The moduli space of three-qutrit states

The moduli space of three-qutrit states Emmanuel Briand,a) Jean-Gabriel Luque,b) and Jean-Yves Thibonc) Institut Gaspard Monge, Université de Marne-la-Vallée, F-77454 Marne-la-Vallée cedex, France Frank Verstraeted) Max-Planck-Institut für QuantenOptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC).We describe the orbit space of the SLOCC group SL共3,C兲⫻3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes. I. INTRODUCTION The invariant theory of trilinear forms over a three-dimensional complex vector space is an old subject with a long history, which, as we shall see, appears even longer if we take into account certain indirect but highly relevant contributions.1–4 This question has been recently revived in the field of Quantum Information Theory as the problem of classifying entanglement patterns of three-qutrit states. Indeed, since the advent of quantum computation and quantum cryptography, entanglement has been promoted to a resource that allows quantum physics to perform tasks that are classically impossible. Quantum cryptography5,6 proved that this gap even exists with small systems of two entangled qubits. Furthermore, it is expected that the study of higher dimensional systems and of multipartite (e.g., 3-partite)states would lead to more applications. A seminal example is the so-called 3-qutrit Aharonov-state, which “is so elegant it had to be useful”:7Fitzi, Gisin, and Maurer7found out that the classically impossible Byzantine agreement problem8can be solved using 3-partite qutrit states. From a more fundamental point a view, the Aharonov state led to nontrivial counterexamples of the conjectures on additivity of the relative entropy of entanglement9and of the output purity of quantum channels.10 Obviously, these results provide a strong motivation for studying 3-partite qutrit states. Furthermore, interesting families of higherdimensional states are perfectly suited to address questions concerning local realism and Bell inequalities (see, e.g., Ref. 11 for a study of three-qutrit correlations). It is therefore of interest to find some classification scheme for three-qutrit states. A possible direction is to look for classes of equivalent states, in the sense that they are equivalent up to local unitary transformations12–14 or local filtering operations (also called SLOCC operations).14–19 In the case of three qubits, especially the last classification proved to yield a lot of insights (the a)Electronic mail: [email protected] b)Electronic mail: [email protected] c)Electronic mail: [email protected] d)Electronic mail: [email protected] classification up to local unitaries has too many parameters left); the reason for that is that in the closure of each generic orbit induced by SLOCC operations, there is a unique state (up to local unitary transformations)with maximal entanglement.14,17 In Ref. 19, a numerical method converging to such a maximally entangled state has been described. It has been experimentally observed that, when applied to a three-qutrit state, this method converged to a very special normal form. We shall provide a formal proof of this property, and then study in some detail the geometry of those normal forms. Precise statements of the results are summarized in the forthcoming section. II. RESULTS Let V=C3and H=V丢V丢Vregarded as a representation of the group G=SL共3,C兲⫻3. The elements of Hwill be interpreted either as three-qutrit states 兩 ␺ 典=兺 i,j,k=0 2 Aijk兩i,j,k典共1兲 or as trilinear forms f=f共x,y,z兲=兺 i,j,k=0 2 Aijkxiyjzk,共2兲 that is, we identify