MONODROMY OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION OF TYPE (3, 6) III. Contents 1. Introduction 1

Transcription

1 MONODROMY OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION OF TYPE (3, 6) III KEIJI MATSUMOTO, TAKESHI SASAKI, AND MASAAKI YOSHIDA Abstract. For the hypergeometric system E(3, 6; α) of type (3, 6), two special cases α 1/2 and α 1/6 are studied in [MSY] and [MSTY2], respectively. The monodromy group of the former is an arithmetic group acting on a symmetric domain, and that of the latter is the unitary reflection group ST34. In this paper, we find a relation between these two groups. Contents 1. Introduction 1 2. Hypergeometric system E(3, 6; α) 1 3. A compactification X of X(3, 6) 2 4. Local property of E(3, 6) 2 5. Monodromy groups 3 6. Groups related to ST 34 and Γ(2) 5 7. Relation between the two monodromy groups 6 8. Concluding remarks 8 References 9 1. Introduction For the hypergeometric system E(3, 6; α) of type (3, 6), two special cases α 1/2 and α 1/6 are studied in [MSY] and [MSTY2], respectively. The monodromy group of the former, say M(1/2), is an arithmetic group acting on a symmetric domain, and that of the latter, say M(1/6), is the unitary reflection group ST34. In this paper, we find a relation between these two groups; roughly speaking, M(1/6) is isomorphic to M(1/2) modulo Hypergeometric system E(3, 6; α) Let X = X(3, 6) be the configuration space of six lines in the projective plane P 2 defined as X(3, 6) = GL 3 (C)\{z M(3, 6) D z (ijk) 0, 1 i < j < k 6}/H 6, where M(3, 6) is the set of 3 6 complex matrices, D z (ijk) is the (i, j, k)-minor of z, and H 6 GL 6 (C) is the group of diagonal matrices. It is a 4-dimensional complex manifold. A matrix z M(3, 6) defines six lines in P 2 : L j : l j := z 1j t 1 + z 2j t 2 + z 3j t 3 = 0, 1 j 6, Date: 21 June Mathematics Subject Classification. 33C80. Key words and phrases. hypergeometric differential equation, monodromy group, Schwarz map, unitary reflection group. 1

2 2 KEIJI MATSUMOTO, TAKESHI SASAKI, AND MASAAKI YOSHIDA where t 1 : t 2 : t 3 is a system of homogeneous coordinates. For parameters α = (α 1,..., α 6 ) ( α i = 3), we consider the integral 6 l j (z) αj 1 dt, dt = t 1 dt 2 dt 3 + t 2 dt 3 dt 1 + t 3 dt 1 dt 2 σ j=1 for a (twisted) 2-cycle σ in P 2 L j. It is a function in z, but not quite a function on X. So for simplicity, we fix local coordinates on X as x 1 x 2, x 3 x 4 and consider such integrals above as functions in x = (x 1, x 2, x 3, x 4 ). Then they satisfy a system of linear differential equations on X, called the hypergeometric differential equation E(3, 6; α) of type (3, 6). The rank (dimension of local solutions) of this system is six. This system can be represented, for example, by (α D 1234 )D 1 u = x 1 (D α 5 )(D 12 + α 2 )u, (α D 1234 )D 2 u = x 2 (D α 6 )(D 12 + α 2 )u, (α D 1234 )D 3 u = x 3 (D α 5 )(D 34 + α 3 )u, (α D 1234 )D 4 u = x 4 (D α 6 )(D 34 + α 3 )u, x 1 (α 5 1 D 13 )D 2 u = x 2 (α 6 1 D 24 )D 1 u, x 3 (α 5 1 D 13 )D 4 u = x 4 (α 6 1 D 24 )D 3 u, x 1 (α 2 + D 12 )D 3 u = x 3 (α 3 + D 34 )D 1 u, x 2 (α 2 + D 12 )D 4 u = x 4 (α 3 + D 34 )D 2 u, x 2 x 3 D 1 D 4 u = x 1 x 4 D 2 D 3 u, where D i = x i / x i, α i...j = α i + + α j, D i...j = D i + + D j. 3. A compactification X of X(3, 6) The configuration space X = X(3, 6) admits an obvious action of the symmetric group S 6 permuting the numbering of the six lines. The Grassmann duality on the Grassmannian Gr(3, 6) induces an involution on X. A system of six lines, representing a point of X, is fixed by if and only if there is a conic tangent to the six lines. The set of fixed points of on X is a 3-dimensional submanifold of X. Indeed it is isomorphic to the configuration space X(2, 6) of six points on P 1. The action of S 6 and that of commutes. There is a compactification X of X on which S 6 acts bi-regularly, such that X/ = P 4 (bi-regular). 4. Local property of E(3, 6) Let X ijk be the set of points in X represented by the system (L 1,..., L 6 ) of six lines such that L i, L j, L k meet at a point. Then X = X 1 i<j<k 6 X ijk. The system E(3, 6) can be considered to be defined on X with regular singularity along X ijk.

