On p-tuples of the Grassmann manifolds

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On p-tuples of the Grassmann manifolds

Joël Rouyer

To cite this version:

Joël Rouyer. On p-tuples of the Grassmann manifolds. 2010. �hal-00491032�

On p-tuples of the Grassmann manifolds

Joël Rouyer June 10, 2010

Abstract

We provide a matrix invarariant for isometry classes of p-tuples of points in the Grassmann manifoldGn K^d (K=RorC). This invariant fully caracterizes thep-tuple. We use it to determine the regularp-tuples ofG2 R^d ,G3 R^d andG2 C^d .

1 Introduction and notation

A triangle (triple of points) of the Euclidean space is fully de…ned, up to isom- etry, by three numbers, namely its side lengths. Of course, three given positive numbers may or may not be the side lengths of some triangle. Indeed, they are if and only if they satisfy the well known triangle inequality. More generally, we can considerp-tuples of points. Once again, a p-tuple of the Euclidean space is fully characterized by the data of the distances determined by each pair of points. The existence criterion is a little less easy, but could be stated in terms of signs of the minors of some matrix build up from the square of the prescribed distances (see for instance [1, p. 239]). The aim of this paper is too discuss analogous matter forp-tuples of points in the Grassmann manifoldGn(K^pn), where K = R or C. It will turn out that in this case, the data of distances between the points of ap-tuple are no longer su¢ cient to characterize it up to a global isometry ofK^pn, and so, we have to de…ne another matrix invariant.

The Grassmann manifold Gn K^d is the set of n-dimensional linear sub- spaces (n-spaces, for aim of shortness) ofK^d. The groupU(d)of isometries of K^dacts onG_n K^{d p}on a natural way. We say that twop-tuples areisometric if and only if they lie in the same orbit under this action. We say that twop-tuples ( ₁; : : : ; _p)and ⁰₁; : : : ; ⁰_p arecongruentif and only ifd( _i; _j) =d ⁰_i; ⁰_j (see below for the de…nition of the geodesic distance d in G_n K^d ) for all i, j 2 f1; : : : ; pg. Of course, two isometric p-tuples are congruent, but the con- verse is not true.

The problem we investigate here is to …nd some numerical invariant for p- tuples up to isometry. The solution is well known forp= 2. Given twon-spaces

1, 2 K^d, we consider the function :P( 1)!Rwhich associates to a line lof 1the angle betweenland its orthogonal projection onto 2. Of course,K^d is endowed with its canonical scalar product (i. e. h(u1; : : : ; ud);(v1; : : : ; vd)i=

Pd

k=1ukvk). So, we can de…ne the angle between two linesKuandKv2P K^d as cos ¹p^jhu;vij

hu;uihv;vi. The stationary values of the function are called the critical angles between 1 and 2. Critical angles allow us to de…ne the so- called geodesic distance between 1and 2as the square root of the sum of the squares of the critical angles.

It can be proven that two pairs ofn-spaces are isometric if and only if they have the same critical angles. Moreover, givennnumbers 1, . . . , n 2 0;₂ , there always exist twon-spaces embedded in K²ⁿ such that these numbers are the critical angles between them.

A convenient way to compute the critical angles between two n-spaces 1,

2 K^d is to equip each of them with an orthonormal basis eⁱ₁; : : : ; eⁱ_n (i = 1;2), and to put the coordinates (in the canonical basis of K^d) of eⁱ_j into the j^th column of ad nmatrixBⁱ. Then the cosines of the critical angles are the singular values of the matrixB₁B₂[9].

1.1 Earlier results

If the case of twon-spaces is for long quite well-understood, there are hitherto very few investigations concerning triples, or more generalp-tuples.

Augustin Fruchard investigated the case triples of real planes (i. e.2-spaces) in [4]. Projecting orthogonally the unit circle of each plane onto the two others yields two ellipses in each plane. The angles between the major axes of these ellipses are called the inner angles. Of course, this notion makes sense if and only if the projection of a unit circle is never a circle,i. e.each pair of planes has two distinct critical angles. A. Fruchard de…ned a generic triangle as a triple for which the inner angles are well de…ned, and for which all the angles (critical and inner) belong to 0; ₂ . He proved that a generic triangle is fully de…ned by the six critical angles, the three inner angles, and four signs"₁; "₂; "₃; "₄= 1, the pack of which he refers to as thesignature of the triangle. We shall give in section 3 an analogous result, but our method leads to an invariant involving only one sign". This is due to the fact that the class of triples we can investigate with our method is slightly larger than the class of generic triples. What appear in A. Fruchard’s work as16distinct continuous families can be glued into two bigger ones. The triples of the interface are those that can be treated by our method, but are not generic.

Another point of view for the study of a p-tuple = ( 1; : : : ; p)is to con- sider the orthogonal projections onto 1, . . . , p. The traces and determinants of various products of these projections are numerical invariants of the isometry class of . Giovani Masala obtained nice results in this way, still in the case of real planes. He gave a list of invariants which fully characterizes ap-tuple. His invariant system allows him to classify the regular quadruples ofG₂ R⁸ . How- ever, although the statements of his theorems di¤er considerably in spirit from those of A. Fruchard, the proofs are more or less similar. In particular, they chie‡y involve the notion of inner angles, and so, let few hope of generalization in higher dimensions [6][7].

Another important investigation is the case of p-tuples of equi-isoclinic n- spaces. Twon-spaces are said to beisoclinicif their critical angles are all equal.

Further,p n-spaces are said to be equi-isoclinic if they are pairwise isoclinic with the same angle. In this quite restrictive case, a theory does exist. A normal Seidel matrix (named after J.J. Seidel in [2]) is a Hermitian np np matrix with the following property: if we see it as ap pmatrix whose coe¢ cients aren nmatrices, then (i) the diagonal coe¢ cients vanish, (ii) the coe¢ cients of the …rst row and the …rst column (except the …rst one which is zero) are identity matrices, and (iii) all other blocks are unitary (or orthogonal ifK=R).

