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    Журнал: Наукові відомості Бєлгородського державного університету. Серія: Математика. фізика
    Наукова стаття на тему 'Reflection groups of rank three and systems of uniformization equations'

    Текст наукової роботи на тему «Reflection groups of rank three and systems of uniformization equations»

    ?y ^ K 517.55

    REFLECTION GROUPS OF RANK THREE AND SYSTEMS OF UNIFORMIZATION EQUATIONS Jiro Sekiguchi

    Department of Mathematics, Faculty of Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo, 184-8588, Japan, e-mail: Этот адрес электронной почты защищён от спам-ботов. У вас должен быть включен JavaScript для просмотра.

    Abstract. In this note, I survey recent progress on systems of uniformization equations along Saito free divisors. In particular, I show concrete forms of such systems along Saito free divisors which are defined as the zero sets of discriminants of complex reflection groups of rank three. A part of the results in this note is a joint work with M. Kato (Univ. Ryukyus)

    Keywords: uniformization equations, discriminants of complex reflection groups, Saito free divisors.

    Introduction

    The notion of Saito free divisors was introduced by K. Saito (cf. [10]). He also formulated and stressed the importance of systems of uniformization equations along such divisors (cf. [9]). In this note, I explain recent progress on systems of uniformization equations along Saito free divisors defined as the zero sets of the discriminants of complex reflection groups of rank three. The hypersurface defined by the discriminant of a real reflection group is a typical example of Saito free divisors. It is known that the discriminant of a complex reflection group of rank three is also a Saito free divisor (cf. [8], [5]). But I do not know whether it is true for the case of arbitrary complex reflection groups. My interests on this subject are to construct (1) Saito free divisors, (2) systems of uniformization equations, and (3) their solutions in a concrete manner. Restricting to the case of three dimensional affine space, I obtained some results on (1), (2). But it is difficult to attack (3) compared with (1), (2). The purpose of this note is to report my results on (1), (2) for the discriminants of irreducible complex reflection groups of rank three. A part of the results of the last three sections are obtained by a joint research with M. Kato (Univ. Ryukyus).

    1 Definition of Saito free divisors

    Let F (x) = F (x1, x2, ..., xn) be a reduced polynomial. Then D = {x G Cn; F (x) = 0} is a (weighted homogeneous) Saito free divisor if (C1) + (C2) hold.

    (C1) There is a vector field

    n

    E = ^ 2 mixidXi

    i = 1

    such that EF = dF, where m1, m2, ..., mn, d are positive integers with 0 < m1 < m2 < | | | <

    ffln.

    (C2) There are vector fields

    Vi = ^ 2 aij (x) dxj (i = 1, 2, ..., n) j = i

    such that

    (I) each aij (x) is a polynomial of x1, x2, ..., xn,

    (Ii) det (aij (x)) = cF (x) for a non-zero constant c,

    (Iii) V1 = E, ViF (x) = ci (x) F (x) for polynomials ci (x),

    (Iv) [E, Vi] = kiVi for some constants ki,

    (V) V% (j = 1, 2, ..., n) form a Lie algebra over R = C [x1, x2, ..., xn]

    We now give examples of Saito free divisors.

    Let

    f (t) = tn + x2tn 2 + x ^ t ™ 3 + | | | + xn-it + xn

    be a polynomial of nth degree and let A (x2, x3, ..., xn) be the discriminant of f (t). Then A = 0 is a Saito free divisor in Cn-1.

    More generally, the zero locus of the discriminant of an irreducible real reflection group is a Saito free divisor.

    Basic reference of this section is [10].

    2 Irreducible complex reflection groups of rank three.

    In this section, we collect some results on irreducible complex reflection groups of rank three. A basic reference on complex reflection groups is Shephard-Todd [16] (see also [8]).

    Reflection groups treated in this section are real reflection groups of types A3, B3, H3 and complex reflection groups of No.24, No.25, No.26, No.27 in the sense of [16]. The real reflection group of type H3 is same as the group No. 23 in [16].

