On deformation and classification of V-systems

The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. We discuss also possible matroidal structures of V-systems in relation with projective geometry and give the catalogue of all known irreducible rank 3 V-systems.


Introduction
The ∨-systems are special finite sets of covectors introduced in [19,20]. The motivation came from the study of certain special solutions of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, playing an important role in 2D topological field theory and N = 2 SUSY Yang-Mills theory [2,10].
Let V be a real vector space and A ⊂ V * be a finite set of vectors in the dual space V * (covectors) spanning V * . To such a set one can associate the following canonical form G A on V : where x, y ∈ V , which establishes the isomorphism The inverse ϕ −1 A (α) we denote as α ∨ . The system A is called ∨-system if the following relations Two matrices A and A representing the same matroid M are said to be projectively equivalent representations of M if A can be obtained from A by a sequence of these operations. Equivalently, one can say that A = CAD, where C is an invertible r × r matrix, and D is a diagonal n × n matrix with non-zero diagonal entries.
Alternatively, one can define the linear dependence matroid on the set X as a family I C of minimal dependent subsets C of X (called circuits) through the following axioms: • The empty set is not a circuit.
• No curcuit is contained in another circuit.
• If C 1 ,C 2 ∈ I C are two circuits sharing an element e ∈ X, then (C 1 ∪ C 2 ) \ e is a circuit or contains a circuit.
The rank of a circuit is defined as the dimension of the vector space spanned by its vectors. Circuits spanning the same d-dimensional subspace can be united in so-called d-flats. A set F ⊆ X is a flat of the matroid M if for all x ∈ X \ F, where r(F) is the rank of the flat F. The matroid can be labelled by listing all d-flats.
As an example consider the positive roots of the B 3 -type system. The corresponding matrix (with the first row giving the labelling) is  (3,4,8,9), (1,2,7,8), (5,6,7,9)} with three and four elements respectively. Together with the 3-flat X this gives the complete list of flats.
Graphically on the projective plane we have A matroid is called simple if it does not contain one-or two-element circuits. For vector matroids this means that no two vectors are proportional.
Number of matroids up to isomorphism grows very rapidly with n = |X|. The following table summarises the results for rank 3-matroids for small n (see [12]). Vector matroids build the class of realisable matroids. The problem of finding a criterion for realisability is known to be NP-hard [14].
Let M be a rank r vector matroid. We say that matroid M is projectively rigid if the space of all its rank r vector realisations Let G be a finite Coxeter group, which is a finite group generated by the hyperplane reflections in a Euclidean space. We say that matroid M is of Coxeter type if it describes the vector configuration of the normals to the corresponding reflection hyperplanes (one for each hyperplane) for such a group. For rank three Coxeter matroids we have the following result. Theorem 1. The matroids of Coxeter types A 3 and B 3 are strongly projectively rigid. The matroid of type H 3 is projectively rigid with precisely two projectively non-equivalent vector realisations.
Proof. Let us prove this first for B 3 case. Since the images a 1 , a 2 , a 3 , a 4 of the elements 1,2,3 and 4 in the projective plane form a projective basis it is enough to prove that the remaining a 5 , a 6 , a 7 , a 8 , a 9 can be constructed uniquely. From the matroid structure we can see that x 5 must be an intersection point of the lines (2-flats) a 1 a 3 and a 2 a 4 . We denote this as using the general lattice theory notation. Similarly we have Similarly one can prove the rigidity in A 3 case (see Fig. 2). In both these cases the space of realisations modulo projective equivalence consists of only one point. The H 3 case is more interesting. Fig. 3 shows the graphic representation of the system H 3 in the real projective plane RP 2 .
Recall that on the projective line RP 1 any three points can be mapped into any other three via the action of the group PGL(2, R). For four distinct points p 1 , p 2 , p 3 , p 4 on the projective line RP 1 with homogeneous coordinates [x i , y i ] there is a projective invariant, namely cross-ratio defined as If none of the y i is zero the cross-ratio can be expressed in terms of the ratios z i = x i y i as follows: Since any projection from a point in the projective plane preserves the cross-ratio of four points we have the equalities (a 6 , a 5 ; a 9 , a 3 ) = (a 4 , a 7 ; a 10 , a 3 ) = (a 5 , a 6 ; a 8 , a 3 ), (a 6 , a 5 ; a 9 , a 3 ) = (a 7 , a 11 ; a 10 , a 3 ) = (a 5 , a 8 ; a 9 , a 3 ).
Using elementary manipulations with cross-ratios one can show that that these equalities imply that x = (a 6 , a 5 ; a 9 , a 3 ) satisfies the equation 2 . If we fix the positions of the four points a 4 , a 5 , a 6 , a 7 forming a projective basis in RP 2 we can first reconstruct a 1 = (a 5 a 4 ) ∧ (a 6 a 7 ), a 2 = (a 5 a 7 ) ∧ (a 6 a 4 ), a 3 = (a 5 a 6 ) ∧ (a 7 a 4 ).
Then using the knowledge of x = (a 6 , a 5 ; a 9 , a 3 ) we can reconstruct a 9 and all the remaining points as a 14 = (a 2 a 9 ) ∧ (a 5 a 4 ), a 12 = (a 7 a 6 ) ∧ (a 2 a 9 ), a 10 = (a 2 a 9 ) ∧ (a 3 a 4 ), a 13 = (a 9 a 4 ) ∧ (a 6 a 7 ), a 8 = (a 2 a 13 ) ∧ (a 3 a 6 ), a 15 = (a 2 a 13 ) ∧ (a 5 a 4 ), Thus we have shown that modulo projective group we have only two different vector realisations of matroid H 3 .  Remark. The existence of two projectively non-equivalent realisations is related to the existence of a symmetry of matroid M(H 3 ), which can not be realised geometrically, see [3]. These two realisations are related by re-ordering of the vectors and thus give rise to the equivalent ∨-systems. For a generic vector matroid this set is actually empty. For example, for n vectors a i in R 3 in general position the ∨-conditions imply that these vectors must be pairwise orthogonal, which is impossible if n > 3.