The unitary (or orthogonal) group acts on the set of normal Seidel matrices by conjugation of each block. The theory provides a way to associate to ap-tuple of equi-isoclinicn-spaces of angle one of the orbits of the Seidel matrices under this action. Moreover, the data of this orbit and the angle fully determine the p-tuple. The existence problem has also a nice solution: given a normal Seidel matrixS and an angle , there exist ap-tuple of equi-isoclinicn-spaces with angle and associated Seidel matrix S if and only if cos is lower than or equal to the opposite of the inverse of the smallest eigenvalue ofS. In case of equality, thep-tuple can be embedded in K^np , where is the multiplicity of the smallest eigenvalue of S; otherwise thep-tuple spansK^np [5][2]. These results were actually stated in the real case, but all the proofs hold verbatim for K=C.

The method we propose in this article will include as particular cases both A. Fruchard’s and J. J. Seidel’s points of view.

1.2 Notation

Throughout this article, we shall have to deal withnp np matrices which are de…ned by n n blocks. IfM is such a matrix, Mij will stand for the n n block consisting of the intersection of the lines (i 1)n+ 1, . . . ,(i 1)n+n and the columns (j 1)n+ 1, . . . , (j 1)n+n. As usual, if M is a n n matrix,Mij is the scalar coe¢ cient which lie on lineiand the column j. IfM andN arenp np matrices,M N stands for thenp np matrix de…ned by (M N)_ij =MijNij,1 i; j p.

IfX is a set of square matrices of ordern, andN is a diagonal matrix of the same size,

M(k)stands for the set ofk kmatrices with coe¢ cients inK.

D(k)stands for the set ofk kdiagonal matrices with coe¢ cients inR. D⁺(k)stands for the set ofk kdiagonal matrices with positive diagonal coe¢ cients.

U(k)stands for the unitary group of degreekifK=C, or the orthogonal group ifK=R.

SU(k)stands for the special unitary group of degree kif K=C, or the special orthogonal group ifK=R.

O(k)stands for the orthogonal group of degreek.

SO(k)stands for the special orthogonal group of degreek.

M(X; p) stands for the set of np np matrices whose diagonal block- coe¢ cients areIn and other coe¢ cients belongs toX.

H(X; p)stands for the subset of Hermitian (symmetric ifK=R) matrices ofM(X; p).

S(N) stands for the set onn n matrices whose singular values are the diagonal coe¢ cients ofN.

D(X; p)stands for the set ofnp np matrices which are block-diagonal, with diagonal block-coe¢ cients belonging toX.

IfM1, . . . ,Mp belongs toM(n)we denote bydiag (M1; : : : ; Mp)the block- diagonal matrix whose diagonal block-coe¢ cients areM1, . . . ,Mp. As usual, if

1, . . . , _n are scalars,diag ( ₁; : : : ; _n)2D(n)stands for the diagonal matrix with diagonal coe¢ cients ₁, . . . , _n.

For M 2 M(n), we denote by [M]_p the square matrix of order np whose block-coe¢ cients are all equal toM, save the diagonal ones which are identity matrices.

2 General theory

2.1 A matrix invariant

Let = ( 1; : : : ; p)be ap-tuple ofGn(K^np). The most obvious matrix invari- ant is a matrix built up from critical angles. We say that a symmetric matrix 2M(D(n); p)is an edge matrix associated to if the diagonal coe¢ cients of ij 2 D(n)are the cosines of the critical angles between i and j. It is clear that ap-tuple is, in general, associated to more than one edge-matrix, for the critical angles can be arranged in any order. If we ask furthermore that the cosines are arranged in non-increasing order of multiplicity, and, for cosines of the same multiplicity, in decreasing order, the obtained matrix will be called the— with the de…nite article— edge matrix of .

We say that ap-tuple ishomogeneous if its edge matrix is such that i. All of the diagonal coe¢ cients of ij are positive (1 i; j p).

ii. The set C ^def= fU 2U(n)jU _ij= _jiUg does not depend on i, j (1 i < j p)

It is clear that C is a subgroup of U(n); we call it the group of . Note that the …rst point ensures that ij is invertible.

If thenangles between any twon-spaces are pairwise distinct and less than

2, thep-tuple will be said to beanisoclinic. It is clear that anisoclinicp-tuples

are homogeneous. Indeed, an homogeneous p-tuple is anisoclinic if and only ifC is the group of diagonal unitary matrices, if and only ifCis Abelian. Moreover, a genericp-tuple of planes is necessarily anisoclinic.

The following Lemma will be useful for further development.

Lemma 1 LetN 2D⁺(n). Assume that U₁,V₁,U₂,V₂2U(n)are such that U₁N V₁ =U₂N V₂. Then there exists a matrixE2U(n)which commutes with N and such thatU₂=U₁E andV₂=V₁E.

Proof. Let M = U1N V₁; we have M M = U1N²U₁ = U2N²U₂. Hence the columns of U1 and U2 are eigenvectors of the same matrix M M . More precisely, if we denote byB_i the block ofUi(i= 1;2) consisting of eigenvectors associated to one given eigenvalue of M M , then the columns of B_i form an orthonormal basis of the eigenspace of M M associated to . Therefore, there exists an unitary matrix E (whose order is the multiplicity of ) such thatB₂ =B₁E . Since the argument holds for any eigenvalue , there exists E2U(n)which commute withN and such thatU2=U1E.

On the other hand, M M =V₁N²V₁ =V₂N²V₂, whence there exists F 2 U(n) such that V₂ = V₁F. Now, U₁N V₁ = U₂N V₂ = U₁N EF V₁, whence EF =I_n.

From now on, we suppose that is homogeneous. Let be the edge matrix of , andC be the group of .

Let eachn-space _k be endowed with an orthonormal basis e^k_i; : : : e^k_j and consider the Gram matrixGof the family

e¹₁; e¹₂; : : : ; e¹_n; e²₁; : : : ; e^p_{n 1}; e^p_n .