    Let G be one of the seven groups and let P1, P2, P3 algebraically independent basic G-invariant polynomials and put kj = deg? (Pj). We may assume that k1 < k2 < k3. Let r be the greatest common divisor of k1, k2, k3 and put kj = kj / r (j = 1, 2, 3). For the later convenience, we write x1, x2, x3 for P1, P2, P3. Let §g (x1, x2, x3) be the discriminant of G expressed as a polynomial of x1, x2, x3.

    In the cases A3, B3, H3, taking G-invariants x1, x2, x3 suitably, FW (A3) (x1, x2, x3),

    FW (B3) (x1, x2, x3), FW (H3) (x1, x2, x3) are discriminants for G up to a constant factor, respectively, where Fa1, FB1, FH1 are the polynomials given in Theorem of [13 ].

    group order k \ - k>. k3 degree (k [, k'2, 4)

    ^ 3 W (A3) 24 2,3,4 12 (2,3,4)

    B3 W (B3) 48 2,4,6 18 (1,2,3)

    H3 W (H3) 120 2,6,10 30 (1,3,5)

    No.24 G33 6 336 4,6,14 42 (2,3,7)

    No.25 ^ 648 648 6,9,12 36 (2,3,4)

    No.26 Gl296 1296 6,12,18 54 (1,2,3)

    No.27 G2I6O 2160 6,12,30 90 (1,2,5)

    The concrete forms of discriminants of W (A3), W (B3), W (H3) are as follows: Type A3: 16x ^ x3 - 4x3x2 - 128x '^ x22 + 144x1x ^ x3 - 27x2 + 256x3.

    Type B3: x3 (x21x'2 - 4x3 - 4x1x3 + 18x1x2x3 - 27x3).

    Type H3: -50x | + (4x \ - ') 0 ./- f./-2) ./ • "| + (4x | x2 + 60./-?.r] + 225 ^ 1 ^ 2) ^ 3 - ^^ 2

    1 1 _ 1 Oo ^^ o ^ 3 _ A ry2

    115x1x2 10x1x2 4x1x2 '

    The discriminants for the groups No.25, No.26 are same as those for W (A3), W (B3) respectively by taking the basic invariants suitably. On the other hand, those for the groups No.24, No.27 will be given in §5, §.6.

    3 Systems of uniformization equations along Saito free divisors.

    Let F (x) = 0 be a Saito free divisor and let Vj (j = 1, 2, ..., n) be basic vector fields logarithmic along F = 0. Let u = u (x1, x2, ... , xn) be an unknown function. Assume that V1u = su for a constant s (= 0) Put u = f (u, V2u, ..., Vnu) and consider the system of differential equations (UE) Vju = Aj (x) u (j = 1, 2, ..., n) where Aj (x) (j = 1, 2, ..., n) are nxn matrices whose entries are polynomials of x.

    If (UE) is integrable, it is called a system of uniformization equations with respect to the Saito free divisor F = 0.

    The system (UE) is written by

    V lu = su

    n

    ViVju = Y, hj (x) Vku (Vi, j) k = i

    where hkj (x) are polynomials of x. There are n number of fundamental solutions of (UEa). Let

    uj (x) (j = 1, 2, ..., n) be fundamental solutions outside the divisor F = 0. Then

    <^ (X) = (ui (x), u2 (x),..., Un (x))

    defines a map of Cn - {F = 0} to Cn. The following two problems are fundamental in the study on systems of uniformization equations.

    PROBLEM 1: Construct fundamental solutions uj (x) (j = 1, 2, ..., n) of (UEa). PROBLEM 2: Construct the inverse of <^ (X) in a concrete manner.

    These two problems are solved in the case W (A3) for a special but interesting system of

    uniformization equations by K. Saito. For the details of the results, see [9].