3 MONODROMY OF HYPERGEOMETRIC SYSTEM 3 The Schwarz map s(α) of the system E(3, 6; α) is defined by linearly independent solutions u 1,..., u 6 as s : X x u 1 (x) : : u 6 (x) P 5, It has exponent α i + α j + α k 1 along X ijk. In the x-coordinates, only 14 among the twenty X ijk are visible; they are given by the equations D(ijk) = 0: D(135) = x 1, D(136) = x 2, D(345) = x 1 1, D(346) = x 2 1, D(125) = x 3, D(126) = x 4, D(245) = x 3 1, D(246) = x 4 1, D(145) = x 1 x 3, D(146) = x 2 x 4, D(256) = x 3 x 4, D(356) = x 1 x 2, D(156) = x 1 x 4 x 2 x 3, D(456) = (x 1 1)(x 4 1) (x 2 1)(x 3 1). 5. Monodromy groups For 1 i < j < k 6, we introduce 6-vectors a ijk and b ijk : a 123 = ( d 123, d 12 c 3, 0, d 1 c 23, 0, 0), b 123 = (1, 0, 0, 0, 0, 0), a 124 = ( d 4 c 12, d 12, d 12 c 4, d 1 c 2, d 1 c 24, 0), b 124 = (1, 1, 0, 0, 0, 0), a 125 = ( d 5 c 12, 0, d 12, 0, d 1 c 2, 0), b 125 = (1, 1, 1, 0, 0, 0), a 126 = (1, 0, 0, 0, 0, 0), b 126 = ( d 126, d 1236 /c 3, d 5 c 126, 0, 0, 0), a 134 = (d 4 c 1, d 34 c 1, d 3 c 14, d 1, d 1 c 4, d 1 c 34 ), b 134 = (0, 1, 0, 1, 0, 0), a 135 = (d 5 c 1, d 5 c 13, d 3 c 1, 0, d 1, d 1 c 3 ), b 135 = (0, 1, 1, 1, 1, 0), a 136 = ( 1/c 2, c 3 /c 2, 0, 0, 0, 0), b 136 = ( d 2, d 1236 /c 3, d 5 c 126, d 1236 /c 3, d 5 c 126, 0), a 145 = (0, d 5 c 1, d 45 c 1, 0, 0, d 1 ), b 145 = (0, 0, 1, 0, 1, 1), a 146 = (0, 1, c 4, 0, 0, 0), b 146 = (d 2 /c 2, d 1456, d 5 c 16, d 3 c 1456, d 5 c 16, d 5 c 16 ), a 156 = (0, 0, 1, 0, 0, 0), b 156 = (d 2 /c 2, d 1456, d 156, d 3 c 1456, d 34 c 156, d 4 c 156 ), a 234 = ( d 4, d 34, d 3 c 4, d 234, d 23 c 4, d 2 c 34 ), b 234 = (0, 0, 0, 1, 0, 0), a 235 = ( d 5, d 5 c 3, d 3, d 5 c 23, d 23, d 2 c 3 ), b 235 = (0, 0, 0, 1, 1, 0), a 236 = (1, c 3, 0, c 23, 0, 0), b 236 = (d 1 /c 1, 0, 0, d 45 c 6, d 5 c 6, 0), a 245 = (0, d 5, d 45, d 5 c 2, d 45 c 2, d 2 ), b 245 = (0, 0, 0, 0, 1, 1), a 246 = (0, 1, c 4, c 2, c 24, 0), b 246 = (d 1 /c 1, d 1 /c 1, 0, d 3 c 456, d 5 c 6, d 5 c 6 ), a 256 = (0, 0, 1, 0, c 2, 0), b 256 = (d 1 /c 1, d 1 /c 1, d 1 /c 1, d 3 c 456, d 34 c 56, d 4 c 56 ), a 345 = (0, 0, 0, d 5, d 45, d 345 ), b 345 = (0, 0, 0, 0, 0, 1), a 346 = (0, 0, 0, 1, c 4, c 34 ), b 346 = (0, d 1 /c 1, 0, d 3456, 0, d 5 c 6 ), a 356 = (0, 0, 0, 0, 1, c 3 ), b 356 = (0, d 1 /c 1, d 1 /c 1, d 3456, d 3456, d 4 c 56 ), a 456 = (0, 0, 0, 0, 0, 1), b 456 = (0, 0, d 1 /c 1, 0, d 12 /c 12, d 456 ), where c j = exp 2πiα j, c ij = c i c j, d ij = c ij 1. The circuit matrix around a loop in X going once around the divisor X ijk is given by R ijk = I 6 t a ijk b ijk. These R ijk (1 i < j < k 6) generate the monodromy group M(α) of the system E(3, 6; α). The monodromy group keeps the form d 1 d 2 d 345 d 1 d 2 d 45 d 1 d 2 d c 3 d 1 d 2 d 45 d 1 d 23 d 45 d 1 d 23 d 5 d 1 d 3 d 45 d 1 d 3 d 5 0 H = d 6 c 34 d 1 d 2 d 5 c 4 d 1 d 23 d 5 d 1 d 234 d 5 c 4 d 1 d 3 d 5 d 1 d 34 d 5 d 1 d 4 d 5 0 c 2 d 1 d 3 d 45 c 2 d 1 d 3 d 5 d 12 d 3 d 45 d 12 d 3 d c 24 d 1 d 3 d 5 c 2 d 1 d 34 d 5 c 4 d 12 d 3 d 5 d 12 d 34 d 5 d 12 d 4 d c 23 d 1 d 4 d 5 0 c 3 d 12 d 4 d 5 d 123 d 4 d 5 invariant: t ŘHR = H, R M(α),

