permutation onthe subscripts upon renumbering the second hyperplane arrangement.Also assume that upon relabeling the antipodal pairs of vectors in U and changing the signs of constant coefficients in ( H mn ) F coherently, ifnecessary, we have δ ( v i ) = v i , ≤ i ≤ n and we have two concurrency arrangements( C n ( nm +1 )) F , (cid:0) ( C n ( nm +1 )) F (cid:1) δ = ( C n ( nm +1 )) F c , c , . . . , c n ) , ( c , c , . . . , c n ) give rise to isomorphic arrange-ments by an isomorphism which is trivial on subscripts.3. Let ( c , c , . . . , c n ) ∈ C, ( c , c , . . . , c n ) ∈ D where C, D represent convexcones in the concurrency arrangements ( C n ( nm +1 )) F , ( C n ( nm +1 )) F respectively.4. Since they are isomorphic by an isomorphism which is identity on sub-scripts, the boundary co-dimension one hyperplanes giving rise to C and D have the same set of subscripts which correspond to the setof m -dimensional simplex polyhedralities present in the isomorphic ar-rangements ( H mn ) F , ( H mn ) F .5. This pairing between C and D extends to a bijection of all the conesin ( C n ( nm +1 )) F and ( C n ( nm +1 )) F using Theorem 7.8 such that each bijectivepair ˜ C, ˜ D of cones correspond to isomorphic arrangements under theisomorphism which is trivial on subscripts.6. Now the important assumption we make is that, both the arrangements( H mn ) F , ( H mn ) F are infinity arrangements with permutations σ , σ bothbeing identity (refer to Definition 7.9). Such an isomorphic bijective pair( ˜ C, ˜ D ) of cones exists.7. For further reasoning we stick to Condition (a) using a reflection on oneof the arrangements if necessary. This we do it mainly to make someobservations independent of the value k = 1 ,
2, that is, independent ofthe two arrangements. This is done as follows.8. For 1 ≤ i < i < . . . < i m < i m +1 ≤ n , let ∆ m H ki H ki . . . H ki m +1 , k =1 , P ki j be the ver-tex opposite to H ki j in the simplex ∆ m H ki H ki . . . H ki m +1 , k = 1 , , ≤ j ≤ m +1. Now we use Proposition 7.4. The vertex orientation [ P i P i . . . P i m +1 ]is positive on the simplex ∆ m H i H i . . . H i m +1 if and only if the vertexorientation [ P i P i . . . P i m +1 ] is positive on the simplex ∆ m H i H i . . . H i m +1 .Here is where we may need to reflect one of the arrangements to maintainuniformity of the agreement (and not disagreement) of vertex orienta-tions of all the similar simplices 1 ≤ i < i < . . . < i m < i m +1 ≤ n inthe arrangements for k = 1 , w ki l ∈ {± v ki l } , ≤ l ≤ ( m + 1) , k = 1 , m H ki H ki . . . H ki m +1 , k = 1 , λ ki l , ≤ l ≤ m + 1 , k = 1 , m +1 (cid:88) l =1 λ ki l w ki l = 0 , λ ki l > δ ( w i l ) = w i l , ≤ l ≤ m + 1 uniformlyfor any such choice 1 ≤ i < i < . . . < i m +1 ≤ n .10. Now to prove Condition (a) on the sign maps S , S it is enough toprove for these two isomorphic infinity arrangements ( H mn ) F , ( H mn ) F .11. Both values S (cid:0) M { i ,i ,...,i m +1 } (cid:1) , S (cid:0) M { i ,i ,...,i m +1 } (cid:1) ntipodal Point Arrangements on Spheres 37represent same signed values because of the following. We refer to Fig-ure 2 and use previous Lemma 7.2).12. The signs of S k , k = 1 , w ki l to v ki l , ≤ l ≤ m + 1 and the vertexorientation [ P ki P ki . . . P ki m +1 ] on the simplex ∆ m H ki H ki . . . H ki m +1 whichdoes not depend on whether k = 1 , S k , k = 1 , − δ for δ ifnecessary that Condition (a) holds. Assume without loss of generalitythat δ is trivial on subscripts.2. Now by applying similar changes to the constants in the cones C, D , wemove the constants to other pair of cones as we already have a bijectionof subscripts given by δ . Now we can assume that one of them, the firstone, is an infinity arrangement with the permutation σ being identity(refer to Definition 7.9) whose convex cone is ˜ C and that Condition (a)holds. Let the moved convex cone for the other be ˜ D .3. Now tracing back the argument, about signs and the normals, againusing Lemma 7.2 we conclude that the second arrangement is also aninfinity arrangement with its associated permutation σ being identityas well. This is done as follows.4. This argument, about tracing back signs and normals, goes throughusing Lemma 7.2, for the pair ( ˜ C, ˜ D ), because we observe the following.In a hyperplane arrangement, if we have a hyperplane such that, theoutward normal of all simplices with this hyperplane face is same, thenthis hyperplane must be a hyperplane at infinity.5. Now both are infinity arrangements and there exists a convex posi-tive bijection. For the pair of cones ( ˜ C, ˜ D ) we conclude using the proofof the forward implication ( ⇒ ) in the proof of the Normal Represen-tation Theorem 1.14 (refer to Section-Proof of the main theorem inC. P. Anil Kumar [2]), that, these two are isomorphic.6. Now we reverse the changes applied to the constants performed in thefirst one in step 2 and hence on the second one accordingly to concludethat both the given initial arrangements with constant coefficients incones C, D are isomorphic.This completes the proof of this theorem. (cid:4)
9. Open Questions: The Enumeration Problem and TheProblem of a Complete List of Representatives
We have solved the classification problem of isomorphism classes of hyper-plane arrangements and the classification problem of isomorphism classes ofnormal systems in any dimension. Now we mention the remaining two ques-tions which are still open.
Question 9.1.
Let F be an ordered field and n, m be positive integers.1. (Enumeration Problem): Enumerate the isomorphism classes of normalsystems in F m of cardinality n .2. (Representation Problem): Construct a complete list of representativesfor the list of isomorphism classes of normal systems in F m of cardinal-ity n . References [1] M. Aigner, G. M. Ziegler,