the basis state 兩ijk典with the monomial xiyjzk.Ifg=共g共1兲,g共2兲,g共3兲兲苸Gis a triple of matrices, we define xi ⬘=兺pgip 共1兲xp,yj ⬘=兺qgjq 共2兲yq,zk ⬘=兺rgkr 共3兲zr, and the coefficients Aijk ⬘by the condition 兺Aijk ⬘xi ⬘yj ⬘zk ⬘=兺Aijkxiyjzk,共3兲 the action of Gon Hbeing defined by g·A=兺Aijk ⬘xiyjzk,共4兲 It has been shown by Vinberg20 that a generic state can be reduced to the normal form Aijk ⬘=u ␦ ijk +w−v 2 ⑀ ijk +w+v 2兩 ⑀ ijk兩共5兲 (where ␦ ijk is the Kronecker symbol and ⑀ ijk the completely antisymmetric tensor)by an appropriate choice of g苸G. Our first result is as follows. Theorem II.1: When applied to a generic 3-qutrit state (1) the numerical algorithm of Ref. 19 converges to a state which is a Vinberg normal form, generically in the same G-orbit as 兩 ␺ 典. As proved in Refs. 14 and 19, the normal form 兩 ␺ ⬘典is unique up to local unitary transformations. More precisely, we have the following. Theorem II.2: A generic state has exactly 648 different normal forms. For special states, this number can be reduced to 216, 72, 27 or 1. Moreover, the coefficients u,v,w of the normal form can be computed algebraically. Theorem II.3: The coefficients of the normal forms are determined, up to a sign, by an algebraic equation of degree 1296, which is explicitly solvable by radicals. To form this equation, we need some notions of invariant theory. A polynomial P共A兲in the coefficients Aijk is an invariant of the action of Gon Hif P共A⬘兲 =P共A兲for all g苸G. These invariants form a graded algebra R(any invariant Pis a sum of homogeneous invariants)and the first issue is to determine the dimension of the space Rdof homogeneous invariants of degree d. The Hilbert series h共t兲=兺 d艌0dim Rdtd共6兲 is known20 h共t兲=1 共1−t6兲共1−t9兲共1−t12兲共7兲 and in fact, one can prove that Ris a polynomial algebra generated by three algebraically independent invariants of respective degree 6, 9, and 12. The modern way to prove this result is due to Vinberg, who obtained it from his notion of Weyl group of a graded Lie algebra, applied to a Z3-grading of the exceptional Lie algebra E6.21 In Sec. III, we shall explain how it can be deduced from the work of Chanler.22 We prove that certain invariants I6,I9, and I12 introduced in Ref. 22 are indeed algebraic generators of Rand explain how to compute them from the numerical values of the coefficients Aijk, by expressing them in terms of transvectants, that is, by means of certain differential polynomials in the form f, rather than in terms of the classical symbolic notation. Given the values of the invariants for a particular state, we show how to form and solve the system of algebraic equations determining the coefficients, u,v,wof the normal form. Let a=I6,b=I12, and c=I18 (a certain polynomial in the fundamental invariants). Then, the symmetric functions of u3,v3, and w3 ␺ =u3+v3+w3, ␹ =u3v3+u3w3+v3w3,␭= 216u3v3w3共8兲 satisfy ␺ 2−12 ␹ −a=0, ␺ 4+␭ ␺ −b=0, ␺ 6−5 2␭ ␺ 3−1 8␭2−c=0. 共9兲 Theorem II.4: The system (9) has generically 1296 solutions 共u,v,w兲,which can be obtained by solving a chain of algebraic equations of degree at most 4. Only 648 of them give the correct sign for I9.The number of solutions (with the correct sign for I9) can be reduced only to 216, 72, 27 or 1. Moreover, the isotropy groups of these degenerate orbits can be determined, and the configuration of the points 共u,v,w兲in C3can be interpreted in terms of the geometry of regular complex polyhedra. The details are given in Sec. VII. III. THE FUNDAMENTAL INVARIANTS In this section, we describe the fundamental invariants, as well as the other concomitants