As a Gram matrix,G is Hermitian and positive semi-de…nite. Moreover, due to the fact that e^k_i; : : : e^k_j is orthonormal, G belongs toH(M(n); p). If you choose some other orthonormal basis e⁰_i^k; : : : e⁰_j^k in k, 1 k p, then there existsV^k 2U(n)such thate⁰_i^k =Pn

j=1e^k_jV_ji^k. It follows that the correspond- ing Gram matrix G⁰ is obtained from G by the formula G⁰ = DGD , where D 2 D(U(n); p) is de…ned by Dii = Vⁱ . This fact motivates the following de…nition: two matrices of H(M(n); p) areequivalent if they are conjugated by a matrix ofD(U(n); p). It follows that two Gram matrices are equivalent if and only if they are the matrices of the samep-tuple ofn-spaces, up to a global isometry.

By construction ofG, the critical angles between iand j are the singular values of G_ij, whence there exist U_ij, V_ij 2U(n) such thatG_ij =U_{ij ij}V_ij. Since G_ij = G_ji, we have V_{ij ij}U_ij = U_{ji ji}V_ji. By Lemma 1, there exists C_ji2Csuch thatU_ij =V_jiC_jiandV_ij =U_jiC_ji. Moreover, these two formulae infer that C_ijC_ji = I_n. It follows that there exist C 2 H(C; p) and U 2 M(U(n); p)such thatG=U C U . A Gram matrix which admits such a decomposition will be called a -Gram matrix.

We say thatM 2S( ij) is ofL-kind (respectively R-kind) if there exists W 2U(n)such that M = W ij (respectively M = ijW). It follows from

Lemma 1 that, if U; V 2U(n) are such that the matrix U ijV is ofL-kind (respectivelyR-kind), thenV 2C (respectivelyU 2C).

We say that a -Gram matrixGisprereduced if forj 2,G_1j is ofL-kind, andG₁₂ = ₁₂. SinceG is Hermitian, the prereduceness implies furthermore thatG_i1 is ofR-kind for i 2, andG₂₁= ₂₁ .

Lemma 2

i. Each -Gram matrix is equivalent to a prereduced Gram matrix.

ii. Two prereduced -Gram matrix G and G⁰ are equivalent if and only if there existsD2D(C; p)such that D₁₁=D₂₂ andG⁰=DGD .

Proof. Chose a decompositionG=U C U ofG. LetD2D(U(n); p) be de…ned by

D₁₁=C₁₂U₁₂

Dii=U_i1 ifi >1, and putG⁰=DGD . We have forj >1,

G⁰_ij =DiiGijD_jj=DiiUij ijCijU_jiUj1. HenceG⁰_1i=D₁₁U_1jC_{1j ij} is ofL-kind andG⁰₁₂= ₁₂.

Assume now thatGandG⁰are prereduced and equivalent, there existsD2 D(U(n); p)such thatG⁰=DGD . By hypothesisGis prereduced, soG_1jis of L-kind (j >1) andG₁₂= ₁₂. It follows that there existU₂=I_n,U₃, ...,U_p2 U(n)such thatG_1j=U_{j 1j}(j >1). SinceG⁰_1j =D₁₁G_1jD_jj=D₁₁U_{j ij}D_jj is ofL-kind,D_jj 2C for allj >1. SinceG⁰₁₂=D_{11 12}D₂₂= ₁₂, by Lemma 1, we haveD11=D222C.

It is clear that a decomposition of a -Gram matrix G of the form U C U is not unique, and worst, there is no canonical way to de…ne one preferred decomposition among all the possible ones. So it will be necessary to arbitrary distinguish such a decomposition. For this purpose, we choose a (not necessary continuous) functions:U(n)!U(n)such that, for all U 2U(n), the restriction of s to the left coset UC is constant, and such that s(U) 2 UC. In other words, we chose in each left cosetUCan arbitrary distinguished elements(U). We assume furthermore thatsis chosen such thats(C) =fIng. Such a map s will be called a split map. We also de…ne : U(n) ! C by (U) = s(U) U, and, for any U 2 U(n), the map Ub : C ! C de…ned by Ub(E) = (EU).

We say that a decompositionG=U C U of a -Gram Gmatrix is s-normal ifU 2M(Ims; p). It is clear that each -Gram matrix has a single s-normal decomposition. We say that a -Gram matrix is reduced if, on the one hand, it is prereduced, and on the other hand, the …rst block row of the matrix C of its s-normal decomposition contains only identity matrix. Note that, sinceC is Hermitian, the latter condition may be equivalently stated on the …rst block column.

Lemma 3

i. Each -Gram matrix is equivalent to a reduced one.

ii. If two reduced -Gram matricesGandG⁰ are equivalent, then there exist E2C such that

G⁰= diag Ud₁₁(E); : : : ;Ud_1p(E) G diag Ud₁₁(E); : : : ;Ud_1p(E) , whereU is the …rst matrix of the s-normal decomposition ofG.

Proof. LetG=U C U be thes-normal decomposition of a prereduced -Gram matrixG. SinceGis prereduced,U_i1=U₁₂=C₂₁=C₁₂=I_n. De…ne D 2 D(C; p) by D_ii = C_1i for i 2. Put G⁰ = DGD ; de…ne U⁰ and C⁰ such that G⁰ = U⁰ C⁰ U⁰ is the s-normal decomposition of G⁰. Since D11=In=D12, by virtue of Lemma 2, the matrixG⁰is prereduced. Moreover, fori 2,G⁰_i1=DiiUi1Ci1 1iU_1iD₁₁=C1iCi1 1iU_1i, whence C_i1⁰ =In.

Now assume thatG=U C U andG⁰ =DGD =U⁰ C⁰ U⁰ are reduced and that the above decompositions ares-normal. By virtue of Lemma 2, D2D(C; p). MoreoverG⁰_i1=Dii i1(D11U1i) andIn=C_1i⁰ =DiiUc1i(D11) , thusD_ii=Uc_1i(D₁₁).