    4 The discriminant of the Coxeter group W (H3) of type H3.

    A part of the argument in the case of W (A3) in [9] is applicable to the case of the Saito free divisor defined by the zero locus of the discriminant of the Coxeter group of type H3. In this section, I will explain the results on this case. For the details of results in this section, see [15]. The discriminant of the polynomial P (t) defined by

    P (t) = t6 + yi t5 + y2t3 + y3t + - y \ - -yry3 (4.1)

    is A2 up to a constant factor, where A = 125y3y | + 864y5 - 1250у4у2уз - 9000Шу3уз + 3125y5y2 + 25000 ^ 2 + 50000y3. (4.2)

    Remark 4.1 The equation P (t) = 0 is essentially same as "Die allgemeine Jacobi'sche Gleichung sechsten Grades" (see p.223 in Klein's book [7]).

    (UEa)

    The polynomial A is regarded as the discriminant of the group W (H3). In fact, the substitution of the variables (y1, y2, y3) with (x1, x2, x3) defined by the relations

    y1 = -4x1

    y2 = 10x1 - 25x2 (4.3)

    y3 = -4xf + 50x1x2 - 50x3

    implies that A coincides with the determinant of the matrix M up to a constant factor, where M is defined by

    x1 3x2 5x3

    M = | 3x2 2x3 + 2x '^ x2 7x1x2 + 2xjx2 | (4.4)

    ~ X \ X11 • 2./-|'./-2 /? 1 • 1;

    and det M is the discriminant of W (H3) (cf. [17]). In the sequel, we always regard P (t) as a polynomial of t and x. The hypersurface defined as the zero set of the polynomial f0 = det M is an example of Saito free divisors. To show this, we define vector fields V0, V1, V2 by

    Vo \ (dXl

    V1 1 = M 1 dx2

    V2 V dx3

    Then we have

    and

    Remark 4.2 We note that

    [Vo, Vi] = 2V1, [Vq, V2] = 4V2,

    [Vi, V2] = (4x ^ x2 + 2x ^) Vo + 4X1X2V1

    Vofo = 15fo,

    Vifo = 2x1fo,

    V2fo = 2xi (2x1 + 5x2) fo

    1 25

    P (-x 1) = (4.5)

    This implies that (-Xi, 5 \ / 5/2 • x2) is a point on the hyperelliptic curve s2 = P (t) on (s, t) plane.

    Consider the system of differential equations

    u \ in

    Vi | V1U I = Bi + 1 + 1 V1U | (I = 0,1,2) (4.6)

    V2U V V2U

    The system (4.6) is a system of uniformization equations along the Saito free divisor f0 (x) = 0. Here Bj are defined as follows:

    f So 0 0

    В1 = 0 2 + So 0

    0 0 4 +

    під

    i

    900

    вз

    1

    900

    1

    450

    (8 + 70s1 - 100sf + 8s0

    2_ J + 35sis0 + 2sl) xl 225- M + (_ 180 + 825s 1 - 750s?

    -90so + 75s1s0) x2 '(-128 + 80si + 100s2 - 128s0

    + 40siso - 32s0) x1 + 10 (-32 - 400s1 + 550s2 + 328s0 -20siso - 8s0) x1x2 + (-4500si + 5625s1 + 1800s0) x2 + (2400 - 3000si + 1200s0) xiX3

    (-128 + 80s1 + 100s2 - 128s0 + 40sis0 - 32s0) xi + 10 (-32 - 400si + 550s2 - 32s0 -20sis0 - 8s0) xix2 + si (-4500 + 5625si) x2 + 100 (24 - 30si + 12s0) xiX3 (-128 + 80si + 100s2 - 128s0 + 40sis0 - 32s0) x ^

    + (80 - 500s1 + 500s2 - 280s0 + 200sis0 - 160s0) x5x2 +25 (-104 - 130si + 325s2 + 40s0 + 25sis0 - 8s2) x2 ^ 2 + 100 (12 - 15si + 24s0) x ^ x3 +50 (60 - 75si + 30s0) x2 ^ 3

    Yg (8 + 5s i + 4sq) x \

    si

    15

    x1

    {+10 (8 + 5 ^ ^ (4 "5Sl + 28о) Ж1

    Tk

    hxi

    (8 + 5si + 4s0) x3 + 10 (2 + 5si + s0) x2

    je (4 - 5si + 2s0) xf

    > T (4 + 5s 1) x2 (4x \ + 5x2) TF

    ((16 - 5si + 8s0) x3

    \ + (40 - 50si + 20s0) x2

    \

    Remark 4.3 In the case s0

    i "

    01 Si

    1, the monodromy group of the system Vjn = Bj + 1n (j =

    0,1, 2) coincides with W (H3). This case is treated in Haraoka-Kato [6].