Classification Problem for ∨-Systems of Given Matroidal Type
In the case when the space R ∨ (M) is known to be non-empty (for example, for all vector matroids M of Coxeter type) we have the question of how to describe this space effectively.
For the case of matroid of Coxeter type A 3 the answer is known [1]. The positive roots of Since matroid A 3 is strongly projectively rigid it is enough to consider the system All the corresponding ∨-systems can be parametrized as with arbitrary positive real c 1 , . . . , c 4 .
Without loss of generality, we may choose c 4 = 1 and consider the restriction of the system onto the hyperplane x 4 = 0. This gives the following parametrisation of the space R ∨ (M(A 3 )) by positive real c 1 , c 2 , c 3 as Consider now the case B 3 , corresponding to the following configuration of vectors in R 3

Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors 549 The following 4-parametric family of ∨-systems of B 3 -type was found in [1]: with arbitrary positive c 1 , c 2 , c 3 and γ such that c i + γ > 0 for all i = 1, 2, 3. Proof. The proof is by direct computations, but we present here the details to show the algebraic nature of ∨-conditions in this example.
Since B 3 matroid is strongly projectively rigid, we can assume that the corresponding ∨-system has the form where all the parameters can be assumed without loss of generality to be positive.
The corresponding form G has the matrix In case 1. the ∨-conditions are just the orthogonality conditions G(α ∨ , β ∨ ) = 0 for the corresponding two covectors α and β in the plane Π. We obtain the system Note that the second factors in all equations are ratios of principal minors of matrix G and thus must be positive, since the form G is positive definite. This implies that α i j = α i j , which reduces the matrix G to In cases 2. and 3. we fix for each plane Π a basis v 1 , v 2 ∈ A ∩ Π. The corresponding dual plane Π ∨ is spanned by v ∨ 1 and v ∨ 2 and the ∨-condition implies the proportionality of the restrictions of the forms G and G Π onto Π ∨ . In our case this proportionality turns out to be equivalent to the following system of equations: Introducing new parameters c i , i = 1, 2, 3 and γ by we can see that these relations imply which leads to the parametrisation (3.2).
For larger matroids the direct analysis of the ∨-conditions is very difficult, so we consider a simpler problem about infinitesimal deformations of ∨-systems.