We de…ne as-pair as an ordered pair (U; C)2M(Ims; p) H(C; p) such that for alli2 f1; : : : ; pg, Ui1=Ci1=C1i=U12=In. IfG=U C p U is as-normal decomposition of a reduced -Gram matrix G associated to , then(U; C)is as-pair, which is said to beassociatedto . We de…ne an action ofCon the set ofs-pairs by E (U; C) = (E U; E C), where

(E U)_ij =s Uc1i(E)Uij (1)

(E C)_ij =Ucij Uc1i(E)Cij Ucji Ud1j(E) . (2) Note that the notation E C is abusive, forE C actually depends on U. A thorough veri…cation shows that E (U; C) is the s-pair corresponding to the Gram matrixD(U C U )D , whereD is the block-diagonal matrix de-

…ned byDii=Uc1i(E). Hence, by virtue of Lemma 3, twos-pairs are associated to the samep-tuples if and only if they lie in the same orbit. An orbit under this action will be called as-orbit. The following Theorem summarize the above considerations.

Theorem 1 Two homogeneousp-tuple ofGn(K^np)are isometric if and only if they have the same edge matrix, and the sames-orbit.

Remark 4 In the case of an p-tuple of equi-isoclinic n-spaces of angle ,

ij = cos ( )In, C =U(n) and s= 1. If (U; C) is a s-pair associated to , thenU = [In]_p. It follows that the p-tuple is wholly determined by the orbit of the matrixC 2H(U(n); p). The matrix C Inp is the normal Seidel matrix mentioned in the introduction. See section 5.2.

Remark 5 The center Z of U(p) (which is clearly a normal subgroup of C)

…xes anys-pair. Hence this action can be seen as an action ofC=Z.

In the special case whereU = [In]_p, we say that thes-pair(U; C)isspecial.

On the other hand, ifU contains exactly2p(i. e.the minimum number) block- coe¢ cients equal to In, then (U; C)is said to be common. Of course, a s-pair may be neither common nor special. Note that, if ones-pair is common, then all the s-pairs of its s-orbit are also common; in this case the corresponding s-orbit andp-tuples are said to be common.

Proposition 6 Assume thatC is Abelian. If thes-pair(U; C)is special, then itss-orbit is a singleton.

Proof. Let U = [I_n]_p; C be a special s-pair and choose E 2 C, we have (E U)_ij=s Ibn(E)In =s(E) =In, whenceE U =U. Moreover(E C)_ij = id_C id_C(E)C_ij(id_C id_C(E)) =EC_ijE =C_ij. HenceE (U; C) = (U; C).

2.2 Regularity

A p-tuple = ( ₁; : : : ; _p) is said to be regular, if it is isometric to ^def=

(1); : : : ; _(p) for any permutation 2 S_p. The aim of this section is to give some characterization of regularp-tuples by mean of itss-pair. Of course, the critical angles between any twon-spaces of a regularp-tuple do not depend on the considered pair ofn-spaces. In other words, thep-tuple admits an edge matrix in which all the non-diagonal block are identical.

Theorem 2 Let = [N]_p be an edge matrix in which all the non-diagonal blocks are equal toN 2D⁺(n). A p-tuple with edge matrix ands-pair(U; C) is regular if and only if there existF₁,F₃, . . . , F_n 2C such that

U2j=s(F1U1j) (3 j p) (3) U1j=s(F1U2j) (3 j p) (4) s C2jU_j2 =s Ud1j(F1)Uj2 (3 j p) (5) s(C_2iU_i2U_ij) =s Uc_1i(F₁)U_ij (3 i6=j p) (6)

dU_j2(C2j) = Ud2j(F1)C2j dUj2 dU1j(F1)

(3 j p) (7) U\_i2U_ij(C_2i)C_ij U\_j2U_ji(C_2j) =Uc_ij Uc_1i(F₁)C_ij Uc_ji dU_1j(F₁)

(3 i6=j p) (8) s(F_kU_1k) =s(U_1k) (3 k p) (9) s(FkU1j) =s(U_1kU1j) (3 k6=j p) (10)

s(FkU2k) =Uk2 (3 k p) (11) s(FkU2j) =Ukj (3 k6=j p) (12) s Ud1k(Fk)Uk2 =s( (U_1k)U2k) (3 k p) (13) s Uc_1i(F_k)U_i2 =s( (U_1kU_1i)U_ik) (3 k6=i p) (14) s Ud_1k(F_k)U_kj =s( (U_1k)U_2j) (3 k6=j p) (15) s Uc_1i(F_k)U_ik =s( (U_1kU_1i)U_i2) (3 k6=i p) (16) s Uc1i(Fk)Uij =s( (U_1kU1i)Uij)

(3 i; j; k p; i6=j; j 6=k; k6=i) (17) Ud2k(Fk)C2k Udk2 Ud1k(Fk) =Ck2Ud2k( (U_1k)) (3 k p) (18)

dU2j(Fk)C2j Udj2 dU1j(Fk) =CkjUdjk( (U_1kU1j))

(3 k6=j p) (19) Udkj Ud1k(Fk)Ckj dUjk dU1j(Fk) =Ud2j( (U_1k))C2jdUj2( (U_1kU1j))

(3 k6=j p) (20) Ucij Uc1i(Fk)Cij Ucji dU1j(Fk) =Ucij( (U_1kU1i))CijUcji( (U_1kU1j))

(3 i; j; k p; i6=j; j 6=k; k6=i). (21) Proof.Let( ₁; : : : ; _p)be ap-tuple of edge matrix = [N]_pands-pair(U; C), wheresis a split map with respect to the groupC of . Fork6= 2, we denote by G^k be the matrix obtained from G ^def= U C U by exchanging the block lines of indices2andk, and the corresponding block columns. ClearlyG^k is a Gram matrix corresponding to the p-tuple (2 k) . SinceS_p is generated by the the transpositions(1 2), (2 3), . . . ,(2 p), is regular if and only if the s-pairs U^k; C^k corresponding toG^k (k= 1, 3,4, . . . ,p) lie in the same orbit as(U; C). From now on,iand j are indices distinct from1and2, and distinct to each other. By de…nition ofG¹ we have

G¹₁₂=N,G_1j¹ =U2jC2jN U_j2,G¹_2j =U1jN, G_ij¹ =UijCijN U_ji. De…ne

G^{10 def}= diag (U₁₂; : : : ; U_n2)G¹diag (U₁₂; : : : ; U_n2) , we obtain

G¹₁₂⁰ =N,G¹_1j⁰ =U₁₂U2jC2jN U_j2Uj2=U2jC2jN, G¹_2j⁰ =U_1jN U_j2,G_ij¹⁰ =U_i2U_ijC_ijN U_jiU_j2. Put

G^{100 def}= diag (C21; : : : ; C2n)G¹⁰diag (C21; : : : ; C2n) ,

we have

G¹₁₂⁰⁰=N,G¹_1j⁰⁰ =U2jN,G¹_2j⁰⁰ =U1jN C2jU_j2 , G¹_ij⁰⁰=C_2iU_i2U_ijC_ijN C_2jU_j2U_ji .