4 4 KEIJI MATSUMOTO, TAKESHI SASAKI, AND MASAAKI YOSHIDA where ˇ is the operator which changes c j to 1/c j. The above facts are shown in [MSTY1, MSTY2]. Note that the lists in these papers contains errors for a 136 and a 145, so we tabulated here the corrected vectors. The scalars a ijk tb lmn have the following properties. Lemma 1. (1) a ijk tb ijk = 1 c i c j c k (1 i < j < k 6). (2) For the other entries, we omit explicit expressions: If a ijk tb lmn is not zero as a rational function of c 1,..., c 5 (c 6 = (c 1 c 5 ) 1 ), we replace the value simply by z. Then the matrix AB := (a ijk tb lmn ) is given as z z z z z z z z z z z z z z z z z z z 0 z z z z z z z z z 0 z z z z z 0 z z z z z z z z z 0 0 z 0 z z 0 0 z 0 z z z z 0 0 z z z z z 0 z z z z z 0 z z 0 0 z z z 0 z z z z z z 0 z z z z z z z z 0 z z z z z z z 0 z z 0 0 z 0 z z 0 z z 0 0 z z 0 z z 0 z z z z z z z z z z 0 z z z z z z z z z z z 0 z z z 0 0 z z 0 z z z z z z 0 0 z z z z 0 0 z z z z z z 0 z z 0 0 z z z 0 z z z z z z z z z z z 0 z z z z z z z z z z 0 z z 0 z z 0 0 z z 0 z z 0 z 0 0 z z 0 z z z z z z z 0 z z z z z z z z 0 z z z z z z 0 z z z 0 0 z z 0 z z z z z 0 z z z z z 0 0 z z z z 0 z 0 0 z z 0 z 0 0 z z z z z z z z z 0 z z z z z 0 z z z z z z z z z 0 z z z z z z z z z z z z z z z z z z z We are interested in the most symmetric cases: c 1 = = c 6 =: c. Since c 6 = 1, we have c = 1, 1, ω, ω, where ω is a primitive third root of unity. Excluding the trivial case c = 1 (for all i, j, k, we have a ijk = 0 or b ijk = 0, and so R ijk = I 6 ), there are three cases. By the explicit expression of AB, we see Lemma 2. When c = 1 and ω, the entries of the matrix AB marked z are not equal to zero. When c = ω, the diagonal elements of the matrix AB are zero, while the other entries marked z are not zero. The case c = ω is of special interest; this will be studied in [SY]. In this paper, we treat the two cases { Case 0 : 1/2 = (1/2,..., 1/2) and (α 1,..., α 6 ) = Case 1 : 1/6 = (1/6, 1/6, 1/6, 1/6, 1/6, 1/6 + 1, 1/6 + 1). In these cases, the circuit matrix R ijk around the divisor X ijk is a reflection of order 2 with respect to the hermitian matrix H; the vector b ijk can be expressed in terms of a ijk and H as b ijk = 2ā ijk H/(a ijk, a ijk ) H, and so each reflection is expressed by a row 6-vector a = a ijk as R ijk = I 6 2 t aāh/(a, a) H,.

