OON DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS V. SCHREIBER AND A.P. VESELOV
Abstract.
The ∨ -systems are special finite sets of covectors which appearedin the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)equations. Several families of ∨ -systems are known, but their classificationis an open problem. We derive the relations describing the infinitesimal de-formations of ∨ -systems and use them to study the classification problem for ∨ -systems in dimension three. We discuss also possible matroidal structuresof ∨ -systems in relation with projective geometry and give the catalogue of allknown irreducible rank three ∨ -systems. Keywords : Root systems; ∨ -systems; WDVV equation.1. Introduction
The ∨ -systems are special finite sets of covectors introduced in [19, 20]. The mo-tivation came from the study of certain special solutions of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, playing an important role in 2Dtopological 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 dualspace V ∗ (covectors) spanning V ∗ . To such a set one can associate the following canonical form G A on V : G A ( x, y ) = (cid:88) α ∈A α ( x ) α ( y ) , where x, y ∈ V , which establishes the isomorphism ϕ A : V → V ∗ . The inverse ϕ − A ( α ) we denote as α ∨ . The system A is called ∨ - system if thefollowing relations(1) (cid:88) β ∈ Π ∩A β ( α ∨ ) β ∨ = να ∨ (called ∨ - conditions ) are satisfied for any α ∈ A and any two-dimensional planeΠ ⊂ V ∗ containing α and some ν , which may depend on Π and α. If Π containsmore than two covectors, then ν does not depend on α ∈ Π and the correspondingtwo forms G A and G Π A ( x, y ) := (cid:88) α ∈ Π ∩A α ( x ) α ( y )are proportional on the plane Π ∨ ⊂ V (see [19, 20]). If Π contains only two covectorsfrom A , say α and β, then we must have G A ( α ∨ , β ∨ ) = 0 . a r X i v : . [ m a t h - ph ] N ov V. SCHREIBER AND A.P. VESELOV
The ∨ -conditions are equivalent to the flatness of the corresponding Knizhnik–Zamolodchikov-type ∨ -connection ∇ a = ∂ a + κ (cid:88) α ∈A (cid:104) α, a (cid:105)(cid:104) α, x (cid:105) α ∨ ⊗ α. The examples of ∨ -systems include all two-dimensional systems, Coxeter config-urations and so-called deformed root systems [11, 17, 19], but the full classificationis an open problem. The main results in this direction can be found in [1, 4, 5, 6, 9].In particular, in [5] it was shown that the class of ∨ -systems is closed under theoperation of restriction, which gives a powerful tool to construct new examples of ∨ -systems.The most comprehensive list of known ∨ -systems together with their geometricproperties can be found in [4, 5]. The main purpose of this paper is to present somearguments in favour of the completeness of this list in dimension three by studyingthe infinitesimal deformations of ∨ -systems.We start with a brief review of the general notions from the theory of matroids[13], which provides a natural framework for the problem of classification of ∨ -systems. A matroidal approach in this context was also used by Lechtenfeld et alin [9].Then we study the infinitesimal deformations of the ∨ -systems of given matroidaltype and derive the corresponding linearised ∨ -conditions. This allows us to showthat the isolated 3D ∨ -systems listed in [4] are indeed isolated.The main question which still remains open is what are possible matroidal struc-tures of ∨ -systems. We discuss this in the context of the projective geometry usingthe analysis of the known ∨ -systems.In the last section we study the property of the corresponding ν -function on theflats of matroid and state the uniqueness conjecture, saying that the matroid andfunction ν on its flats uniquely determine the corresponding ∨ -system.In the Appendix we give the catalogue of all known ∨ -systems in dimension threetogether with the corresponding matroids and ν -functions.2. Vector Configurations and Matroids
The combinatorial structure of the vector configurations can be described usingthe notion of matroid. The theory of matroids was introduced by Whitney in 1935,who was looking for an abstract notion generalising the linear dependence in thevector space.We review some standard notions from this theory following mainly Oxley [13].A matroid
M is a pair ( X , I ), where X is a finite set and I is a collection ofsubsets S of X (called the independent sets of M ) such that: • I is non-empty • For any S ∈ I , any S (cid:48) ⊂ S one has S (cid:48) ∈ I . • If A, B ∈ I , | A | = | B | +1 then ∃ x ∈ A \ B such that B ∪ { x } ∈ I .The rank of the matroid M is defined as r ( M ) = max I ∈I {| I |} . More generally,the rank of the subset S ⊂ X is defined as r ( S ) = max I ∈I {| I | : I ⊆ S } . A direct sum of matroids M = ( X , I ) and M = ( X , I ) is defined as M ⊕ M = ( X ∪ X , { I ∪ I : I ∈ I , I ∈ I } ) . A matroid is called connected if it can not be represented as a direct sum.
N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 3 The most important class of matroids for us consists of vector matroids. Let A be a real r × n matrix, X = { , , . . . , n } be the set of column labels of A , and I bethe collection of subsets S of X, for which the columns labelled by S are linearlyindependent over R . Then ( X , I ) is a matroid, which is called rank r vector matroid and denoted by M [ A ] . The following operations on matrix A do not affect the corresponding vectormatroid M [ A ] :(1) Elementary operations with the rows,(2) Multiplication of a column by a non-zero number.Two matrices A and A (cid:48) representing the same matroid M are said to be projec-tively equivalent representations of M if A (cid:48) can be obtained from A by a sequenceof these operations. Equivalently, one can say that A (cid:48) = CAD , where C is aninvertible r × r matrix, and D is a diagonal n × n matrix with non-zero diagonalentries.Alternatively, one can define the linear dependence matroid on the set X asa family I C of minimal dependent subsets C of X (called circuits ) through thefollowing axioms: • The empty set is not a circuit. • No curcuit is contained in another circuit. • If C , C ∈ I C are two circuits sharing an element e ∈ X , then ( C ∪ C ) \ e is a circuit or contains a circuit.The rank of a circuit is defined as the dimension of the vector space spanned byits vectors. Circuits spanning the same d -dimensional subspace can be united inso-called d -flats. A set F ⊆ X is a flat of the matroid M if r ( F ∪ { x } ) = r ( F ) + 1for all x ∈ X \ F, where r ( F ) is the rank of the flat F. The matroid can be labelledby listing all d -flats.As an example consider the positive roots of the B -type system. The corre-sponding matrix (with the first row giving the labelling) is A = − − − . Here matroid M is defined on the set X = { , , , , , , , , } , with 2-flats { (4 , , , (6 , , , (4 , , , (1 , , } and { (3 , , , , (1 , , , , (5 , , , } with three and four elements respectively. Together with the 3-flat X this gives thecomplete list of flats.Graphically on the projective plane we have V. SCHREIBER AND A.P. VESELOV
263 8 4 9571
Figure 1.