The matrixG¹⁰⁰ is reduced, it follows that

U_1j¹ =U2j,U_2j¹ =U1j,U_i2¹ =s C2jU_j2 U_ij¹ =s(C2iU_i2Uij),C_2;j¹ =Ud_j2(C2j) C_i;j¹ =U\_i2U_ij(C_2i)C_ij U\_j2U_ji(C_2j) .

Now, is isometric to (1 2) if and only if there exist F1 2 C such that U¹; C¹ =F1 (U; C),i. e. such that the relations (3), . . . , (8) hold.

Letk be …xed in f3; : : : ; pg. From now on, i and j are moreover supposed to be distinct fromk. By de…nition ofG^k, we have

G^k₁₂=U1kN,G^k_1j =U1jN, G^k_1k =N,G^k_2j=UkjCkjN U_jk, G^k_2k=Uk2Ck2N U_2k,G^k_kj=U2jC2jN U_j2,G^k_ij =UijCijN U_ji. Put

G^{k0 def}= diag (U_1k; I_n; : : : ; I_n)G^kdiag (U_1k; I_n; : : : ; I_n) , we obtain

G₁₂^k0 =N, G^k_1j⁰ =U_1kU_1jN,G_1k^k0 =U_1kN, G_2j^k0 =U_kjC_kjN U_jk, G^k_2k⁰ =Uk2Ck2N U_2k, G^k_kj⁰ =U2jC2jN U_j2,G_ij^k0 =UijCijN U_ji.

LetD be the block-diagonal matrix de…ned byD11=D22=In,Dkk = (U_1k) andD_jj= (U_1kU_1j). PutG^k00def= DG^kD , we have

G^k₁₂⁰⁰=N,G^k_1j⁰⁰=s(U_1kU_1j)N,G^k_1k⁰⁰=s(U_1k)N,

G^k_2j⁰⁰=UkjCkjN( (U_1kU1j)Ujk) , G^k_2k⁰⁰=Uk2Ck2N( (U_1k)U2k) , G^k_kj⁰⁰= (U_1k)U_2jC_2jN( (U_1kU_1j)U_j2) ,

G^k_ij⁰⁰= (U_1kU1i)UijCijN( (U_1kU1j)Uji) . The matrixG^k00 is reduced, whence

U_1k^k =s(U_1k),U_1j^k =s(U_1kU_1j),U_2k^k =U_k2,U_2j^k =U_kj,

U_k2^k =s( (U_1k)U2k), U_i2^k =s( (U_1kU1i)Uik),U_kj^k =s( (U_1k)U2j), U_ik^k =s( (U_1kU1i)Ui2),U_ij^k =s( (U_1kU1i)Uij),

C_2k^k =C_k2Ud_2k( (U_1k)) ,C_2j^k =C_kjUd_jk( (U_1kU_1j)) , C_kj^k =Ud_2j( (U_1k))C_2jUd_j2( (U_1kU_1j)) ,

C_ij^k =Ucij( (U_1kU1i))CijUcji( (U_1kU1j)) .

Now, is isometric to (2 k) if and only if there exist Fk 2 C such that U^k; C^k =E (U; C),i. e. such that the relations (9), . . . , (21) hold.

We say that a s-pair is regular, if it is associated to some regular p-tuple.

By Theorems 2 and 4 (see below), it is equivalent to say that it satis…es the relations (3), . . . , (21). As-orbit will be said to be regular if one (or equivalently each) of itss-pair is regular.

Proposition 7 A regularp-tuple is either common of special.

Proof. Follows from relations (3), (11), (12) and (15).

Theorem 3 Let be a special anisoclinic p-tuple with edge matrix = [N]_p ands-pair [In]_p; C . Then is regular if and only if there existE 2C such thatE²=I_n andC_ij =E for2 i6=j p

Proof. Put U_ij =I_n in the relations (3), . . . , (21). The relations (3), . . . , (6), (9), . . . , (17) and (21) are always satis…ed. The others become

C_2j =C_2j, (3 j p) (22)

C_2iC_ijC_2j =C_ij, (3 i6=j p) (23)

C2k =Ck2, (3 k p) (24)

C_2j =C_kj, (3 k6=j p) (25)

Ckj=C2j, (3 k6=j p) (26)

From (25), Ckj does not depends on k = 2; : : : ; p, k 6= j. From (22), Ckj

is Hermitian. On the other hand, C is also Hermitian, whence Ckj does not depends onj either. Conversely, if all theCij are equal for2 i6=j p, the relations (22), . . . , (26) are satis…ed.

2.3 The existence problem

The problem of existence of ap-tuple with prescribed edge matrix ands-pair (U; C) admits an obvious solution: such a p-tuple will exist if and only if the matrixU C U is positive semi-de…nite. The criterion can be restated in signs of some minors of the matrix, and so appears to be as easy as the Euclidean case. It seems di¢ cult to expect a better one in such a general theory. However, the criterion is no so easy to use in practice, so it will be convenient to have some su¢ cient conditions. We shall prove in particular that a p-tuple with a small enough edge matrix (i. e. large enough prescribed critical angles) always exists. We says that a matrix M is lower than or equal to a matrix N, and writeM N, if and only if each scalar coe¢ cient ofM is lower than or equal to its corresponding coe¢ cient inN.