    We consider the system VjU = Bj + 1u (j = 0,1, 2) with s0 = -2, ^ = 0. Then we obtain

    V0v = - 2v

    ViViV = 0

    V2V1v = 0

    V2V2v = -4xf (3x2 + 2x1x3) v + x2 (4xf + 5x2) V1v

    (4.7)

    Theorem 4.4 (cf. [15]) The function v (x) defined by

    v (x)

    P (t) -1 / 2dt

    0

    1

    0

    1

    0

    0

    1

    is a solution of (4.7).

    The proof of this theorem is given by an argument similar to the case of type A3.

    If u (x) is a solution of (4.7) such that V1u = 0, then u is a solution of

    {V0u = -2u

    V1u = 0 (4.8)

    {V22 + 4x1 (3x2 + 2x1x3)} u = 0

    Taking two paths C1, C2 appropriately and define

    Wj (x) = I ^ i (t) dt (j = 1, 2)

    JCj

    Then each Wj (x) is also a solution of (4.7) and in this manner we can construct solutions of (4.8). In this case it is not clear whether solutions of (4.8) are expressed by special functions or not. Moreover PROBLEM 2 (the construction of the inverse mapping) is still open.

    5 The discriminant of the group G336, Shephard-Todd notation No.24.

    In this case, we begin with defining the polynomial

    P (t) = t '-? (Ci - 1) ^ 215 -? (Ci - l) x3t4 - 7 (ci + 4) ^ 213 - 14 (ci + 2) x2x3t2 + | {(3ci - 7) 4 - {c \ + 5); r3} i + | (7ci - 131) ^ 2 ^ 3 x7

    (Cl = -7)

    The discriminant of P (t) is f0 up to a constant factor, where

    f0 = 2048x "x3 - 22016x2x3 + 60032x:] x5 - 1728x1 + 256x7x7 - 1088x4xijx7 -1008x2x ^ x7 + 88x2x3x7 - x7

    and f0 is the discriminant of the complex reflection group G336. (The polynomial f0 is same as the one shown in p.262 of the paper of A. Adler in the book "The Eightfold Way" by x2 ^ f, x3 V, x7 ^ C. The polynomial P (t) is given in p.406 of GMA of F. Klein, Band II.) Define vector fields V0, V1, V2 by

    l (Vo, Vi, V2) = Ml (dX2, dX3, dX7)

    Then V0, V1, V2 form the generators of logarithmic vector fields along f0 = 0. Here

    (2x2 3x3 7x7 \

    x'l ~~ hx "-1 ^ 2 (28 ^ 2 ^ 3 _ 128rr | + 3x2x7) I

    7x7 -56x2 (2x2 - 13x2) 28 (32x6 - 40x2x3 - 84x ^ + 59x2x3x7) J

    Put

    A0 = {{s0,0,0}, {0, s0 + 4,0}, {0,0, s0 + 5}};

    A1 = {{0,1,0}, {1/162 * x2 * (4 * (- 1 + c4-s0) * (8 + c4 + 2 * s0) * x2 ~ 3-3 * (24 + 43 * c4 + 5 * c4 ~ 2 + 24 * s0 +

    19 * c4 * s0) * x3 ~ 2), 1/9 * (- 10 + c4-4 * s0) * x2 ~ 2, c4 * x3 / 504},

    {-7 / 54 * (8 * (- 152 + 37 * c4 + 7 * c4 ~ 2-172 * s0-14 * c4 * s0-38 * s0 ~ 2) * x2 ~ 3 * x3-18 * (8 * c4 + c4 ~ 2 + 76 * s0) * x3 ~ 3 + 3 * (8 + c4 + 38 * s0) * x2 * x7),

    -14/3 * (- 20 + 5 * c4-38 * s0) * x2 * x3, -1 / 9 * (8 + c4 + 2 * s0) * x2 ~ 2}> ;