Graphic representation of B -matroid: lines corre-spond to rank-2 flatsA 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 | . Thefollowing table summarises the results for rank 3-matroids for small n (see [12]). n realisable matroids. The problem of finding acriterion for realisability is known to be
N P -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 R ( M ) = { A : M = M [ A ] } / ∼ modulo projective equivalence is discrete and strongly projectively rigid if it consistsof only one point (which means that modulo projective equivalence M has a uniquevector realisation).Let G be a finite Coxeter group, which is a finite group generated by the hyper-plane reflections in a Euclidean space. We say that matroid M is of Coxeter type ifit describes the vector configuration of the normals to the corresponding reflectionhyperplanes (one for each hyperplane) for such a group. For rank three Coxetermatroids we have the following result.
Theorem 1.
The matroids of Coxeter types A and B are strongly projectivelyrigid. The matroid of type H is projectively rigid with precisely two projectivelynon-equivalent vector realisations.Proof. Let us prove this first for B case. Since the images a , a , a , a of theelements 1,2,3 and 4 in the projective plane form a projective basis it is enough toprove that the remaining a , a , a , a , a can be constructed uniquely. From thematroid structure we can see that x must be an intersection point of the lines(2-flats) a a and a a . We denote this as a = ( a a ) ∧ ( a a )using the general lattice theory notation. Similarly we have a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) , N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 5 a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) . Similarly one can prove the rigidity in A case (see Fig. 2). In both these cases thespace of realisations modulo projective equivalence consists of only one point. Figure 2.
Graphic representation of A -matroidThe H case is more interesting. Fig. 3 shows the graphic representation of thesystem H in the real projective plane R P .Recall that on the projective line R P any three points can be mapped intoany other three via the action of the group P GL (2 , R ) . For four distinct points p , p , p , p on the projective line R P with homogeneous coordinates [ x i , y i ] thereis a projective invariant, namely cross-ratio defined as( p , p ; p , p ) = ( x y − x y )( x y − x y ) ( x y − x y )( x y − x y ) . 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: ( z , z ; z , z ) = ( z − z )( z − z ) ( z − z )( z − z ) . Since any projection from a point in the projective plane preserves the cross-ratioof four points we have the equalities( a , a ; a , a ) = ( a , a ; a , a ) = ( a , a ; a , a ) , ( a , a ; a , a ) = ( a , a ; a , a ) = ( a , a ; a , a ) . Using elementary manipulations with cross-ratios one can show that that theseequalities imply that x = ( a , a ; a , a ) satisfies the equation x − x − x = √ and x = −√ .If we fix the positions of the four points a , a , a , a forming a projective basisin R P we can first reconstruct a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) . V. SCHREIBER AND A.P. VESELOV
Then using the knowledge of x = ( a , a ; a , a ) we can reconstruct a and all theremaining points as a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) ,a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) , a = ( a a ) ∧ ( a a ) ,a = ( a a ) ∧ ( a a ) . Thus we have shown that modulo projective group we have only two different vectorrealisations of matroid H . (cid:3) Figure 3.
Graphic representation of H -matroid Remark.
The existence of two projectively non-equivalent realisations is relatedto the existence of a symmetry of matroid M ( H ), which can not be realised geo-metrically, see [3]. These two realisations are related by re-ordering of the vectorsand thus give rise to the equivalent ∨ -systems.3. Classification Problem for ∨ -Systems of Given Matroidal Type For any ∨ -system A ⊂ V ∗ one can consider the corresponding matroid M ( A ),which encodes a combinatorial structure of A . Conversely, having a matroid M one can look for ∨ -system realisations A of M with given combinatorial structure M ( A ) = M. Let R ∨ ( M ) be the set of all such realisations modulo group G = GL ( V ∗ ) oflinear automorphisms of V ∗ . If vector matroid M = M ( A ) is strongly projectively rigid then all its vectorrealisations modulo G have the form A (cid:48) = AD , or in terms of the columns a i , i =1 , . . . , n of A , a (cid:48) i = x i a i , i = 1 , . . . , n N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 7 with arbitrary non-zero parameters x i . The ∨ -conditions form a system of nonlinearalgebraic relations on the parameters x i ∈ R \ R ∨ ( M ) as an open setof a real algebraic variety.For a generic vector matroid this set is actually empty. For example, for n vectors a i in R in general position the ∨ -conditions imply that these vectors mustbe pairwise orthogonal, which is impossible if n > . In the case when the space R ∨ ( M ) is known to be non-empty (for example, forall vector matroids M of Coxeter type) we have the question of how to describethis space effectively.For the case of matroid of Coxeter type A the answer is known [1]. The pos-itive roots of A system are e i − e j , ≤ i < j ≤
4, where e i , i = 1 , . . . , R . Since matroid A is strongly projectively rigid it is enough to consider the system(2) A = { µ ij ( e i − e j ) , ≤ i < j ≤ } . Theorem 2. [1]
The system (2) satisfies the ∨ -conditions if and only if the param-eters satisfy the relations µ µ = µ µ = µ µ . All the corresponding ∨ -systems can be parametrized as A ( c ) = (cid:8) √ c i c j ( e i − e j ) , ≤ i < j ≤ (cid:9) , with arbitrary positive real c , . . . , c . Without loss of generality, we may choose c = 1 and consider the restriction ofthe system onto the hyperplane x = 0 . This gives the following parametrisation ofthe space R ∨ ( M ( A )) by positive real c , c , c as A ( c ) = (cid:40) √ c i c j ( e i − e j ) , ≤ i < j ≤ √ c i e i , i = 1 , , . Consider now the case B , corresponding to the following configuration of vectorsin R B = (cid:40) e i ± e j , ≤ i < j ≤ ,e i i = 1 , . . . , . The following 4-parametric family of ∨ -systems of B -type was found in [1]:(3) B ( c, γ ) = (cid:40) √ c i c j ( e i ± e j ) , ≤ j < i ≤ (cid:112) c i ( c i + γ ) e i , ≤ i ≤ c , c , c and γ such that c i + γ > i = 1 , , . Theorem 3.
Formula (3) gives all rank 3 ∨ -systems of matroid type B . Proof.