Deformations of ∨-Systems
For projectively rigid matroids M one can always reduce any deformation to such a form. Let ξ α =μ α (0). We are going to derive the conditions on ξ α , which can be considered as linearised ∨-conditions for such deformations.
Consider now the ∨-conditions. For any two-dimensional plane containing only two covectors we have Differentiating it in t we havė where here and below byα ∨ Thus we have and thus Substituting this into (4.2) we have the first linearised ∨-condition: for α, β being the only two covectors in a plane Π we have Let now Π be a two-dimensional plane containing more than two covectors from A (and hence from A t . Then from the ∨-conditions there exists ν = ν(Π) ∈ R such that for any [5]). Now assuming that A depends on t as above and differentiating with respect to t at t = 0 we have as before for any α, Since this is true for all α, β ∈ Π ∩ A we have the second linearised ∨-condition: for any plane Π containing more than two covectors from A we have where the sign ∼ means proportionality. Thus we have proved Case by case check of the ∨-systems from the Appendix leads to the following Theorem 5. All rank three vector matroids corresponding to known irreducible 3D ∨-systems are projectively rigid. The H 3 matroid is the only one, which is not strongly projectively rigid.
Let us show that the largest known case (H 4 , A 1 ) is strongly projectively rigid. We will use the labelling of the points shown at the last figure of the paper. A direct computation shows that in case of the classical systems A 3 and B 3 the linear system (4.2),(4.5) has corank four in agreement with the results of the previous section.
The analysis of the linearised ∨-conditions for the families D 3 (t, s), F 3 (t), G 3 (t) and (AB 4 (t), A 1 ) 1,2 shows that these families of ∨-systems can not be extended.
Consider, for example, the family of ∨-systems D 3 (t, s) from [5] with with real parameters s,t such that |s − t| < 1, s + t > 1. Matrices G and X have the form For the three covectors α 5 , α 6 , α 7 the first linearised ∨-conditions X(α ∨ i , α ∨ j ) = 0, i, j = 5, 6, 7 are equivalent to For the planes with more than two covectors we have the linear system (t + 1)(−t(s(ξ 1 + ξ 4 − 2(ξ 5 + ξ 6 )) + ξ 3 (3s + 2) − 2(ξ 5 + ξ 7 )) + s(−s(ξ 1 + 2ξ 6 ) + ξ 3 A check with Mathematica shows that the co-rank of the total system is three for every admissible values of s and t. The free parameters correspond to two deformation parameters s and t and the uniform scaling of the system. This approach with the use of Mathematica (see the programme in Appendix B to [15]) allows us to prove that the isolated examples of ∨-systems from the list [5] are indeed isolated.