obtained by Chanler,22 in a form suitable for calculations, in particular for their numerical evaluation (see also Refs. 23 and 24). As already mentioned, we shall identify a three-qutrit state 兩 ␺ 典苸Hwith a trilinear form f共x,y,z兲=兺 1艋i,j,k艋3Aijkxiyjzk共10兲 in three ternary variables. To construct its fundamental invariants, we shall need the notion of a transvectant, which is defined by means of Cayley’s omega process (see, e.g., Ref. 25). Let f1,f2, and f3be three forms in a ternary variable x=共x1,x2,x3兲. Their tensor product f1 丢f2丢f3is identified with the polynomial f1共x共1兲兲f2共x共2兲兲f3共x共3兲兲in the three independent ternary variables x共1兲,x共2兲and x共3兲. We use the “trace” notation of Olver26 to denote the multiplication map f1丢f2丢f3→f1f2f3, that is, tr f1共x共1兲兲f2共x共2兲兲f3共x共3兲兲=f1共x兲f2共x兲f3共x兲.共11兲 Cayley’s operator ⍀xis the differential operator ⍀x= 冨 ⳵ ⳵ x1 共1兲 ⳵ ⳵ x1 共2兲 ⳵ ⳵ x1 共3兲 ⳵ ⳵ x2 共1兲 ⳵ ⳵ x2 共2兲 ⳵ ⳵ x2 共3兲 ⳵ ⳵ x3 共1兲 ⳵ ⳵ x3 共2兲 ⳵ ⳵ x3 共3兲 冨 .共12兲 Now, we consider three independent ternary variables x,y, and ztogether with the associated dual (contravariant)variables ␰ =共 ␰ 1, ␰ 2, ␰ 3兲, ␩ =共 ␩ 1, ␩ 2, ␩ 3兲, ␨ =共 ␨ 1, ␨ 2, ␨ 3兲[that is, ␰ iis the linear form on the xspace such that ␰ i共xj兲= ␦ ij]. Aconcomitant of fis, by definition, a polynomial Fin the Aijk,x,y,z, ␰ , ␩ , ␨ , such that if g=共g1,g2,g3兲苸SL共3,C兲3, then, with A⬘,x⬘, etc., as above, F共A⬘;x⬘,y⬘,z⬘; ␰ ⬘, ␩ ⬘, ␨ ⬘兲=F共A;x,y,z; ␰ , ␩ , ␨ 兲.共13兲 The algebra of concomitants admits only one generator of degree 1 in the Aijk, which is the form fitself. Other concomitants can be deduced from fand the three absolute invariants P ␣ =兺 ␰ ixi,P ␤ =兺 ␩ jyj, and P ␥ =兺 ␨ kzk, using transvectants. If F1,F2, and F3are three 6-tuple forms in the independent ternary variables x,y,z, ␰ , ␩ , and ␨ , one defines for any 共n1,n2,n3兲 ⫻共m1,m2,m3兲苸N3⫻N3the multiple transvectant of F1,F2, and F3by 共F1,F2,F3兲m1m2m3 n1n2n3=tr ⍀x n1⍀y n2⍀z n3⍀ ␰ m1⍀ ␩ m2⍀ ␨ m3兿 i=1 3 Fi共x共i兲,y共i兲,z共i兲; ␰ 共i兲, ␩ 共i兲, ␨ 共i兲兲.共14兲 For convenience, we will set 共F1,F2,F3兲n1n2n3=共F1,F2,F3兲000 n1n2n3. The concomitants of degree 2 given by Chanler22 can be obtained using these operations, Q ␣ =共f,f,P ␤ P ␥ 兲011,共15兲 Q ␤ =共f,f,P ␣ P ␥ 兲101,共16兲 Q ␥ =共f,f,P ␣ P ␤ 兲110.共17兲 The invariant I6is then I6=1 96共Q ␣ ,Q ␣ ,Q ␣ 兲011 200 =1 96共Q ␤ ,Q ␤ ,Q ␤ 兲101 020 =1 96共Q ␥ ,Q ␥ ,Q ␥ 兲110 002.共18兲 There is an alternative expression using only the ground form f, I6=1 1152共f2,f2,f2兲222.共19兲 Now, in degree 3 the covariants B ␣ ,B ␤ , and B ␥ of Ref. 22 are B ␣ =共f,f,f兲011,共20兲 B ␤ =共f,f,f兲101,共21兲 B ␥ =共f,f,f兲110.共22兲 The other concomitants found by Chanler can be written in a similar way, C ␣␤ =1 4共f,f,fP ␤ 兲110,共23兲 C ␤␣ =1 4共f,f,fP ␣ 兲110,共24兲 C ␣␥ =1 4共f,f,fP ␥ 兲101,共25兲 C ␥␣ =1 4共f,f,fP ␣ 兲101,共26兲 C ␤␥ =1 4共f,f,fP ␥ 兲011,共27兲 C ␥␤ =1 4共f,f,fP ␤ 兲011,共28兲 D ␣ =−2共fP ␤ ,fP ␥ ,f兲111,共29兲 D ␤ =2共fP ␣ ,fP ␥ ,f兲111,共30兲 D ␥ =−2共fP ␣ ,fP ␤ ,f兲111,共31兲 E ␣ =共Q ␣ ,f,P ␣ 兲100,共32兲 E ␤ =共Q ␤ ,f,P ␤ 兲010,共33兲 E ␥ =共Q ␥ ,f,P ␥ 兲001,共34兲 G ␣ =−3 8共fP ␤ ,fP ␥ ,f兲011 +5 16共fP ␤ P ␥ ,f,f兲011,共35兲 G ␤ =−3 8共fP ␣ ,fP ␥ ,f兲101 +5 16共fP ␣ P ␥ ,f,f兲101,共36兲 G ␥ =−3 8共fP ␣ ,fP ␤ ,f兲110 +5 16共fP ␣ P ␤ ,f,f兲110,共37兲 H=1 2共fP ␣ ,fP ␤ ,fP ␥ 兲111.共38兲 Here, we have combined the concomitants of degrees 0, 1, and 2 into independent concomitants of degree 3. Next, we have chosen the scalar factors so that the syzygies given by Chanler22 hold in the form H+E ␣ −E ␥ +D ␤ P ␤ =0, 共39兲 H+E ␤ −E ␣ +D ␥ P ␥ =0, 共40兲 H+E ␥ −E ␤ +D ␣ P ␣ =0, 共41兲 3C ␣␤ −B ␥ P ␤ =0, 共42兲 3C ␤␣ −B ␥ P ␣ =0, 共43兲 3C ␣␥ −B ␤ P ␥ =0, 共44兲 3C ␥␣ −B ␤ P ␣ =0, 共45兲 3C ␤␥ −B ␣ P ␥ =0, 共46兲 3C ␥␤ −B ␣ P ␤ =0, 共47兲 6G ␣ −3Q ␣ f+B ␣ P ␤ P ␥ =0, 共48兲 6G ␤ −3Q ␤ f+B ␤ P ␣ P ␥ =0, 共49兲 6G ␥ −3Q ␥ f+B ␥ P ␣ P ␤ =0. 