The proof is by direct computations, but we present here the details to showthe algebraic nature of ∨ -conditions in this example.Since B matroid is strongly projectively rigid, we can assume that the corre-sponding ∨ -system has the form A = α ij ( e i + e j ) , ≤ i < j ≤ (cid:101) α ij ( e i − e j ) , ≤ j < i ≤ β i e i , ≤ i ≤ , V. SCHREIBER AND A.P. VESELOV where all the parameters can be assumed without loss of generality to be positive.To write down all ∨ -conditions consider all two-dimensional planes containingat least two vectors v , v ∈ A . There are 3 different types of such planes Π:(1) < e , e ± e >, < e , e ± e >, < e , e ± e >, (2) < e , e , e ± e >, < e , e , e ± e >, < e , e , e ± e >, (3) < e − e , e − e , e − e >, < e − e , e + e , e + e >,< e − e , e + e , e + e >, < e − e , e + e , e + e > . The corresponding form G has the matrix G = α + α + (cid:101) α + (cid:101) α + β α − (cid:101) α α − (cid:101) α α − (cid:101) α α + α + (cid:101) α + (cid:101) α + β α − (cid:101) α α − (cid:101) α α − (cid:101) α α + α + (cid:101) α + (cid:101) α + β In case 1. the ∨ -conditions are just the orthogonality conditions G ( α ∨ , β ∨ ) = 0for the corresponding two covectors α and β in the plane Π . We obtain the system (cid:101) α (cid:101) α + (cid:101) α (cid:101) α + (cid:101) α (cid:101) α − α α − α (cid:101) α − α (cid:101) α ) − α β + (cid:101) α β − α β + (cid:101) α β = 02( (cid:101) α (cid:101) α + (cid:101) α (cid:101) α + (cid:101) α (cid:101) α − α α − α (cid:101) α − α (cid:101) α ) − α β + (cid:101) α β − α β + (cid:101) α β = 02( (cid:101) α (cid:101) α − α α − α (cid:101) α − α (cid:101) α + (cid:101) α (cid:101) α + (cid:101) α (cid:101) α ) − α β + (cid:101) α β − α β + (cid:101) α β = 02( α α + α (cid:101) α + α (cid:101) α − α α − α (cid:101) α − α (cid:101) α ) − α β + (cid:101) α β + α β − (cid:101) α β = 02( α α + α (cid:101) α + α (cid:101) α − α (cid:101) α − α (cid:101) α − α α ) − α β + (cid:101) α β + α β − (cid:101) α β = 02( α α + α (cid:101) α + α (cid:101) α − α (cid:101) α − α (cid:101) α − α α ) − α β + (cid:101) α β + α β − (cid:101) α β = 0 , which can be reduced to ( − α + (cid:101) α )( α + α + (cid:101) α + (cid:101) α + β − ( α − (cid:101) α ) ( α + α + (cid:101) α + (cid:101) α + β ) ) = 0( − α + (cid:101) α )( α + α + (cid:101) α + (cid:101) α + β − ( α − (cid:101) α ) ( α + α + (cid:101) α + (cid:101) α + β ) ) = 0( − α + (cid:101) α )( α + α + (cid:101) α + (cid:101) α + β − ( α − (cid:101) α ) ( α + α + (cid:101) α + (cid:101) α + β ) ) = 0 . 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 impliesthat α ij = (cid:101) α ij , which reduces the matrix G to G = α + α ) + β α + α ) + β
00 0 2( α + α ) + β . In cases 2. and 3. we fix for each plane Π a basis v , v ∈ A ∩ Π . The corre-sponding dual plane Π ∨ is spanned by v ∨ and v ∨ and the ∨ -condition implies theproportionality of the restrictions of the forms G and G Π onto Π ∨ . In our case thisproportionality turns out to be equivalent to the following system of equations: α α + α )+ β − α α + α )+ β = 0 α α + α )+ β − α α + α )+ β = 0 α α + α )+ β − α α + α )+ β = 0 . Introducing new parameters c i , i = 1 , , γ by c i := α ij α ik α jk , γ := β − c c . N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 9 we can see that these relations imply α ij = c i c j , β i = 2 c i ( c i + γ ) , which leads to the parametrisation (3). (cid:3) For larger matroids the direct analysis of the ∨ -conditions is very difficult, so weconsider a simpler problem about infinitesimal deformations of ∨ -systems.4. Deformations of ∨ -Systems Let A = { α } ⊂ V ∗ be a ∨ -system realisation of matroid M. Consider its smooth scaling deformation A ( t ) of the form(4) A ( t ) = { α t } , α t = µ α ( t ) α, µ α (0) = 1 . For projectively rigid matroids M one can always reduce any deformation to sucha form.Let ξ α = ˙ µ α (0). We are going to derive the conditions on ξ α , which can beconsidered as linearised ∨ -conditions for such deformations.Let G t ( x, y ) := G A ( t ) ( x, y ) = (cid:88) α ∈A α t ( x ) α t ( y )with G = G = G A ( t ) and consider its derivative˙ G t ( x, y ) = (cid:88) α ∈A ˙ α t ( x ) α t ( y ) + (cid:88) α ∈A α t ( x ) ˙ α t ( y ) , which at t = 0 gives ˙ G ( x, y ) = 2 X, where X = (cid:88) α ∈A ξ α α ( x ) α ( y ) . Consider now the ∨ -conditions.For any two-dimensional plane containing only two covectors we have G t ( α ∨ t , β ∨ t ) = 0 . Differentiating it in t we have(5) ˙ G t ( α ∨ t , β ∨ t ) + G ( ˙ α ∨ t , β ∨ t ) + G ( α ∨ , ˙ β ∨ t ) = 0 , where here and below by ˙ α ∨ t we mean ddt ( α ∨ t ) . To find G ( ˙ α ∨ t , β ∨ t ) note that by definition of α ∨ t G t ( α ∨ t , v ) = α t ( v ) for any fixedvector v ∈ V. Differentiating this with respect to t we have˙ G t ( α ∨ t , v ) + G t ( ˙ α ∨ t , v ) = ˙ α t ( v )which for t = 0 gives 2 X ( α ∨ t , v ) + G ( ˙ α ∨ t , v ) = ξ α α ( v ) . Thus we have G ( ˙ α ∨ , v ) = ξ α α ( v ) − X ( α ∨ , v ) . and thus G ( ˙ α ∨ , β ∨ ) = ξ α α ( β ∨ ) − X ( α ∨ , β ∨ ) = − X ( α ∨ , β ∨ )since α ( β ∨ ) = G ( α ∨ , β ∨ ) = 0 by the ∨ -conditions.Substituting this into (5) we have the first linearised ∨ -condition: for α, β beingthe only two covectors in a plane Π we have(6) X ( α ∨ , β ∨ ) = 0 . 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 suchthat for any α ∈ Π ∩ A , v ∈ V we have(7) G Π ( α ∨ , v ) = νG ( α ∨ , v ) , where G Π ( x, y ) = G Π A ( x, y ) = (cid:80) α ∈ Π ∩A α ( x ) α ( y ) (see [5]). Now assuming that A depends on t as above and differentiating with respect to t at t = 0 we have asbefore for any α, β ∈ A ∩ Π˙ G Π ( α ∨ , β ∨ )+ G Π ( ˙ α ∨ , β ∨ )+ G Π ( α ∨ , ˙ β ∨ ) = ˙ νG ( α ∨ , β ∨ )+ ν ˙ G ( α ∨ , β ∨ )+ νG ( ˙ α ∨ , β ∨ )+ νG ( α ∨ , ˙ β ∨ ) . But from (7) we have G Π ( ˙ α ∨ , β ∨ ) = νG ( ˙ α ∨ , β ∨ ) and G Π ( α ∨ , ˙ β ∨ ) = νG ( α ∨ , ˙ β ∨ ).Since ˙ G Π = 2 X Π , where X Π ( x, y ) = (cid:88) α ∈ Π ∩A ξ α α ( x ) α ( y ) , we have 2 X Π ( α ∨ , β ∨ ) = ˙ νG ( α ∨ , β ∨ ) + 2 νX ( α ∨ , β ∨ ) , or, eventually(8) 2( X Π − νX )(( α ∨ , β ∨ ) = ˙ νG ( α ∨ , β ∨ ) . 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(9) X Π − νX ∼ G | Π ∨ , where the sign ∼ means proportionality.Thus we have proved Theorem 4.