Theorem 4 For any integers n 1, p 2 there exits a positive number , which depends only onn and p, such that for any s-pair (U; C) and any edge- matrix 2M(D⁺(n); p) lower than or equal to[diag( ; : : : ; )]_p, there exist ap-tuple which is associates to them.

Proof. IfH is an Hermitian matrix, we denote byl(H)its smallest eigenvalue.

It is well known that the map l hereby de…ned is continuous. For any s-pair (U; C)and edge matrix , we have

l(U C U ) min

V2M(U(n);p) K2H(U(n);p)

l(V K V )^def= L( ).

SinceU(n)is compact, it is clear that the minimum exists and that the function Lis continuous. Moreover, L [0]_p = 1, whence there exists such that, for any edge matrix [diag( ; : : : ; )]_p,L( ) 0. Hence, for anys-pair(U; C), and any satisfying the above inequality, we have l(U C U ) 0, i. e.U C U is a Gram matrix.

Remark 8 Augustin Fruchard proved in [4] that, forK=R,p= 3andn= 2,

1

3 is the best possible value for .

The second Theorem of this section states that, roughly speaking, if we choose a s-pair and the critical angles of one edge of the p-tuple, then it will exist, provided that the other critical angles are large enough. We need for its proof the (rather well-known) following Lemma. As for its proof, we refer, for instance, to [8].

Lemma 9 Let A; B; C; D be four square matrices of order k. Let M be the matrix of order2k de…ned by

M = A B

C D .

IfCD=DC, thendetM = det(AD BC).

Theorem 5 Let N 2 D⁺(n) be such that N_ii 2 ]0;1[ (1 i n). Choose distinct indices i₀ and j₀ in f1; : : : ; pg. Let E( ) be the set of np np edge matrix such that _i0j0 = _j0i0 =N and _ij diag ( ; : : : ; )fori6=j and fi; jg 6=fi₀; j₀g. LetC be the group of unitary matrices commuting withN and sa split map with respect to C. For any s-pair(U; C) there exists >0 such that, for any 2 E( ), ap-tuple with Gram matrix U C U exists.

Proof. Let 0 2 M(D(n); p) be such that ( 0)_i0j0 = ( 0)_j0i0 = N and ( 0)_ij = 0 2M(n) for1 i6=j p, fi; jg 6=fi0; j0g. Since the eigenvalues of a matrix depends continuously of its coe¢ cients, it is su¢ cient to prove that G=U C 0 U is positive de…nite. PutA ^def= Gi0j0 =Ui0j0Ci0j0N U_j0i0, we haveAA =U_i²_{0j0 i}²_0j0U_i0j0. Rearranging lines and columns, and by virtue

of Lemma 9, we have

det (G xI_n) =

(1 x)In A 0 0

A (1 x)I_n 0 0

0 0 . .. . .. ...

... ... . .. (1 x)I_n 0

0 0 0 (1 x)I_n

= (1 x)^{(p 2)n} (1 x)In A A (1 x)In

= (1 x)^{(p 2)n}det (1 x)²In Ui0j0

2i0j0U_i0j0

= (1 x)^{(p 2)n}det (1 x)²I_n ²_i0j0

= (1 x)^{(p 2)n} Yn i=1

(1 i x) (1 + i x).

It follows that the eigenvalues of G are 1, 1 i and 1 + i, and so are all positive.

The last theorem of this section concerns special regularp-tuple.

Theorem 6 Let 1; : : : ; n be pairwise distinct numbers of]0;1[. Let Cbe the group of diagonal unitary matrices of order n. LetE be an element of C such that E² =I_n (i. e.E 2 D(n) and E_ii = 1). Let C 2 H(C; p) be de…ned by C_1j =I_n (1 j p) andC_ij =E (2 i6=j p). Then the (special regular anisoclinic)p-tuple with edge matrix [diag ( ₁; : : : ; _n)]_p ands-pair [I_n]_p; C exists if and only if, for each indexi, either E_ii= 1or _{i p}¹₁.

Proof. Put N ^def= diag ( 1; : : : ; n)andG=C [N]_p. Thep-tuple with edge matrix[N]_pands-pair [In]_p; C exists if and only ifGis positive semi-de…nite.

Note thatGand[N E]_pare conjugated bydiag (E; I_n; : : : ; I_n)and thus have the same spectrum. Denote byJ_p thep pmatrix whose diagonal coe¢ cients are 0and the others are1. we have[N E]_p=I_np+J_p N E, and so the eigenvalues of G are the product of eigenvalues of J_p by eigenvalues of N E, plus 1. The computation of the eigenvalues ofJ_pis classical, they are 1 (with multiplicity p 1) and p 1. Hence the eigenvalues of G are 1 1E11, . . . , 1 pEpp

(multiplicity p 1) and 1 + (p 1) 1E11, . . . , 1 + (p 1) pEpp. The p…rst eigenvalues are non-negative, the non-negativeness of the other ones yields the conditionsEii = 1or i 1

p 1.

3 The real anisoclinic case

In this section we consider only real anisoclinicp-tuples. In this case, the group C is the group of diagonal matrices whose diagonal coe¢ cients are 1, or in

other words, the group generated by the re‡ections with respect to the hyper- planes orthogonal to the vectors of the canonical basis. ObviouslyCis of order 2ⁿ, and the center Zof O(n)is f I_ng, hence the s-orbits have at most2^{n 1} s-pairs.

3.1 Case of planes

In this section, we only consider p-tuples of planes, i. e.n = 2. We denote byR( ) the rotation of angle and by S the symmetry with respect to the axisR(1;0). It is well known thatO(2) =fR( ); R( )S j 2R=2 Zg. The group C has four elements, namely I2; S, and the left coset of R( ) is fR( ); R( + ); R( )S; R( + )Sg. We de…nesby

s(R( )S)=s(R( )) =R( ), if 2[0; [, s(R( )S)=s(R( )) =R( + ), otherwise.