共50兲 One can remark that a basis of the space of the concomitants of degree 3 found by Chanler can be constructed using only transvections and products from smaller degrees, f3,Q ␣ f,Q ␤ f,Q ␥ f,B ␣ ,B ␤ ,B ␥ ,D ␣ ,D ␤ ,D ␥ ,E ␣ .共51兲 The knowledge of these concomitants allows one to construct the invariants I9and I12, I9=1 576共E ␣ ,E ␤ ,E ␤ 兲111 111,共52兲 I12 =1 124 416共B ␣ f,B ␣ f,B ␣ f兲411.共53兲 These expressions, which can be easily implemented in any computer algebra system, will prove convenient to compute the specializations discussed in the sequel. IV. NORMAL FORM AND INVARIANTS It will now be shown that a generic state can be reduced to the normal form Aijk =u ␦ ijk +w−v 2 ⑀ ijk +w+v 2兩 ⑀ ijk兩,共54兲 where ⑀ ijk is the alternating tensor, or, otherwise said, that the generic trilinear form f共x,y,z兲is equivalent to some Nuvw共x,y,z兲=u共x1y1z1+x2y2z2+x3y3z3兲+v共x1y3z2+x2y1z3+x3y2z1兲 +w共x1y2z3+x2y3z1+x3y1z2兲.共55兲 For such a state, the local density operators are all proportional to the identity. This property is automatically satistfied by the limiting state obtained from the numerical method of Ref. 19, and implies maximal entanglement as well. Since this algorithm amounts to an infinite sequence of invertible local filtering operations, the genericity of Vinberg’s normal form, together with the previously mentioned properties, implies convergence to a Vinberg normal form for a generic input state, that is, our Theorem II.1 (see also Refs. 27 and 28). This normal form is in general not unique, and the relations between the various Nuvwin a given orbit is an interesting question, which will be addressed in the sequel. Although, the validity of this normal form follows from Vinberg’s theory,21 it can also be proved in other ways, some of them being particularly instructive. We shall detail one of these possibilities, which will give us the opportunity to introduce some important polynomials, playing a role in the algebraic calculation of the normal form and in the geometric discussion of the orbits. The shortest possibility, although not the most elementary, relies on the results of Ref. 22, and starts with computing the invariants of Nuvw. We then use a few results of algebraic geometry, which can be found in Ref. 29. Let us denote by Ck⬅Ck共u,v,w兲共k=6,9,12兲the values of the Ik on Nuvw. Direct calculation gives, denoting by mpqr the monomial symmetric functions of u,v,w (sum of all distinct permutations of the monomial upvqwr), C6=m共6兲−10m共3,3兲,共56兲 C9=共u3−v3兲共u3−w3兲共v3−w3兲,共57兲 C12 =m共12兲+4m共9,3兲+6m共6,6兲+ 228m共6,3,3兲.共58兲 It is easily checked by direct calculation that the Jacobian of these three functions is nonzero for generic values of 共u,v,w兲. Actually, its zero set consists of 12 planes, whose geometric significance will be discussed below. Let us denote by ␸ :H→ 共I6,I9,I12兲 C3, the map sending a trilinear form to its three invariants, so that 共C6,C9,C12兲= ␸ 共Nuvw兲. Let S=兵Nuvw兩共u,v,w兲苸C3其be the three-dimensional space of normal forms. The nonvanishing of the Jacobian proves that ␸ induces a dominant mapping from Sto C3 (that is, the direct image of any nonempty open subset of Scontains a nonempty open subset of C3). Note that the independence of C6,C9,C12 implies the independence of I6,I9,I12. Now, Chanler22 has shown that I6,I9,I12 separate the orbits in general position. This proves that the field of rational invariants of Gis freely generated by I6,I9,I12 (Ref. 29, Lemma 2.1). As