    A2 = {{0,0,1},

    {-7 / 54 * (8 * (- 152 + 37 * c4 + 7 * c4 ~ 2-190 * s0-14 * c4 * s0-38 * s0 ~ 2) * x2 ~ 3 * x3-18 * c4 * (8 + c4) * x3 ~ 3 + 3 * (8 + c4 + 8 * s0) * x2 * x7), - 14/3 * (- 152 + 5 * c4-38 * s0) * x2 * x3,

    -1/9 * (2 + c4 + 2 * s0) * x2 ~ 2>,

    {98/9 * (48 * (- 24 + 5 * c4 + c4 ~ 2-36 * s0-c4 * s0) * x2 ~ 5 + 4 * (- 440 + 97 * c4 + 19 * c4 ~ 2-658 * s0 -89 * c4 * s0-722 * s0 ~ 2) * x2 ~ 2 * x3 ~ 2 + 3 * (8 + c4 + 38 * s0) * x3 * x7),

    -1176 * (- 2 + c4) * (2 * x2 ~ 3-x3 ~ 2), 14/3 * (190 + 5 * c4 + 76 * s0) * x2 * x3}>;

    There is a system of differential equation of rank three defined by

    Vj

    u

    Viu

    V2u

    Aj

    u

    Viu

    V2u

    (J = 0,1, 2)

    This system has two parameters s0, c4.

    Substituting s0 = - 1, c4 = 0 in Aj, we obtain Aj0);

    A (°)

    A0

    -10 0 0 3 0

    0 0 4

    Ai

    (0)

    0

    0

    ___2 2

    3 2

    i \ г: \. гл - 76; r | + 5X2X7) -84 ^ 2 ^ 3

    0

    0

    _2r2 3 2

    A

    (0)

    The system

    0

    0

    0

    532x2x3

    1

    0

    196 (32x2 - 112x2x2 - 5x3x7) 2352 (2x2 - x \) 532x2x3

    u

    Vj I Viu V2u

    Aj

    (0)

    u

    Viu

    V2u

    (J = 0,1, 2)

    has a quotient which is defined by V1u = 0. Assuming V1u = 0, the system for out to be

    u

    V2u

    turns

    V0

    Vi

    V2

    u

    V2u

    u

    V2u

    u

    V2u

    -1 0 u

    0 4) \ V2u

    0

    0

    ^ (8х'2Хз - 76x1 + 5 ^ 2 ^ t) "1 ^ 2

    01 196 (32x5 - 112x ^ 2x \ - 5x3x7) 532x2x3

    u

    V2u

    u

    V2u

    (5.1)

    We now study the restriction of the system (5.1) to the hyperplane x2 = 0. Then we obtain an ordinary differential equation

    dl7 +

    18x7

    7 (1728x7 + x7)

    dX7 +

    10x7

    3 \ ^ x7

    49 (1728x7 + x?)

    u = 0

    1

    2

    One of its solutions is

    >21'21'3 '1728x3,

    Similarly as the restriction to x3 = 0 of (5.1), we obtain an ordinary differential equation

    2 2564+ 11 ^ | 3 .

    • T7 7 ^ (2564-4) 'T7 49 (-256 ^ 2 + x2)'

    One of its solutions is

    -i / o / 1 3 3 x-

    : I: 2 F 77'77; 7;

    14 '14 7 256x2

    Remark 5.1 We note that, in the case c4 = -9, s0 = 1/2, the system of differential equations has a monodromy group isomorphic to G336. This case is treated in [6].