Of course, the interval[0; [ should be understood as an interval of the circle R=2 Z. It’s clear that this function sis a split map. We de…ne :C!Cby ( I₂) = I₂and ( S) = S. IfV 2ImsnfI₂g, thenVb = andVc = . Moreover, ifAis a the rotation of angle , we denote byA⁰the rotation of angle

.

Proposition 10 Let (U; C)be a s-pair associated to a common p-tuple of real planes. Then its orbit contains exactly twos-pairs. Moreover, the others-pair ( ~U ;C)~ is related to(U; C)according the following relations

U~ij=U_ij⁰ , 1 i n,2 j n,i6=j, (i; j)6= (1;2) C~_ij=C_ij, 3 i6=j n

C~_2j= C_2j, 3 j n.

It follows that each commons-orbit contains exactly ones-pair such thatC232 fI2; Sg. Thiss-pair and the edge matrix fully characterized thep-tuple.

Proof. Apply the formulae (1) and (2) withE=S.

3.1.1 Triangles

As-pair will depends on three rotations ofIms(the block-coe¢ cientsU13,U23

andU32) and one element ofC(the block-coe¢ cientC23=C32). Thes-orbit of thes-pair(U; C)has at most two elements which are(U; C)andS (U; C). By a straightforward veri…cation(S C)₂₃="23"32"13C23, where"ij = 1ifUij =I2

and 1 otherwise. It follows that, if the number of identity blocks in U is even, there are exactly two elements in the orbit of(U; C), and exactly one with C232 fI2; Sg. Thiss-pair is a numerical invariant which fully characterizes the p-tuple.

If some block U_ij equals I₂, then(S U)_ij =s Uc_1i(S) =I₂, and if some blockUij does not equalI2,(S U)_ij = U_ij. It follows that there are exactly

16 s-orbits which are singletons, those whose matrix U contains only identity blocks, and those whose matrixU contains one identity block and two blocks equal toR ₂ . The other orbits have two elements. For , , 2 ₂;₂ and E2C, we denote byO(E; ; ; )thes-orbit of thes-pair

(U; C) = 0

@ 0

@ I2 I2 R ₂+

I2 I2 R ₂ +

I2 R ₂ + I2

1 A;

0

@ I2 I2 I2

I2 I2 E I₂ E I₂

1 A

1 A.

Strictly speaking, if one of the angle , , is equal to ₂, then the above pair (U; C)is not as-pair. In this case,O(E; ; ; )should be de…ned as the orbit of as-pair corresponding to the -Gram matrixU C U , where is an edge matrix small enough to ensure thatU C U is positive semi-de…nite.

We have the following equalities:

O(E; ; ; ) =O( E; ; ; ) O E; 2; ; =O E;

2; ; O E; ;

2; =O E; ; 2; O E; ; ;

2 =O E; ; ; 2 .

It follows that the space of orbits has two connected components (E= I₂and E= S); each of them is homeomorphic to a cube on which the points of each face are identi…ed by the symmetry with respect to the center of the face.

Remark 11 A generic triple, as de…ned in [4], corresponds to an orbitO( E;

; ; ), with 0<j j, j j, j j< ₂. There is, in such an orbit, exactly one s-pair (U; C)such that C₂₃ belongs tofI₂; Sg. So the triple is characterized by its edge matrix, one sign"= detC₂₃ and the angles , , 2 0;₂ .

As stated in the introduction, Augustin Fruchard has proven that a generic triple is determined by its critical angles, its inner angles !₁,!₂, !₃ 2 0;₂ , and its signature ("₁; "₂; "₃; "₄) 2 f 1g⁴. Using the canonical form (see [4, p.130]), one can compute a Gram matrix from these invariants, and derive the s-pair. A. Fruchard’s invariants and ours are related by the following formulae

2 = (2!₁ )"₁"₂"₃"₄ 2 = (2!2 )"1"2"4

2 = (2!3 )"1"3

"="₁"₂.

3.1.2 Regular anisoclinicp-tuples in G₂ R^2p

From Proposition 7, a regularp-tuple is either common or special. The case of specialp-tuples is settled by Theorem 3 and Theorem 6. So we now consider the case of common regularp-tuples. In this case, we have necessarilyp= 3or p= 4, as stated by the

Theorem 7 Let be an anisoclinic regular p-tuple of real planes. If p > 4, then is special.

Proof. Let be regular, common, anisoclinicp-tuple of real planes withp 5.

Let (U; C) be one of its two s-pairs. Since is common, the relation (18) becomes (Fk)C2kFk = C2k (k= 3; : : : ; p), whence Fk = S; we can thus choseFk=S. The formula (17) becomess U_ij =Uij; hence for3 i6=j p, Uij = R ₂ (the hypothesis p > 4 is used here, for if p 4, there is no triples of distinct indicesi; j; k 2 f3; : : : ; pg, and so, the formula (17) gives no relation). The relations (12) becomes s U_2j = Ukj, whence U2j = R ₂ . Put" = detF1. The relation (3) yieldsU1j ="R "₂ . At last, (10) becomes s U_1j =s(U_1kU_1j),i. e.R ₂ =I₂, which is absurd.

The case of triangles is given by the

Theorem 8 Let be an anisoclinic, regular, common triple of real planes.

Then admits a unique s-pair which is either of the form 0

@ 0

@ I₂ I₂ A I₂ I₂ A I₂ A⁰ I₂

1 A;

0

@ I₂ I₂ I₂ I₂ I₂ S I₂ S I₂

1 A

1 A

or of the form 0

@ 0

@ I2 I2 A⁰ I2 I2 A I2 A⁰ I2

1 A;

0

@ I2 I2 I2

I2 I2 I2

I2 I2 I2

1 A

1 A, whereA=R( ),0< < .

Remark 12 The Theorem 8 can be restated using the invariants , , , "

de…ned in Remark 11: a common triangle of planes is regular if and only if

"= = . Each family of regular triangles corresponds to a diagonal of one the two cubic connected components of the space ofs-orbits. The endpoints of these diagonals correspond to the special regular triangles.