The deformations of ∨ -systems of the form (4) are described by thelinear ∨ -conditions (6), (9). For projectively rigid matroidal types this describes allinfinitesimal deformations of a given ∨ -system. Case by case check of the ∨ -systems from the Appendix leads to the following Theorem 5.
All rank three vector matroids corresponding to known irreducible3D ∨ -systems are projectively rigid. The H matroid is the only one, which is notstrongly projectively rigid. Let us show that the largest known case ( H , A ) is strongly projectively rigid.We will use the labelling of the points shown at the last figure of the paper. Fixthe positions of the four points 6 , , ,
30 forming a projective basis in R P . Afterthis all the remaining points can be reconstructed uniquely as follows:31 = (25 , ∧ (6 , ,
29 = (25 , ∧ (6 , ,
12 = (6 , ∧ (27 , , , ∧ (29 , ,
17 = (25 , ∧ (6 , ,
28 = (17 , ∧ (12 , , , ∧ (12 , ,
24 = (17 , ∧ (25 , , , ∧ (28 , ,
23 = (3 , ∧ (27 , ,
11 = (27 , ∧ (28 , ,
19 = (3 , ∧ (4 , , , ∧ (11 , ,
16 = (7 , ∧ (1 , ,
20 = (16 , ∧ (4 , , , ∧ (25 , ,
21 = (25 , ∧ (7 , ,
10 = (7 , ∧ (4 , ,
26 = (7 , ∧ (4 , ,
14 = (4 , ∧ (11 , ,
18 = (21 , ∧ (28 , ,
22 = (7 , ∧ (3 , ,
15 = (7 , ∧ (24 , , , ∧ (10 , , , ∧ (21 , ,
13 = (7 , ∧ (12 , , , ∧ (10 , . N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 11 A direct computation shows that in case of the classical systems A and B thelinear system (5),(8) has corank four in agreement with the results of the previoussection.The analysis of the linearised ∨ -conditions for the families D ( t, s ), F ( t ), G ( t )and ( AB ( t ) , A ) , shows that these families of ∨ -systems can not be extended.Consider, for example, the family of ∨ -systems D ( t, s ) from [5] with A = √ √ s + t − − − √ (cid:113) s − t +1 t − − √ (cid:113) − s + t +1 s with real parameters s, t such that | s − t | < , s + t > . Matrices G and X havethe form G = s + t + 1) 0 00 s + t +1) t
00 0 s + t +1) s X = ξ + ξ + ξ + ξ + 2 ξ ( s + t − ξ − ξ − ξ + ξ ξ − ξ + ξ − ξ ξ − ξ − ξ + ξ ξ + ξ + ξ + ξ + s +1) t ξ − ξ ξ + ξ − ξ − ξ ξ − ξ + ξ − ξ ξ + ξ − ξ − ξ ξ + ξ + ξ + ξ + − s + t +1) s ξ . For the three covectors α , α , α the first linearised ∨ -conditions X ( α ∨ i , α ∨ j ) = 0, i, j = 5 , , ξ + ξ − ξ − ξ = 0 ,ξ − ξ − ξ + ξ = 0 ,ξ − ξ + ξ − ξ = 0 , which imply that ξ = ξ = ξ = ξ . For the planes with more than two covectors we have the linear system( s + t )( ξ ( s + t +1)+ ξ ( s + t − t ( ξ (2 s +3)+ ξ (2 s − − s ( ξ + ξ )+ ξ + ξ ))+ t ( − ( ξ + ξ − ξ )) − s ( s ( ξ + ξ − ξ ) + ξ − ξ + 2( ξ + ξ )) + 2 ξ ) = 0 , ( s + t )(( s − ξ − ξ + s ( ξ − ξ )) + t ( ξ + ξ − ξ − ξ + s ( ξ + ξ − ξ − ξ ))+ t ( − ξ + ξ )) = 0 , ( s + 1) t ( s ( ξ − ξ + ξ + 3 ξ − ξ + ξ )) + ξ + 3 ξ + ξ − ξ − ξ + ξ )) + ( s + 1) t ( ξ − ξ + 2 ξ ) + ( s − − ξ ( s − − ξ ( s −
1) + 2 ξ s − ξ ) = 0 , ( s + 1) t ( ξ + s ( ξ + ξ − ξ − ξ ) + ξ − ξ − ξ ) + ( s − ξ + s ( ξ − ξ ) − ξ )+ ( s + 1) t ( − ( ξ − ξ )) = 0 , ( t + 1)( − t ( s ( ξ + ξ − ξ + ξ )) + ξ (3 s + 2) − ξ + ξ )) + s ( − s ( ξ + 2 ξ ) + ξ + ξ (3 s −
1) + 2( ξ + ξ )) − ξ s + ξ s ( t −
3) + ( t − ) + t ( ξ − ξ ) + ξ − ξ = 0 , ( t + 1)( t ( ξ + s ( ξ + ξ − ξ − ξ ) + ξ − ξ − ξ ) + ( s − ξ + s ( ξ − ξ ) − ξ ) + t ( ξ − ξ ) = 0 , ( t + 1)( ξ ( s ( t −
3) + ( t − ) − t ( s ( ξ + ξ − ξ + ξ )) + ξ (3 s + 2) − ξ + ξ )) − s ( s ( ξ − ξ + 2 ξ ) + ξ + ξ − ξ − ξ + ξ )) + t ( ξ − ξ ) + ξ − ξ ) = 0 , ( t + 1)( t ( ξ + s ( ξ + ξ − ξ − ξ ) + ξ − ξ − ξ ) + ( s − ξ + s ( ξ − ξ ) − ξ ) + t ( ξ − ξ )) = 0 , ( s + 1) t ( ξ ( s − − s ( ξ + 3 ξ + ξ − ξ + ξ )) − ξ + ξ − ξ + 2( ξ + ξ ))+ (cid:0) s − (cid:1) ( ξ ( s −
1) + ξ ( s − − ξ s + 2 ξ ) + ( s + 1) t ( − ( ξ − ξ + 2 ξ )) = 0 , ( s + 1) t ( ξ + s ( ξ + ξ − ξ − ξ ) + ξ − ξ − ξ ) + ( s − ξ + s ( ξ − ξ ) − ξ ) + ( s + 1) t ( − ( ξ − ξ )) = 0 , ( s + t )( t ( ξ (2 s + 3) + ξ (2 s −
1) + ξ − s ( ξ + ξ ) + ξ + ξ )) + s ( − s ( ξ + ξ − ξ )+ ξ − ξ + ξ )) + t ( − ( ξ + ξ − ξ )) − ξ s + 3 ξ s + ξ ( s + t + 1) − ξ + 2 ξ ) = 0 , ( s + t )( t ( s ( ξ + ξ − ξ − ξ ) + ξ + ξ − ξ − ξ ) + t ( ξ − ξ ) + ( s − s ( ξ − ξ )+ ξ − ξ )) = 0 . A check with Mathematica shows that the co-rank of the total system is threefor every admissible values of s and t . The free parameters correspond to twodeformation parameters s and t and the uniform scaling of the system.This approach with the use of Mathematica (see the programme in Appendix Bto [15]) allows us to prove that the isolated examples of ∨ -systems from the list [5]are indeed isolated. Theorem 6.