    6 The discriminant of the group G2160, Shephard-Todd notation No.27.

    Consider the polynomial

    P (t) = t6 + y1t5 + y2t4 + Pst3 + Vit2 + V5t + Ve-Substitute Vj (j = 1, 2, ..., 6) by xj (j = 1, 2, ..., 6); y1 = x1

    y2 = (5/16) * (9 + sr) * x2, y3 = (5/64) * (11 + 3 * sr) * x1 * x2, y4 = (5/512) * (37 + 45 * sr ) * x2 ~ 2,

    y5 = (61 + 5 * sr) * (- 64 * x1 ~ 3 * x2 + 373 * x1 * x2 ~ 2 + 15 * sr * x1 * x2 ~ 2 + 2 * x3)) / 12288, y6 = (- 279 + 145 * sr) * (- 512 * x1 ~ 4 * x2 + 2864 * x1 ~ 2 * x2 ~ 2 +

    1425 * x2 ~ 3 + 135 * sr * x2 ~ 3 + 16 * x1 * x3) / 3538944,

    where sr2 = -15. Then / q the discriminant of the polynomial P (t), where

    f0 = 65536x11x2 - 1765376x? x3 + 17406016x [xi - 73887360x5x5 + 107371008x3x2 + 34338816x1x2 - 4096x1x2x3 + 96640x6x2x3 - 707952x4x3x3 + 1622592x '^ x'4x3 + 186624x5x3 + 64x5x2 - 1584x1x2x2 + 7128x1x ^ x3 + 9x3

    up to a constant factor.

    Remark 6.1 By direct computation, we find that

    (3 - 5sr) ^ \ 5 (-45 + 11sr) (39 - sr) _2

    P 1 = --- ЇІ52-- 12 - "Tie" *

    This means that

    / 5 (-45 + llsr) \

    V ЇЇ52 J

    (3 - 5sr) / 5 (-45 + 11sr) \ 1/3 f (39 - sr)

    * I '------ 7777 ------- 1 * 2 -------- 777 ---- * 1

    72 V тисячі сто п'ятьдесят два J I 216

    is a point on the trielliptic curve s3 = P (t) on (s, t) plane.

    2

    7

    i

    The polynomial f0 is regarded as the discriminant of the complex reflection group No.27. In particular, f0 is obtained as the determinant of the matrix

    M

    (X1 2x2 5x3 \

    x2 ^ 2 (144 ^! ^ | - Xs) 4g (640rr®x2 - 938844 + 3660044

    -19872x ^ - 28x3x3 + 307xix2x3) (65920; rf ^ 2 - 88709244 + 28861 У + 16416x2 + 139x1x3) + 367632x2 - 2692x1x3 + 20533x1x2x3) J

    x3 4 (_ 19204 * 2 + 872444 (659204 ^ 2 - 88709244 + 288612044

    It can be shown that f0 coincides with the polynomial gk13 in my notation by a weight preserving coordinate change in the notation of my note. We define vector fields V0, V1, V2 by

    'V0 \ j dXl

    Vi I = M I dx2

    V2) V dx3

    Then Vo, V1, V2 form generators of the logarithmic vector fields along the set f0 = 0 in the (x1, x2, x3) -space. By direct computation, we have

    [Vi, V2] = - (32004 ^ 2 - 1641244 - 18056; ri4 - 804 ^ 3 _ 307 ^ 2 ^ 3) Vo 540

    8 1

    -----(4744 - 41024 ^ 2 + 72094) Vi ------- # 1 (64 _ 73 ^) V2

    135 54

    We consider the system of differential equations

    u \ in

    Vj | Viu I = Aj I Viu | (J = 0,1, 2)

    V2U) v V2U

    where A0, A1, A2 are matrices of rank three defined as follows. A0 = {{s0,0,0}, {0,3 + s0,0}, {0,0,4 + s0}};

    A1 = {{0,1,0}, {1/2099520 * (320 * (1 + 1728 * h1-4 * s0) * (3 + 864 * h1 + s0) * x1 ~ 6-120 * (47+ 58752 * h1-280 * s0) * (3 + 864 * h1 + s0) * x1 ~ 4 * x2 + 36 * (2115+ 2967840 * h1 + 679311360 * h1 ~ 2-15870 * s0-2515104 * h1 * s0- 6125 * s0 ~ 2) * x1 ~ 2 * x2 ~ 2-3888 * (15 + 30240 * h1 + 7464960 * h1 ~ 2-175 * s0 +