5 MONODROMY OF HYPERGEOMETRIC SYSTEM 5 where (a, a ) H := āh t a. From now on, everything related to the case 0 has 0 on ones top-right, and put nothing for the case 1. For example, while c 0 j = 1, d 0 j = 2, d 0 ij = 0,..., c j = ω, d j = ω 2, d ij = ω 2 1,... (ω 2 + ω + 1 = 0), and for the case 0, the hermitian matrix is H 0, the roots are a 0 ijk, and the reflections are Rijk 0, while for the case 1, they are H, a ijk and R ijk. Lemma 2 implies Fact 1. For vectors a ijk and a 0 ijk, we have a ijk H a lmn if and only if a 0 ijk H 0 a 0 lmn. Fact 2. ([MSY]) The reflections Rijk 0 generate the principal congruence subgroup Γ(2) with respect to H 0. Fact 3. ([MSTY2]) The reflections R ijk generate the finite complex reflection group ST34 (Shephard-Todd registration number 34, order = ). These groups Γ(2) and ST34 will be studied in the next section. 6. Groups related to ST 34 and Γ(2) 6.1. Arithmetic groups. The invariant form H 0 is an integral symmetric matrix unimodularly equivalent to ( ) 0 1 U U ( I 2 ), U =, 1 0 where I k denotes the unit matrix of degree k. The symmetric domain H is defined to be a component of We set {z C 6 t zh 0 z = 0, t zh 0 z > 0} P 5. O H 0(Z) = {g GL 6 (Z) t gh 0 g = H 0 }, Γ = {g O H 0(Z) g keeps each of two connected components}, Γ(2) = {g Γ g I 6 mod 2}, Γ(3) = {g Γ g I 6 mod 3}, Γ(6) = Γ(2) Γ(3) = {g Γ g I 6 mod 6}. These groups act properly discontinuously on H. Note that I 6 Γ(2), I 6 / Γ(3), Γ(6). It is shown in [MSY] that the group Γ(2) is generated by the reflections R 0 a = I 6 2a t ah 0 /(a, a) H 0 with respect to the roots a of norm N(a) := (a, a) H 0 = 1 and that Γ is generated by reflections with respect to the roots of norm 1 and 2. Now we define the subgroup Γ(1) of Γ generated by reflections with respect to the roots of norm 1 and the products of two reflections with respect to the roots of norm 2. Note that [Γ, Γ(1)] = 2, Γ(2) Γ(1).

6 6 KEIJI MATSUMOTO, TAKESHI SASAKI, AND MASAAKI YOSHIDA 6.2. Finite groups ([Atlas]). The invariant hermitian form H is negative definite. It is shown in [MSTY2] that the twenty reflections R ijk generates a unitary reflection group often called ST34. It is a reflection group in GL 6 (Z[ω]) with structure where ST 34 = 6.G.2, 6 stands for the cyclic group of order 6, G stands for the simple group P SU 4 (3), 2 stands for the cyclic group of order 2, 6 is a normal subgroup of ST 34 with ST 34/(6) G.2 and 6.G is a normal subgroup of ST 34 with ST 34/(6.G) 2. Note that 6 corresponds to the group ωi 6 generated by the scalar matrix ωi 6, and 2 corresponds to det = ±1, i.e., S(ST 34)/ ω is isomorphic to the simple group G, where S(ST 34) denotes the subgroup of ST 34 with det(g) = 1. We set GO 6 (3) = {g GL 6(F 3 ) t ghg = H}. It is known that there exist two kinds of non-degenerate quadratic forms on (F 3 ) 6 with Witt defect 0 and 1. Our H gives the form with Witt defect 1. It is shown in [Atlas] that this group has the structure GO 6 (3) = 2.G.(22 ). Note that the center of GO 6 (3) is {±I 6} and (2 2 ) corresponds to the characters det(g) and # 2 (g), where # 2 (g) means the spinor norm which is the number of reflections with N(v j ) = 2 modulo 2 when g is expressed as a product of reflections R 0 v j with N(v j ) = 1, 2. We set GΩ 6 (3) = {g GO 6 (3) # 2(g) = 0}. Since I 6 Γ(2), we have # 2 ( I 6 ) = 0. Note that the kernel of the natural map is Γ(3). p : Γ GO 6 (3) 7. Relation between the two monodromy groups Proposition 1. The correspondence R ijk R 0 ijk induces a homomorphism of ST 34/Z onto Γ(2)/N, where Z is the group generated by ωi 6 (index 2 subgroup of the center ci 6 of ST34), and N is a normal subgroup of Γ(2) included in Γ(6). Proof. We first show that we can choose a set of generators of ST34 as GenRef := {R 346, R 245, R 124, R 123, R 126, R 156 }. Set and a 1 = a 346, a 2 = a 245, a 3 = a 124, a 4 = a 123, a 5 = a 126, a 6 = a 156 R 1 = R 346, R 2 = R 245, R 3 = R 124, R 4 = R 123, R 5 = R 126, R 6 = R 156.