Proof of Theorem 8. Let (U; C) be the s-pair associated to an anisoclinic regular common triple of real planes, such that E ^def= C23 2 fI2; Sg. Put A=U23,"1= detF1and"= detE. As seen in the proof of Theorem 7, we can chooseF3=S in the relations (3), . . . , (21). By straightforward computation, those of the relations (3), . . . , (21) which are not always satis…ed become

A=s(F₁U₁₃) U13=s(F1A) s(EU₃₂) =s(F₁U₃₂)

"="1

A⁰ =U32

U₃₂⁰ =A,

whence the conclusion.

Now let us examine the case of quadruples.

Theorem 9 Given two distinct angles ; 2 0;₂ , there exist at most four common regular quadruples ofG2 R⁸ with critical angles and . They admit as-pair which is either of the form

0 BB

@ 0 BB

@

I2 I2 A A⁰ I2 I2 A A⁰ I2 A⁰ I2 A I₂ A A⁰ I₂

1 CC A;

0 BB

@

I2 I2 I2 I2

I2 I2 S S I2 S I2 S I₂ S S I₂

1 CC A

1 CC A or of the form

0 BB

@ 0 BB

@

I₂ I₂ A⁰ A I₂ I₂ A A⁰ I₂ A⁰ I₂ A I₂ A A⁰ I₂

1 CC A;

0 BB

@

I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂ I₂

1 CC A

1 CC A,

where = 1 andA=R _{2 6} .

Proof. Let (U; C) be the s-pair associated to an anisoclinic regular common quadruple of real planes, such that C23 2 fI2; Sg. Put A = U23. As seen in the proof of Theorem 7, we can choose Fk =S for k 3 in relations (3), . . . , (21). In particular, (11), (12) and (16) become respectively U_k2 = U_2k⁰ (3 k p),U_kj =U_2j⁰ (3 j 6=k p) andU_ik⁰ =U_i2 (3 i6=k p). Hence U34=U42=AandU24=U33=U43=A⁰. Put B^def= R( )^def= U13, by (3), we have U14=B⁰ and eitherB =A or B =A⁰. With this notation, those of the relations (3), . . . , (21) which are not always satis…ed become

A=s(F₁B) (27)

s(C23A⁰ ) =s( (F1)A⁰) (28)

s(C₂₄A ) =s( (F₁)A) (29)

s(C23A⁰ A)= s( (F1)A) (30) s(C₂₄A A⁰)= s( (F₁)A⁰) (31)

(C23)= "C23 (32)

(C24)= "C24 (33)

A[⁰ A(C23)= A A[⁰(C24) (34)

B⁰= s(B⁰ B) (35)

C23= C43 B⁰ B , (36)

where"= detF1. From (35), we get that3 = 0 (mod ), hence = _{2 6} for = 1. The angles of A⁰ A and A A⁰ are opposite to each other and

do not vanish, whence A[⁰ A = and A A[⁰ = and thus (34) becomes (C₂₃) = (C₂₄), whenceC₂₃= C₂₄. Now we discuss two cases.

Case 1: F₁ =I₂. By (27), A=B. The relations (32) implies C₂₃ = S.

Since, by choice of thes-pair,C₂₃ 2 fS; I₂g, we haveC₂₃ =S and C₂₄ = S.

The equation (36) becomesC34= R ₂ S= S. With these values ofCandU, all the relations (27), . . . , (36) are satis…ed.

Case 2: F1=S. By (27),A =B⁰. The relations (32) implies C23 = I2. Since, by choice of thes-pair,C232 fS; I2g, we haveC23 =I2 andC24= I2. The equation (36) becomes C34 = R ₂ I2 = I2. With these values ofC andU, all the relations (27), . . . , (36) are satis…ed.

3.2 Anisoclinic regular p-tuples of real 3-spaces.

We recall that a rotationR2SO(3)is associated to a pair of mutually opposite quaternions in the following way: if we identifyR³with the set of pure imaginary quaternions, the rotation of angle and unitary axis vectoru2R³is represented by the unitary quaternions q = cos ₂ + sin₂u . Then, for any vector of v 2 R³, i. e.for any pure imaginary quaternion v, we have R(v) = qvq.

In order to avoid confusion with indices, we denote by bold characters (i, j, k) the vectors of the canonical basis of R³. Denote by R"0"1"2"3 ("0, "1, "2,

"3 2 f0; 1g) the set of those rotations which are associated to a quaternion numberq=a0+a1i+a2j+a3k, such that"i= sgn (ai)(i= 0, . . . ,3). Obviously R _{"0 "1 "3 "3} =R_"0"1"3"3 and the40elements set

P^def= fR"0"1"3"3j"0; "1; "2; "3= 1;0;1g

is a partition ofSO(3). In order to shorten the notation, we shall only write the sign (+or ) when one of the indices "_k equals 1. We also de…ne R_z (z= 0, 1, 2, 3) as the union of those R_"0"1"3"3 such that exactly z indices among "₀, . . . ,"₃equal zero. Clearly

Q^def= fRkjk= 0; : : :3g is also a partition ofSU(3), coarser thatP.

In the case of anisoclinic tuples of real3-spaces,C= fI₃; S_i; S_j; S_kg, where S_q(q=i;j;k) is the half-turn of axisq, or in other words, the rotation associated to the quaternionq. It is easy to check that the right or left multiplication by S_q (q=i;j;k) respects the partitionP ofSO(3). More precisely we have

S_iR_"0"1"2"4 =R _{"1 "0 "3"2} R_"0"1"2"4S_i=R _{"1 "0 "3 "2} S_jR_"0"1"2"4 =R _{"2"3 "0 "1} R_"0"1"2"4S_j=R _{"2 "3 "0 "1} S_kR_"0"1"2"4=R _{"3 "2 "1 "0} R_"0"1"2"4S_k=R _{"3 "2 "1 "0}.

(37)

In particular each set R_z 2 Q is globally invariant under multiplications by elements ofC^+def= C\SO(3). Now let us examine the left cosets. IfU belongs to someR_"2P ("2 f 1;0g⁴), then the three other elements of the cosetUC,