    28512 * h1 * s0) * x2 ~ 3 + 45 * (3 + 864 * h1-35 * s0) * x1 * x3), - 1/810 * x1 * ((55+ 42768 * h1 + 40 * s0) * x1 ~ 2-3 * (255 + 83376 * h1 + 175 * s0) * x2), - 1/6 * h1 * (x1 ~ 2-6 * x2)}, {1/656100 * (- 25280 * (1 + тисячі сімсот двадцять вісім * h1-4 * s0) * (3 + 864 * h1 + s0) * x1 ~ 7 + 120 * (9879 + 16458336 * h1 + 3920596992 * h1 ~ 2-42907 * s0-16232832 * h1 * s0-17560 * s0 ~ 2) * x1 ~ 5 * x2-36 * (123660 + 210 094 560 * h1 + 50250378240 * h1 ~ 2-721860 * s0-210720096 * h1 * s0-308135 * s0 ~ 2) * x1 ~ 3 * x2 ~ 2 + 1944 * (345 + 3546720 * h1 + 992839680 * h1 ~ 2-14815 * s0 + 6258816 * h1 * s0-5775 * s0 ~ 2) * x1 * x2 ~ 3-45 * (147+ 42336 * h1-1625 * s0) * x1 ~ 2 * x3-3645 * (5 + 1440 * h1 + 44 * s0) * x2 * x3), (4 * (316 * (- 15 + 25704 * h1 + 10 * s0) * x1 ~ 4 -3 * (- 16645 + 15303168 * h1 + 8125 * s0) * x1 ~ 2 * x2-2430 * (56+ 1440 * h1-11 * s0) * x2 ~ 2)) / 2025, (x1 * (2 * (-85 + 42768 * h1-20 * s0) * x1 ~ 2-3 * (- 715 + 166 752 * h1-175 * s0) * x2)) / 1620}};

    A2 = {{0,0,1}, {1/656100 * (- 25280 * (1 + 1728 * h1-4 * s0) * (3 + 864 * h1 + s0) * x1 ~ 7 + 120 * (9879 +

    16458336 * h1 + 3920596992 * h1 ~ 2-75307 * s0-16232832 * h1 * s0-17560 * s0 ~ 2) * x1 ~ 5 * x2-36 * (123660 + 210 094 560 * h1 + 50250378240 * h1 ~ 2-1275765 * s0-210720096 * h1 * s0-308135 * s0 ~ 2) * x1 ~ 3 * x2 ~ 2 + 1944 * (345 + 3546720 * h1 + 992839680 * h1 ~ 2-3530 * s0 + 6258816 * h1 * s0-5775 * s0 ~ 2) * x1 * x2 ~ 3-45 * (147 + 42336 * h1-3785 * s0) * x1 ~ 2 * x3-6075 * (3 + 864 * h1-35 * s0) * x2 * x3), 4 * (632 * (15 + 12852 * h1 + 5 * s0) * x1 ~ 4-3 * (24375+ 15303168 * h1 + 8125 * s0) * x1 ~ 2 * x2-2430 * (- 33 + 1440 * h1- 11 * s0) * x2 ~ 2) / 2025, x1 * (2 * (5 + 42768 * h1-20 * s0) * x1 ~ 2-3 * (15 + 166752 * h1-175 * s0) * x2) / 1620}, {1/820125 * (4 * (1997120 * (1 + 1728 * h1-4 * s0) * (3 + 864 * h1 + s0) * x1 ~ 8-120 * (680901 + 1246968864 * h1 + 302650380288 * h1 ~ 2-3612193 * s0-1421635968 * h1 * s0-1027000 * s0 ~ 2) * x1 ~ 6 * x2 +