7 MONODROMY OF HYPERGEOMETRIC SYSTEM 7 Note that the inner products of the six roots are given as 2 c c c 0 ((a i, a j ) H ) i,j=1,...,6 = c c c The six reflections are related as R 1 R 2 R 3 R 5 R 6 \ 3 / This diagram reads: If two reflections R, R GenRef are joined by an edge, then (RR ) 3 = I, otherwise they commute. The node with label 3 means the following: and {(a 3, a 4 ) H (a 4, a 5 ) H (a 5, a 3 ) H } 2 = c 2 R 4 (R 3 R 4 R 5 ) 2 (R 3 R 5 R 4 ) 2 = I. (= third root of unity), The structure theorem for ST34 established in [Shephard] asserts that the six reflections with the above relations form a set of generating reflections. Moreover it is shown that a generator of the center of ST34 is given as (R 1 R 2 R 3 R 4 R 5 R 6 ) 7 = ci. We next show the corresponding relations for the reflections R 0 1 = R 0 346, R 0 2 = R 0 245, R 0 3 = R 0 124, R 0 4 = R 0 123, R 0 5 = R 0 126, R 0 6 = R hold modulo 6: For R a, R b GenRef, if (R a R b ) 2 = I then (R 0 ar 0 b )2 = I. (Note that we do not need modulo 6.) For R a, R b GenRef, if (R a R b ) 3 = I then (R 0 ar 0 b )3 I mod 6. For the node with label 3, using the same notational convention as above, For the center, we have (R 0 3R 0 4R 0 5) 2 (R 0 3R 0 5R 0 4) 2 I mod 6. (R 0 1R 0 2R 0 3R 0 4R 0 5R 0 6) 7 I mod 6. All the above relations can be shown by computation. Theorem 1. (1) N = Γ(6) and (2) Γ(1) = Γ(2), Γ(3) and ST 34/ ωi 6 Γ(2)/Γ(6) Γ(2), Γ(3) /Γ(3). Γ/Γ(3) GO 6 (3), Γ(1)/Γ(3) GΩ 6 (3). Proof. (1) Orders of ST 34/ ωi 6 and GΩ6 (3) are equal to 4 G. Consider the following maps ST 34 ϕ Γ(2)/N f 1 Γ(2)/Γ(6) f 2 Γ(2), Γ(3) /Γ(3) f 3 GΩ6 (3), where f 1, f 2 are naturally defined and f 3 is given by the natural projection p : Γ(2), Γ(3) GΩ 6 (3). Note that f 1 is surjective and its kernel is Γ(6)/N, f 2 is bijective, and that f 3 is injective.