There are no non-trivial deformations of the ∨ -systems ( E , A × A ) , ( E , A × A ) , ( E , A × A ) , ( E , A × A ) , ( E , A × A ) , ( E , A × A ) , ( H , A ) and H . Matroidal Structure of ∨ -Systems and Projective Geometry The main part of the classification problem is to characterise the correspondingclass of possible matroids. This question was addressed by Lechtenfeld et al in [9].They developed a Mathematica program, which generates simple and connectedmatroids of a given size of the ground set X . If a generated matroid has a vectorrepresentation, they have checked first if the orthogonality ∨ -conditions are possibleto satisfy before verification of the ∨ -conditions for the non-trivial planes (all 2-flats). For matroids with n <
10 elements the orthogonality conditions are strongenough 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 unreason-ably large computer time required. This means that we need a more conceptualapproach, which is still missing.In this section we collect some partial observations based on the analysis of theknown 3D ∨ -systems and projective geometry.We start with the notion of extension and degeneration for ∨ -systems.Let A , A ⊂ V ∗ be two ∨ -systems. If A ⊂ A we call A an extension of A . N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 13 Let ∨ -system A = A t depend on the parameter t. Assume that for some t = t one or more of the covectors α ∈ A t vanishes. In that case the system (cid:101) A = lim t → t A ( t )is called degeneration of A ( t ) . A reverse process we will call regeneration .In the tables below we give the list of all extensions and degenerations for knownthree-dimensional ∨ -systems from the catalogue in the Appendix. Table 1.
Extensions of known 3D ∨ -systems. ∨ -system Extension The added covectors A F ( t ) { , , , , , , } A ( AB ( t ) , A ) { , , , } ( E , A ) ( E , A × A ) { , , , , , , , , } G ( ) ( E , A × A ) { , , , , , } H ( H , A ) { , , , , , , , , , , , , , , } Table 2.
Degenerations of known 3D ∨ -systems. ∨ -system Degeneration The vanishing covectors F ( t ) lim t → F ( t ) ∼ B ( √ { , , , } F ( t ) lim t →∞ F ( t ) ∼ D (1 , { , , , , , } B ( c, c, c ; γ ) lim γ → c B ( c, c, c ; γ ) ∼ A { , , } ( AB ( t ) , A ) lim t →∞ ( AB ( t ) , A ) ∼ D (1 , { , , } ( AB ( t ) , A ) lim t → ( AB ( t ) , A ) ∼ B ( √ { } ( AB ( t ) , A ) lim t → √ ( AB ( t ) , A ) ∼ ( E , A ) { } t ( AB ( t ) , A ) lim t →∞ ( AB ( t ) , A ) ∼ B ( √ { , } G ( t ) lim t → G ( t ) ∼ ( E , A ) { , , } D ( t, s ) lim t → ( s +1) D ( t, s ) ∼ A { } B ( c , c , c , γ ) lim c → − γ B ( c , c , c , γ ) ∼ ( E , A ) { } B ( c , c , c , γ ) lim c ,c → − γ B ( c , c , c , γ ) ∼ D (1 , { , } More relations between ∨ -realisable matroids can be seen using projective geom-etry, in particular projective duality. We will demonstrate this on few examples.We start with the matroid of the ∨ -system of type A . In projective geometry (seee.g. [8]) it is known as the simplest configuration (6 ) consisting of four lines withthree points on each line and two lines passing through every point. Its projectivedual is a complete quadrangle (4 ) consisting of four points, no three of whichare collinear and six lines connecting each pair of points (see figure 4). If we extendthe dual configuration by adding the remaining three points of intersections of lines(the points marked white in the graphic), we come to the projective configurationof seven points and six lines, corresponding to the matroid of the ∨ -system of type D . dual Figure 4.
The projective configuration of A type, its dual andthe extended configuration corresponding to ∨ -system of type D .We can proceed the construction by taking the dual of the new obtained con-figuration 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 -type ∨ -system (see figure 5). dual Figure 5. D configuration, its projective dual and the extendedconfiguration of matroidal type B . The next step of the construction is demonstrated in figure 6. The dual configu-ration was obtained from the configuration D by adding all missing lines passingthrough any pair of points. Applying Desargue’s theorem to two marked triangleswe 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 ( t ) , A ) (see system 9.6 in the Appendix). Figure 6. B -configuration and its schematic extended projectivedual, corresponding to ∨ -system of type ( AB ( t ) , A ) . Due toDesargue’s theorem the three added (white) points are collinear. N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 15 One can check that the adding of three intersection points with red lines andthree lines connecting them pairwise leads to the configuration of F type (seesystem 9.10). However, if we add also the three intersection points with dotted linethen we come to the configuration which can be shown to be not ∨ -realisable.Although this relation with projective configurations and theorems in projectivegeometry looks quite promising, we see that the extension procedure is not straight-forward and does not guarantee the ∨ -realisability of the resulting configuration.We conclude with the following conjecture about 2-flats with precisely fourpoints. Recall that four points A, B, C, D on a projective line form a harmonicrange if the cross-ratio (
A, B ; C, D ) = − . The corresponding pencil of four lineson a plane is called harmonic bundle.
The B configuration provides a geometricway to construct harmonic ranges: on Fig. 1 the points 3,4,9,8 always form a har-monic range. Note that the covectors 8 and 9 are orthogonal and determine thebisectors for the lines corresponding to covectors 3 and 4. Case by case check ofthe known 3D ∨ -systems suggests that the same is true in general. Conjecture 1.