a consequence, ␸ is a rational quotient (Ref. 29, Sec. 2.4)for the action of Gon H(actually, this also implies that ␸ is a categorical quotient, by Ref. 29, Proposition 2.5 and Theorem 4.12, using that ␸ 兩Sis surjective, whence also ␸ ). There exists a nonempty open subset Y0of C3such that the fiber of ␸ over each of its points is the closure of an orbit (Ref. 29, Proposition 2.5). Let then U0= ␸ −1共Y0兲. This set cuts Ssince ␸ 兩S is dominant. Let U1be the union of all orbits having maximal dimension (a nonempty open set, the function dimension of the orbit being lower semicontinuous). It is easy to see that U1intersects S(for instance at u=1,v=1,w=−1, whose orbit has dimension 24=dim G, as may be checked by direct calculation). Let S0=U1艚S, a dense open subset of S. The set ␸ −1 ␸ 共S0兲thus contains a dense open subset U2of H. One then checks that U0艚U1艚U2(a dense open subset, as an intersection of dense open subsets of an irreducible space)is contained in GS. This proves GS =H, that is, the normal form Nuvwis generic. Let us remark that the above discussion also proves, thanks to Igusa’s theorem (Ref. 29, Theorem 4.12)that C关H兴G=C关I6,I9,I12兴, that is, the algebra of invariants is freely generated by Chanler’s invariants. Is is also possible to give a direct proof of the normal form by using the same technique as in Ref. 22. Chanler’s method relies on the geometry of plane cubics, which will play a prominent role in the sequel. V. THE FUNDAMENTAL CUBICS The trilinear form f共x,y,z兲can be encoded in three ways by a 3⫻3 matrix of linear forms Mx共x兲,My共y兲, and Mz共z兲, defined by f共x,y,z兲=tyMx共x兲z=txMy共y兲z=txMz共z兲y共59兲 and the classification of trilinear forms amounts to the classification of one of these matrices, say Mx共x兲up to left and right multiplication by elements of SL共3,C兲and action of SL共3,C兲on the variable x. The most immediate covariants of fare the determinants of these matrices X共x兲= det Mx共x兲=1 6B ␣ ,共60兲 Y共y兲= det My共y兲=1 6B ␤ ,共61兲 Z共z兲= det Mz共z兲=1 6B ␥ .共62兲 These are ternary cubic forms, and for generic fthe equations X共x兲=0, etc., will define nonsingular cubics (elliptic curves)in P2. It is shown in Ref. 24 that whenever one of these curves is elliptic, so are the other two ones, and moreover, all three are projectively equivalent. Actually, one can check by direct calculation that they have the same invariants. When f=Nuvw, these three cubics have even the same equation and are in the Hesse canonical form30 X共x兲=− ␾ 共x1 3+x2 3+x3 3兲+ ␺ x1x2x3=Y共x兲=Z共x兲,共63兲 where we introduced, following the notation of Ref. 2, ␾ =uvw, ␺ =u3+v3+w3.共64兲 The Aronhold invariants of the cubics (63)are given by 64S=− ␾ 共 ␺ 3+共6 ␾ 兲3兲,共65兲 66T=共6 ␾ 兲6+20共6 ␾ 3兲 ␺ 3−8 ␺ 6.共66兲 These are of course invariants of f. We recognize that 64S=−C12, and we introduce an invariant I18 such that C18=I18共Nuvw兲=66T. The three cubics have the same discriminant 64S3+T2, known to be proportional to the hyperdeterminant of f(see Refs. 31 and 32), which we normalize as ⌬=27共64S3+T2兲.共67兲 Then ⌬=C⬘12 3, where C12 ⬘is the product of 12 linear forms C12 ⬘=uvw共u+v+w兲共␧u+v+w兲共u+␧v+w兲共␧2u+␧v+w兲共u+␧2v+w兲 ⫻共␧u+␧v+w兲共␧2u+v+w兲共␧u+␧2v+w兲共␧2u+␧2v+w兲,共68兲 where ␧=e2i ␲ /3, so that C12 ⬘=0 is the equation in P2of the twelve lines containing 3⫻3 the nine inflection points of the pencil of cubics, u3+v3+w3+6mu vw=0, 共69兲 obtained from X,Y,Zby treating the original variables as parameters. We note also that the Jacobian of C6,C9,C12 is proportional to C⬘12 2. VI. SYMMETRIES OF THE NORMAL FORMS In this section, we will prove Theorem II.2. That is, a generic fhas 648 