Matroidal Structure of ∨-Systems and Projective Geometry
The main part of the classification problem is to characterise the corresponding class of possible matroids. This question was addressed by Lechtenfeld et al in [9]. They developed a Mathematica program, which generates simple and connected matroids of a given size of the ground set X. If a generated matroid has a vector representation, they have checked first if the orthogonality ∨conditions are possible to satisfy before verification of the ∨-conditions for the non-trivial planes (all 2-flats). For matroids with n < 10 elements the orthogonality conditions are strong enough to identify all matroids corresponding to ∨-systems in dimensions three. All the identified ∨-systems turned out to be part of the list in [4]. For larger matroids this approach seems unworkable because of the unreasonably large computer time required. This means that we need a more conceptual approach, which is still missing.
In this section we collect some partial observations based on the analysis of the known 3D ∨-systems and projective geometry.
We start with the notion of extension and degeneration for ∨-systems. Let A 1 ,A 2 ⊂ V * be two ∨-systems. If A 2 ⊂ A 1 we call A 1 an extension of A 2 . Let ∨-system A = A t depend on the parameter t. Assume that for some t = t 0 one or more of the covectors α ∈ A t 0 vanishes. In that case the system A = lim . A reverse process we will call regeneration.
In the tables below we give the list of all extensions and degenerations for known threedimensional ∨-systems from the catalogue in the Appendix.
We start with the matroid of the ∨-system of type A 3 . In projective geometry (see e.g. [8]) it is known as the simplest configuration (6 2 4 3 ) consisting of four lines with three points on each line and two lines passing through every point. Its projective dual is a complete quadrangle (4 3 6 2 ) consisting of four points, no three of which are collinear and six lines connecting each pair of points (see figure  4). If we extend the dual configuration by adding the remaining three points of intersections of lines (the points marked white in the graphic), we come to the projective configuration of seven points and six lines, corresponding to the matroid of the ∨-system of type D 3 .
dual Fig. 4. The projective configuration of A 3 type, its dual and the extended configuration corresponding to ∨-system of type D 3 .
We can proceed the construction by taking the dual of the new obtained configuration and extending it by adding the missing points of intersections of lines. The result is the configuration of nine points and seven lines realisable as B 3 -type ∨-system (see figure 5). The next step of the construction is demonstrated in figure 6. The dual configuration was obtained from the configuration D 3 by adding all missing lines passing through any pair of points. Applying Desargue's theorem to two marked triangles we see that the white marked points of the extended configuration are collinear. The new configuration of 10 points and 10 lines is self-dual and corresponds to the ∨-system of type (AB 4 (t), A 1 ) 2 (see system 9.6 in the Appendix).
One can check that the adding of three intersection points with red lines and three lines connecting them pairwise leads to the configuration of F 3 type (see system 9.10). However, if we add also the three intersection points with dotted line then we come to the configuration which can be shown to be not ∨-realisable.
Although this relation with projective configurations and theorems in projective geometry looks quite promising, we see that the extension procedure is not straightforward and does not guarantee the ∨-realisability of the resulting configuration.
We conclude with the following conjecture about 2-flats with precisely four points. Recall that four points A, B,C, D on a projective line form a harmonic range if the cross-ratio (A, B;C, D) = −1. The corresponding pencil of four lines on a plane is called harmonic bundle. The B 3 configuration provides a geometric way to construct harmonic ranges: on Fig. 1 the points 3,4,9,8 always form a harmonic range. Note that the covectors 8 and 9 are orthogonal and determine the bisectors for the lines corresponding to covectors 3 and 4. Case by case check of the known 3D ∨-systems suggests that the same is true in general. A , i = 1, . . . , 4, then the corresponding four lines form a harmonic bundle with two orthogonals.

ν-Function, Uniqueness and Rigidity Conjectures
Let M be a matroid and A be its ∨-system realisation. Such a realisation defines the ν-function on the 2-flats of M, where ν is the coefficient in the ∨-conditions (1.1) corresponding to the plane Π representing the flat.