Let A be a ∨ -system and Π A ⊂ V ∗ be two-dimensional planecontaining exactly four covectors α i ∈ A , i = 1 , . . . , , then the corresponding fourlines 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 definesthe ν -function on the 2-flats of M , where ν is the coefficient in the ∨ -conditions (1)corresponding to the plane Π representing the flat. Conjecture 2. (Uniqueness Conjecture)
An irreducible ∨ -system A is uniquelydetermined 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 uniquelydetermined by its matroid M and the corresponding ν -function on its flats. If the function ν is fixed under deformation then ˙ ν = 0 and the corresponding ∨ -conditions are(10) X ( α ∨ , β ∨ ) = 0for α, β be the only two covectors in the plane, and(11) X Π − νX = 0 | Π ∨ for any plane Π containing more than two covectors from A . Conjecturally this should imply that X = cG corresponding to the global scalingof 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 exten-sions of ∨ -systems from the previous section. Conjecture 4. (Extension Conjecture)
For any irreducible ∨ -system and itsextension the values of the ν -functions on the corresponding flats are proportional. One can check that this is true for all known cases. For example, for the extension H ⊂ ( H , A ) we have the set of values { / , / } = 3 × { / , / } . Now we present some results about ν -functions for ∨ -systems.First we give the following, more direct geometric way to compute ν (Π) . Theform 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 froma ∨ -system A (12) ν (Π) = 12 (cid:88) α ∈ Π ∩A | α | . Proof.
From the ∨ -conditions (1.1) we have (cid:88) α ∈ Π ∩A α ∨ ⊗ α | Π ∨ = ν (Π) I | Π ∨ . Taking the trace of both sides gives (12). (cid:3)
Let
A ⊂ V ∗ be a ∨ -system generating V ∗ and consider the set F A of 2-flatsin 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 x Π , Π ∈ F A is admissible if for each α ∈ A (13) (cid:88) Π ∈F A : α ∈ Π x Π = 1 . Theorem 9.
For every admissible set of weights we have (14) (cid:88) Π ∈F A x Π ν (Π) = n , where n is the dimension of V. Proof.
We have (cid:88) β ∈A β ∨ ⊗ β = (cid:88) β ∈A ( (cid:88) Π ∈F A : β ∈ Π x Π ) β ∨ ⊗ β = (cid:88) Π ∈F A x Π (cid:88) β ∈ Π ∩A β ∨ ⊗ β. From the ∨ -condition (cid:88) β ∈ Π ∩A β ∨ ⊗ β = ν (Π) P Π , where P Π is the orthogonal projector onto Π ∨ . Taking trace and using the fact that (cid:80) β ∈A β ∨ ⊗ β = Id we obtain (14). (cid:3) We call (14) the universal relation for values of function ν. For the ∨ -system of type A the universal relation completely describes the setof all possible functions ν. Indeed, one can easily see from Fig. 2 that x Π = 1 / (cid:88) Π ∈F A ν (Π) = 3 . This gives us three free parameters, which are exactly three parameters of defor-mation.
N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 17 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 ( t )-type (this is however theonly exception among known 3D ∨ -systems).The list of all known ∨ -systems in dimension three with the corresponding ν -functions is given in the Appendix.7. Concluding Remarks
Although the problem of classification of ∨ -systems seems to be very hard, indimension three it does not look hopeless. As we have seen, matroid theory providesa natural framework for the problem of classification of ∨ -systems. For a givenmatroidal structure the ∨ -conditions define a set of algebraic relations on the vectorrealisations. In case when matroid is strongly projectively rigid we have one freeparameter for each vector, which makes possible the full classification of ∨ -systemswith small number of vectors.The main problem is to describe all possible matroidal types, which we believeform a finite list in any dimension. The results of Lechtenfeld et al [9] show that thedirect computer approach is probably unrealistic for ∨ -systems with more than 10covectors, while we have already in dimension three an example with 31 covectors(system ( H , A ), see 7.16 in the Appendix). In dimension three we have an in-triguing relation with the theory of configurations on the projective plane and withthe theorems in projective geometry, which also suggests that the final list shouldbe finite.In the theory of matroids and graphs many families have been proved to be closedunder taking minors, thus giving a possibility to reduce the problem of classificationto the identification of the forbidden minors. We hope that a similar approach couldbe fruitful for classification of ∨ -systems.Another result from matroid theory, which could be relevant, is Seymour’s de-composition theorem [18], which states that all regular matroids can be build upin a simple way as sums of certain type of graphic matroids, their duals, and onespecial matroid on 10 elements. Our analysis of degenerations and extensions of ∨ -systems suggests a possibility of a similar result for the ∨ -realisable matroids.8. Acknowledgements
We are grateful to Olaf Lechtenfeld and especially to Misha Feigin for usefuldiscussions. 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).
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Integrability: the Seiberg-Witten and Whitham equations (Edinburgh,1998), Gordon and Breach (2000), 125–135. Appendix. Catalogue of all Known Real -Dimensional ∨ -Systems Each 3D ∨ -system A is presented below by the matrix with columns givingthe 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 oforthogonal pairs, 2-flats, the form G and the values of ν -function. The ordering ofthe list is according to the number of covectors in the system. The parameters areassumed 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 threetaken from [5]. N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 19 B ( ) A (c) A D (t,s) (AB (t), A ) (AB (t), A ) F (t) G (t) (E ,A ) (E , A ) (E , A ) (E , A x A )
7 1 3 2 (E , A x A )
7 1 3 1 (E , A ) (E , D ) (E , A x D )
8 1 4 (E , A ) (E , A x A )
6 1 2 (E , A x A )
7 1 22 (E , A x A )
8 2 3 (H , A ) (E , A x A )
8 1 32 (E , A x A )
8 2 12 (E , A x A )
8 1 23 H (E , A x A )
8 1 4 (E , A ) (E , D )
PA (t,t,1,1) B ( γ ;c) Figure 7.