different normal forms [the points 共u,v,w兲for which this number is reduced will be studied in Sec. VII]. To prove the theorem, we remark that the Hilbert series (7)is also the one of the ring of invariants of G25, the group number 25 in the classification of irreducible complex reflection groups of Shephard and Todd.4This group, which we will denote for short by K, has order 648. It is one of the groups considered by Maschke2in his determination of the invariants of the symmetry group of the 27 lines of a general cubic surface in P3(a group with 51 840 elements, which is related to the exceptional root system E6). To define K, we first have to introduce Maschke’s group H, a group of order 1296, which is generated by the matrices of the linear transformations on C3given in Table I. This group contains in particular the permutation matrices, and simultaneous multiplication by ±␧k, since E2=−B. The subgroup Kis the one in which odd permutations can appear only with a minus sign. It is generated by A,C,D,E. Then, as proved by Maschke, the algebra of invariants of Kin C关u,v,w兴is precisely C关C6,C9,C12兴. Hence, we can conclude that Kis the symmetry group of the normal forms Nuvw. There was another, equally natural possibility leading to the same Hilbert series. The symmetry group Lof the equianharmonic cubic surface ⌺:z0 3+z1 3+z2 3+z3 3=0 acting on the homogeneous coordinate ring C关⌺兴has as fundamental invariants the elementary symmetric functions of the zi 3, the first one being 0 by definition, so that the Hilbert series of C关⌺兴Lcoincides with (7). Moreover, Lis also of order 648, but it is known that it is not isomorphic to K. Taking into account the results of Sec. IV, we see that S=兵Nuvw兩共u,v,w兲苸C3其共70兲 is what is usually called a Chevalley section of the action of Gon H, with Weyl group K(see Ref. 29, p. 174). VII. THE FORM PROBLEM This section contains the proofs of Theorems II.3 and II.4. Klein (see Ref. 33)has introduced and investigated the notion of “Formenproblem” associated to a finite group action. This is the following: given the numerical values of the invariants, compute the coordinates of a point of the corresponding orbit. In our case, we shall see that the problem of finding the parameters 共u,v,w兲of the normal form of a given generic f, given the values of the invariants, can be reduced to a chain of algebraic equations of degree at most 4, hence sovable by radicals. Let a=I6,b=I12 and c=I18 (we start with I18, because C18 is a symmetric function of u3,v3,w3, and at the end of the calculation, select the solutions which give the correct sign for C9, which is alternating). What we have to do is to determine the elementary symmetric functions e1= ␺ ,e2= ␹ ,e3 = ␾ 3of u3,v3,w3. Let ␭=216 ␾ 3. Then, ␺ 4+␭ ␺ −b=0, 共71兲 ␺ 6−5 2␭ ␺ 3−1 8␭2−c=0. 共72兲 Eliminating ␭from these equations, we get a quartic equation for ␺ 2, 27 ␺ 8−18b ␺ 4−8c ␺ 2−b2=0. 共73兲 The discriminant (with respect to ␺ )of this polynomial is proportional to D=b2共b3−c2兲4. When it is nonzero, we get eight values for e1, each of which determines univocally e2and e3. Hence, we obtain eight cubic equations whose roots are the possible values of u3,v3,w3. This gives eight sets, whence 8⫻6=48 triples, each of which providing generically 27 values of 共u,v,w兲, in all 48 ⫻27=1296 triples corresponding to the given values of a,b,c, among which exactly 1296/2 =648 give the correct sign for I9. The common discriminant of the eight cubics is ␦ =a3−3ab TABLE I. The generators of H. AB C D E u⬘vuu u 1/i冑3共u+v+w兲 v⬘ww ␧v␧v1/i冑3共u+␧v+␧2w兲 w⬘uv␧2w␧w1/i冑3共u+␧2v+␧w兲