Conjecture 2. (Uniqueness Conjecture) An irreducible ∨-system A is uniquely determined modulo linear group GL(V * ) by its matroid M and the corresponding ν-function on its flats.
A weaker version of the conjecture is

Conjecture 3. (Rigidity Conjecture) An irreducible ∨-system A is locally uniquely determined by its matroid M and the corresponding ν-function on its flats.
If the function ν is fixed under deformation thenν = 0 and the corresponding ∨-conditions are for α, β be the only two covectors in the plane, and for any plane Π containing more than two covectors from A . Conjecturally this should imply that X = cG corresponding to the global scaling of the system. Case by case check from the list in the Appendix leads to the following Theorem 7. Both conjectures are true for all known ∨-systems in dimension three.
We have also the following conjecture based on the analysis of the list of extensions of ∨systems from the previous section. First we give the following, more direct geometric way to compute ν(Π). The form G A on V defines the scalar product on V * and thus the norm |α|, α ∈ V * . Theorem 8. For every plane Π ⊂ V * containing more than two covectors α from a ∨-system A Taking the trace of both sides gives (6.3).
Let A ⊂ V * be a ∨-system generating V * and consider the set F A of 2-flats in the corresponding matroid, which the same as the set of 2D planes Π ⊂ V * containing more than two covectors from A .
We say that the set of weights Theorem 9. For every admissible set of weights we have where n is the dimension of V.
where P Π is the orthogonal projector onto Π ∨ . Taking trace and using the fact that ∑ β ∈A β ∨ ⊗ β = Id we obtain (6.5).
We call (6.5) the universal relation for values of function ν.
For the ∨-system of type A 3 the universal relation completely describes the set of all possible functions ν. Indeed, one can easily see from Fig. 2 that x Π = 1/2 is the only admissible weight system, which leads to the universal relation This gives us three free parameters, which are exactly three parameters of deformation.
However, in general universal relations are not strong enough to describe possible ν-functions. Moreover, a ∨-system A may not have admissible weights x Π at all. For instance, this is the case for the ∨-system of D 3 (t)-type (this is however the only exception among known 3D ∨-systems).
The list of all known ∨-systems in dimension three with the corresponding ν-functions is given in the Appendix.

Concluding Remarks
Although the problem of classification of ∨-systems seems to be very hard, in dimension three it does not look hopeless. As we have seen, matroid theory provides a natural framework for the problem of classification of ∨-systems. For a given matroidal structure the ∨-conditions define a set of algebraic relations on the vector realisations. In case when matroid is strongly projectively rigid we have one free parameter for each vector, which makes possible the full classification of ∨-systems with small number of vectors.
The main problem is to describe all possible matroidal types, which we believe form a finite list in any dimension. The results of Lechtenfeld et al [9] show that the direct computer approach is probably unrealistic for ∨-systems with more than 10 covectors, while we have already in dimension three an example with 31 covectors (system (H 4 , A 1 ), see 7.16 in the Appendix). In dimension three we have an intriguing relation with the theory of configurations on the projective plane and with the theorems in projective geometry, which also suggests that the final list should be finite.
In the theory of matroids and graphs many families have been proved to be closed under taking minors, thus giving a possibility to reduce the problem of classification to the identification of the forbidden minors. We hope that a similar approach could be fruitful for classification of ∨-systems.
Another result from matroid theory, which could be relevant, is Seymour's decomposition theorem [18], which states that all regular matroids can be build up in a simple way as sums of certain type of graphic matroids, their duals, and one special matroid on 10 elements. Our analysis of degenerations and extensions of ∨-systems suggests a possibility of a similar result for the ∨-realisable matroids.

Acknowledgements
We are grateful to Olaf Lechtenfeld and especially to Misha Feigin for useful discussions. We thank also an anonymous referee for a very thorough job.
The work of APV was partly supported by the EPSRC (grant EP/J00488X/1).

Appendix. Catalogue of all Known Real 3-Dimensional ∨-Systems
Each 3D ∨-system A is presented below by the matrix with columns giving the covectors of the system (the first row is simply the labelling of the covectors). We give the graphical representation of the corresponding matroid with the list of orthogonal pairs, 2-flats, the form G and the values of ν-function. The ordering of the list is according to the number of covectors in the system. The parameters are assumed to be chosen in such a way that all the covectors are real and non-zero. Below is a schematic way to present all known ∨-systems in dimension three taken from [5].