The map of all known 3-dimensional ∨ -systems from [5].We use here the notations from [4, 5]. In particular, for a Coxeter group G andits parabolic subgroup H ( G, H ) denotes the corresponding ∨ -system given by therestriction procedure [4]. When the type of the subgroup does not fix the subgroupup to a conjugation the index 1 or 2 is used to distinguish them.The ∨ -systems of type AB , G and D are related to the exceptional gener-alised root systems AB (1 , G (1 ,
2) and D (2 , , λ ) appeared in the theory of basicclassical Lie superalgebras [16, 17]. ∨ -systems A ( c , c , c ) . A = √ c −√ c c −√ c c √ c √ c c −√ c c √ c √ c c √ c c G = c (1 + c + c ) − c c − c c − c c c (1 + c + c ) − c c − c c − c c c (1 + c + c ) I = { (1 , , (2 , , (3 , }I = (1 , , ν = c + c c (1 , , ν = c + c c (2 , , ν = c + c c (4 , , , ν = c c , c = c + c + c . N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 21 ∨ -system D ( t, s ) . A = (cid:112) s + t − − − (cid:113) s − t +1) t − − (cid:113) t − s +1) s G = 2(1 + s + t ) t
00 0 s I = { (5 , , (5 , , (6 , }I = (1 , , , (3 , , ν = s + t s + t (1 , , , (2 , , ν = s s + t (1 , , , (2 , , ν = t s + t ∨ -system ( E , A ) . A = √ − − −
12 12 − − √
12 12 √ G = 4
12 0 00 3 00 0 1 I = { (1 , , (2 , , (4 , , (5 , }I = (cid:40) (2 , , , (1 , , ν = (6 , , , (8 , , , (1 , , , (8 , , ν = I = (cid:110) (8 , , , ν =
23N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 23 ∨ -systems of B ( c , c , c , γ ) . A = (cid:112) c ( c + γ ) 0 00 (cid:112) c ( c + γ ) 00 0 (cid:112) c ( c + γ ) · · ·· · · √ c c √ c c −√ c c −√ c c √ c c √ c c −√ c c √ c c √ c c √ c c √ c c √ c c
987 5 426 1 3 G = 2( c + c + c + γ ) c c
00 0 c I = { (1 , , (1 , , (2 , , (2 , , (3 , , (3 , } Four 3-point lines: I = { (4 , , , (4 , , , (5 , , , (7 , , } , ν = (cid:80) c i γ + (cid:80) i =1 c i ) Three 4-point lines: I = { (1 , , , , (1 , , , , (2 , , , } , ν j = γ − c j + (cid:80) c i γ + (cid:80) i =1 c i ) , j=1,2,3 ∨ -system ( E , A ) . A = √ √ √ √ −√ −√ √ √ −√ √ − √ √ − √ −
10 0 0 0 √ √ √ √ G =
24 0 00 12 00 0 4 I = { (1 , , (1 , , (2 , , (2 , , (5 , , (7 , }I = (cid:40) (10 , , , (4 , , , (9 , , , (1 , , , (4 , , , (9 , , , ν = (4 , , , ν = I = { (4 , , , , (9 , , , , (6 , , , } , ν =
23N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 25 ∨ -system ( AB ( t ) , A ) . A = √ √ t +1 √ t +1 t √ √ ( t +1) √ − √ t +1 √ t +1 t √ √ ( t +1) √ − √ t +1 − √ t +1 t √ √ ( t +1)
975 2 4 1638 10 G = 6(1 + 2 t )(1 + t )(1 + 4 t ) t t t t t t t t t I = { (1 , , (2 , , (3 , , (5 , , (7 , , (9 , }I = (1 , , , (2 , , , (3 , , ν = t t ) (4 , , , (4 , , , (5 , , ν = t t ) (5 , , ν = t ) I = { (1 , , , , (1 , , , , (2 , , , } , ν =
236 V. SCHREIBER AND A.P. VESELOV ∨ -system ( AB ( t ) , A ) . A = (cid:112) t + 1) 0 0 √ √ t √ t √ t t t t (cid:112) t + 1) 0 √ −√ t − t t − t t (cid:113) t − t +1 t √ − t √ t t − t − t G = 6 t t
00 0 t +2 t t I = { (2 , , (3 , , (3 , , (4 , , (4 , , (5 , , (5 , }I = (4 , , , (4 , , , (5 , , , (5 , , ν = t t ) (1 , , , (1 , , ν = t t ) (3 , , , (3 , , ν = t (1+2 t ) I = (2 , , , , (2 , , , ν = (1 , , , ν = t t ) (1 , , , ν = t t )N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 27 ∨ -system G ( t ) . A = √ t + 1 0 √ t + 1 (cid:113) t − (cid:113) t − (cid:113) t − √ t + 1 √ t + 1 − (cid:113) t − (cid:113) t − (cid:113) t − (cid:113) t − − −
513 3 12 711910 624 1 8 G = t ) 2(1 + 2 t ) 02(1 + 2 t ) 4(1 + 2 t ) 00 0 3(2 + t ) I = { (4 , , (4 , , (4 , , (5 , , (5 , , (5 , , (6 , , (6 , , (6 , }I = (cid:40) (2 , , , (2 , , , (3 , , , (1 , , , (1 , , , (3 , , ν = t t ) (4 , , , (6 , , , (6 , , , (5 , , , (5 , , , (4 , , ν = t t ) I = { (2,7,10,11),(1,7,8,9),(3,7,12,13) } , ν = t t I = { (1 , , , , , } ,ν = t t ∨ -system ( E , A × A ) . A = √ √ √ − √ − √ √ (cid:113) − (cid:113) − (cid:113) (cid:113) √ −√ √ √ − √ √ − √ (cid:113) − (cid:113) (cid:113) − (cid:113) √ √ √ √ (cid:113) (cid:113) (cid:113) (cid:113) G = 9 I = { (1 , , (1 , , (2 , , (2 , , (3 , , (4 , , (6 , , (7 , , (8 , , (9 , }I = (5 , , , (7 , , ν = (10 , , , (11 , , ν = (6 , , , (7 , , ν = (6 , , , (10 , , , (7 , , , (11 , , ν = I = (cid:40) (5 , , , , (11 , , , ν = (4 , , , ν = I = { (11 , , , , , (6 , , , , } , ν =
23N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 29 ∨ -system F ( t ) . A = √ t + 2 0 0 1 1 1 1 0 0 t √ t √ t √ t √ √ t + 2 0 1 − t √ − t √ t √ − t √
20 0 √ t + 2 0 0 1 − − t √ t √ − t √ − t √
11 9 12 1524 6 710 8 13 G = (6 + 12 t ) I I = { (4,11),(4,13),(5,10),(5,12),(6,13),(7,10),(7,11),(8,11),(8,12),(9,10),(9,13) }I = (cid:110) (4 , , , (4 , , , (5 , , , (5 , , , ν = t I = (1 , , , , (1 , , , , (2 , , , , ν = t )3+6 t (1 , , , , (1 , , , , (2 , , , , (2 , , , , (3 , , , , (3 , , , , ν = t t Coxeter ∨ -system H . A = φ − φ − φ φ φ φ φ − φ φ φ φ φ − φ φ − φ φ φ φ − φ φ φ φ − φ − − φ φ φ φ where φ is the golden ratio φ = √ . G = 10(3 + √ I = (cid:40) (1 , , (1 , , (2 , , (4 , , (4 , , (5 , , (5 , , (6 , , (6 , , (7 , , (7 , , (8 , , (9 , , (10 , , (11 , . I = (cid:40) (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (3 , , , (4 , , , (5 , , , (6 , , , (7 , , ν = I = (cid:40) (1 , , , , , (1 , , , , , (2 , , , , , (2 , , , , , (3 , , , , , (3 , , , , ν =
12N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 31 ∨ -system ( E , A × A ) . A = √ √ √ √ − √ −√ √ −√
10 0 0 √ √
10 0 2 √