8 8 KEIJI MATSUMOTO, TAKESHI SASAKI, AND MASAAKI YOSHIDA Put f = f 3 f 2 f 1, and consider its kernel M. Since M is normal in Γ(2)/N ST 34/ ω, there are few possibilities. Since the image of f has enough many elements, M does not contain G. We can easily see that I 6 is mapped to I 6 by f. By comparing the orders of Γ(2)/N and GΩ 6 (3), we conclude M = I 6 and f is bijective. Thus we conclude that f 3 is surjective, f 1 is injective and N = Γ(6). (2) It is clear that Γ(2), Γ(3) Γ(1). By the definitions of Γ(1) and GΩ6 (3), we can regard Γ(1)/Γ(3) as a subgroup of GΩ6 (3) by the natural projection p. Since f 3 is surjective, p( Γ(2), Γ(3) ) Γ(2), Γ(3) /Γ(3) GΩ6 (3) p(γ(1)). Thus we have Γ(2), Γ(3) Γ(1). 8. Concluding remarks 8.1. Geometric interpretation. Since the domain H is simply connected, the Schwarz map s(1/2) : X H can be thought of the universal branched covering branching along X ijk with index 2. The Schwarz map s(1/6) also branches along X ijk with index 2. Thus, if M( P 5 ) denotes the image of this Schwarz map, the composed map s(1/6) s(1/2) 1 : H M is single-valued. morphism Moreover, the theorem above implies that this map induces a H/Γ(6) M An elliptic analogue. Recall the original hypergeometric differential equation E(a, b, c) : x(1 x)u + {c (a + b + 1)x}u abu = 0, and the Schwarz map s(a, b, c) : C {0, 1} x u(x) : v(x) P 1, where u and v are linearly independent solutions of E(a, b, c). It is classically well known that the projective monodromy group of E(1/2, 1/2, 1) is conjugate to the elliptic modular group Γ 1 (2), where Γ 1 (k) = {g SL 2 (Z) g I 2 mod k}/center, which is a free group, and acts properly discontinuously and freely on the upper half-plane H 1 = {τ C Iτ > 0}, and the Schwarz map s(1/2, 1/2, 1) gives the developing map of the universal covering H 1 C {0, 1} inducing the isomorphism C {0, 1} = H 1 /Γ 1 (2). On the other hand, the projective monodromy group of E(1/6, 1/6, 1/3) is the tetrahedral group. Note that we have isomorphisms Γ 1 (2)/Γ 1 (6) = Γ 1 (1)/Γ 1 (3) = tetrahedral group. Thus our main theorem can be thought of a generalization of this famous fact. Furthermore, this is not only an analogue: if we restrict the equations E(3, 6; 1/2) and E(3, 6; 1/6) to the singular strata X ijk, X ijk X lmn,..., we will end up with

9 MONODROMY OF HYPERGEOMETRIC SYSTEM 9 a 1-dim stratum, on which the monodromy groups of the two restricted equations (to both of which the Clausen formula 3F 2 (2a, a + b, 2b; a + b + 1/2, 2a + 2b; x) = F (a, b; a + b + 1/2; x) 2 for the hypergeometric functions is applicable) are related as the above elliptic cases. Acknowledgment. The authors are grateful to Professor Eiichi Bannai, who kindly helped them work with finite groups, and to the referee for careful reading and comments. References [Atlas] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, The ATLAS of Finite Groups, Oxford University Press, [MSY] K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a 4- parameter family of K3 surfaces and the Aomoto-Gel fand hypergeometric function of type (3,6), Internat. J. of Math., 3 (1992), [MSTY1] K. Matsumoto, T. Sasaki, N. Takayama and M. Yoshida, Monodromy of the hypergeometric differential equation of type (3, 6) I, Duke Math. J., 71 (1993), [MSTY2] K. Matsumoto, T. Sasaki, N. Takayama and M. Yoshida, Monodromy of the hypergeometric differential equation of type (3, 6) II, Ann. Scuola Norm. Sup. Pisa, Serie IV., 20 (1993), [Shephard] G.C. Shephard, Unitary groups generated by reflections, Canad. J. Math., 5(1953), [ST] G.C. Shephard and J.A. Todd, Finite unitary reflection groups, Canad. J. Math., 6(1954), [SY] T. Sasaki and M. Yoshida, Schwarz map of E(3, 6) for the cubic parameters (4/3, 1/3, 1/3, 1/3, 1/3, 1/3), in preparation. [Yo] M. Yoshida, Hypergeometric Functions, My Love, Aspects of mathematics E32, Vieweg Verlag, Wiesbaden, (Matsumoto) Department of Mathematics, Hokkaido University, Sapporo , Japan address: matsu@math.sci.hokudai.ac.jp (Sasaki) Department of Mathematics, Kobe University, Kobe , Japan address: sasaki@math.kobe-u.ac.jp (Yoshida) Faculty of Mathematics, Kyushu University, Fukuoka , Japan address: myoshida@math.kyushu-u.ac.jp