25 52 − √ − √ √ −√ √ −√
10 0 0
12 12 √ √ · · ·· · ·
13 14 15 16 √ −√
10 0 0 √ √ − √ √ √ √
102 1 √ √ G = 30 I = { (1 , , (2 , , (3 , , (3 , , (3 , , (4 , , (4 , , (4 , , (6 , , (6 , , (6 , , (7 , , (7 , , (13 , , (14 , }I = (5 , , , (5 , , , (7 , , ν = (7 , , ν = (3 , , , (4 , , , (4 , , , (3 , , , (11 , , , (2 , , ν = (4 , , , (3 , , , (3 , , ν = I = { (7 , , , , (12 , , , , (14 , , , , (13 , , , , (13 , , , , (13 , , , } , ν = I = { (15 , , , , , (12 , , , , , (11 , , , , } , ν = ∨ -system ( E , A × A ) . A = √ √ √ (cid:113) √
32 32 − − √ √ − √ √ − − − √ − (cid:113)
12 12 √ · · ·· · ·
14 15 16 17 − √ − √ √ √ √ − √ √ − √ √ − √ √ √ N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 33 G = 30 I = { (1 , , (2 , , (3 , , (3 , , (3 , , (4 , , (4 , , (4 , , (5 , , (5 , , (9 , , (9 , , (10 , , (10 , , (11 , , (12 , }I = { (5 , , , (5 , , , (7 , , , (7 , , } ν = { (3 , , , (4 , , , (4 , , , (7 , , , (7 , , , (8 , , , (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (3 , , } ν = I = (cid:40) (8 , , , ν = (1 , , , , (1 , , , , (2 , , , , (5 , , , , (5 , , , , (6 , , , ν = I = { (1 , , , , , , (2 , , , , , } , ν = ∨ -system ( E , A × A ) . A = √ √ √ √ − √
10 0 2 √ − √ − √ √
20 0 2 − − √ − √ √ − √ · · ·· · ·
14 15 16 17 √ √ √ √ √ − √ − √ √ √ −√ √ −√ G = 30 I = { (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (3 , , (3 , , (3 , , (4 , , (4 , , (4 , , (5 , , (5 , , (6 , , (6 , }I = (7 , , , (7 , , , (9 , , , (9 , , ν = (9 , , , (9 , , , (7 , , , (7 , , ν = (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (3 , , , (4 , , , (4 , , ν = I = { (1 , , , , (1 , , , , (1 , , , , (2 , , , , (2 , , , , (3 , , , , (5 , , , } , ν = I = { (6 , , , , , , (5 , , , , , } , ν = ∨ -system ( E , A × A ) . A = √ √ √ √ √ √ √ − √ √ √ √ √
30 3 3 0 −√ −√ √ √ √ √ √ · · ·· · ·
16 17 18 192 √ √
61 0 √ √
63 2 √ √ √ N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 35
47 10 12 14 1713 11 6 2165 18 9 19 1 8 3 15 G = 30 I = { (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (3 , , (3 , , (3 , , (4 , , (4 , , (4 , , (5 , , (5 , , (6 , , (6 , , (10 , , (11 , , (12 , , (13 , , (14 , , (15 , , (16 , , (17 , } I = (cid:40) (3 , , , (3 , , , (4 , , , (4 , , , (7 , , , (7 , , ν = (7 , , , (2 , , , (2 , , , (1 , , , (1 , , , (7 , , ν = I = (cid:40) (1 , , , , (8 , , , , (8 , , , ν = (3 , , , , (5 , , , , (6 , , , ν = I = (cid:40) (9 , , , , , (9 , , , , , (2 , , , , , (2 , , , , , (1 , , , , , (1 , , , , ν = I = { (5 , , , , , } ,ν = ∨ -system ( E , A × A ) . A = √ √ √ √ √ √ √ − √ √ √ √ √
30 3 3 0 −√ −√ √ √ √ √ √ · · ·· · ·
16 17 18 192 √ √
61 0 √ √
63 2 √ √ √ N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 37 G = 30 I = { (1 , , (1 , , (4 , , (4 , , (4 , , (5 , , (5 , , (6 , , (6 , , (7 , , (7 , , (8 , , (8 , , (9 , , (9 , , (13 , , (14 , , (14 , , (15 , , (15 , , (16 , }I = (1 , , , (1 , , , (2 , , , (3 , , , (5 , , , (6 , , ν = (7 , , , (8 , , , (3 , , , (3 , , , (2 , , , (2 , , ν = (1 , , , (1 , , , (4 , , , (4 , , , (5 , , , (6 , , ν = I = (cid:40) (1 , , , , (4 , , , , (5 , , , , (6 , , , , (7 , , , , (8 , , , ν = I = (cid:40) (2 , , , , , ν = (2 , , , , , , (1 , , , , , , (3 , , , , , ν = ∨ -system ( H , A ) . A = √ √ √ √ a a a a b b b b √ √ − √ − √
22 12 12 − − a a − a − a b √ − √ √ − √ b − b b − b −
12 12 − a · · · ...
17 18 19 20 21 22 23 24 25 26 27
12 12 12 a √ a √ b √ b √ (cid:112) b √ (cid:112) b √ b − b − b b √ − b √ a √ a √ a (cid:112) b √ − a (cid:112) b √ − a a − a b √ − b √ a √ − a √ ... · · ·
28 29 30 310 0 2 a (cid:112) b √ − a (cid:112) b √ (cid:112) b √ (cid:112) b √ a (cid:112) b √ − a (cid:112) b √ (cid:112) b √ (cid:112) b √ with a = √ and b = − √ .
116 287113019 31 20 8 425 21 9 27 15 2 22 26 12 5 23 24 17 3 29 10 18 6 14 13 G = I I = { (1 , , (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (2 , , (3 , , (3 , , (3 , , (3 , , (4 , , (4 , , (4 , , (5 , , (5 , , (5 , , (6 , , (6 , , (6 , , (7 , , (7 , , (7 , , (8 , , (8 , , (8 , , (9 , , (9 , , (9 , , (10 , , (10 , , (10 , , (11 , , (11 , , (11 , , (12 , , (12 , , (13 , , (13 , , (13 , , (14 . , (14 , , (12 , , (14 , , (15 , , (15 , , (15 , , (16 , , (16 , , (16 , , (17 , , (17 , , N DEFORMATION AND CLASSIFICATION OF ∨ -SYSTEMS 39 (17 , , (18 , , (18 , , (18 , , (19 . , (19 . , (19 , }I = { (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (3 , , , (4 , , , (4 , , , (4 , , , (5 , , , (17 , , , (8 , , , (18 , , , (16 , , , (19 , , , (5 , , , (5 , , , (6 , , , (6 , , , (6 , , , (7 , , , (7 , , , (7 , , , (9 , , , (10 , , , (11 , , , (12 , , , (13 , , , (14 , , , (15 , , } ν = { (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (3 , , , (8 , , , (11 , , , (9 , , , (10 , , } ν = I = { (1 , , , , , (1 , , , , , (2 , , , , , (2 , , , , , (3 , , , , , (3 , , , , } ν = I = { (1 , , , , , , (1 , , , , , , (2 , , , , , , (4 , , , , , , (5 , , , , , , (5 , , , , , , (5 , , , , , , (6 , , , , , , (6 , , , , , , (6 , , , , , , (4 , , , , , , (4 , , , , , , (7 , , , , , , (7 , , , , , , (7 , , , , , } ν = Department of Mathematical Sciences, Loughborough University, LoughboroughLE11 3TU, UK
E-mail address : [email protected]
Department of Mathematical Sciences, Loughborough University, LoughboroughLE11 3TU, UK and Moscow State University, Moscow 119899, Russia
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