    36 * (6514065 + 14746523040 * h1 + 3706696028160 * h1 ~ 2-67397805 * s0-17324851104 * h1 * s0-16957205 * s0 ~ 2) * x1 ~ 4 * x2 ~ 2-972 * (- 235 065 + 392 096 160 * h1 + 132420925440 * h1 ~ 2 -3712385 * s0 + 1286813088 * h1 * s0-1072500 * s0 ~ 2) * x1 ~ 2 * x2 ~ 3 + 1574640 * (5 + 1440 * h1 -11 * s0) * (- 5 + 8640 * h1 + 33 * s0) * x2 ~ 4 + 45 * (4503 + 1296864 * h1-144155 * s0) * x1 ~ 3 * x3 + 3645 * (740 + 213120 * h1 + 5891 * s0) * x1 * x2 * x3 )), - 16 * (49928 * (- 25 + 60048 * h1) * x1 ~ 5-192 * (- 44815 + 84795282 * h1) * x1 ~ 3 * x2-1620 * (4469 + 1982880 * h1) * x1 * x2 ~ 2 + 180225 * x3) / 10125, -8 * (632 * (- 10 + 6426 * h1-5 * s0) * x1 ~ 4-3 * (- 16250 + 7651584 * h1-8125 * s0) * x1 ~ 2 * x2-2430 * (22 + 720 * h1 + 11 * s0) * x2 ~ 2) / 2025}};

    Remark 6.2 A1, A2, A3 contain parameters s0, h1. The determination of A1, A2, A3 was accomplished by Masayuki Noro (Kobe Univ.).

    The case s0 = |, hi = - ^ §4

    In this case the monodromy group of the system of differential equations becomes GWo-This case is treated in [6].

    The case so = - 3, hi = 0

    In this case there is a quotient of the system above. In fact,

    is a quotient of the system Vjn = Ajn (j = 0,1, 2) defined above, where Bj (j = 0,1, 2) are

    matrices of rank two defined below:

    B0 = {{- 3,0}, {0,1}};

    B1 = {{1/162 * x1 * (13 * x1 ~ 2-162 * x2), 0},

    {(-98592 * x1 ~ 7 + 1926304 * x1 ~ 5 * x2-10970316 * x1 ~ 3 * x2 ~ 2 + 17754552 * x1 * x2 ~ 3 -15066 * x1 ~ 2 * x3 + 30861 * x2 * x3) / 43740, (- 1350 * x1 ~ 3 + 15390 * x1 * x2) / 43740}};

    B2 = {{0,1}, {- 1/164025 * (4 * (- 6490640 * x1 ~ 8 + 180 214 176 * x1 ~ 6 * x2-999084132 * x1 ~ 4 * x2 ~ 2 +

    712058040 * x1 ~ 2 * x2 ~ 3 + 1244595456 * x2 ~ 4-2995542 * x1 ~ 3 * x3 + 665577 * x1 * x2 * x3)),

    -4 * (511920 * x1 ~ 4-3948750 * x1 ~ 2 * x2 + 4330260 * x2 ~ 2) / 164025}};

    Bibliography

    1. A.G.Aleksandrov. Milnor numbers of nonisolated Saito singularities. Funct. Anal. Appl., 21 (1987), 1-9.

    2. A. G. Aleksandrov, Nonisolated hypersurfaces singularities. Advances in Soviet Math., 1 (1990), 211-246.

    3. A. G. Aleksandrov. Moduli of logarithmic connections along free divisor. Contemp. Math., 314 (2002), 2-23.

    4. A. G. Aleksandrov, J. Sekiguchi, Free deformations of hypersurface singularities, To appear in Kokyuroku.

    5. D. Bessis, J. Michel. Explicit presentations for exceptional braid groups, Experimental Math., 13 (2004), 257-266.

    6. Y. Haraoka, M. Kato. Systems of partial differential equations with finite monodromy groups, preprint.

    7. F. Klein. Vorlesungen iiber das Ikosaeder, Birkhauser Verlag, Basel-Boston-Berlin, 1993.

    8. P. Orlik, H. Terao. Arrangements of Hyperplanes, Grundlehren der mathematisches Wissenschaften 300, Berlin: Springer-Verlag, 1992.

    9. K. Saito. On the uniformization of complements of discriminant loci. RIMS Kokyuroku 287 (1977), 117-137.

    10. K. Saito. Theory of logarithmic differential forms and logarithmic vector fields. J. Faculty of Sciences, Univ. Tokyo 27 (1980), 265-291.

    11. K. Saito. On a linear structure of the quotient variety by a finite reflection group. Publ. Res. Inst. Math. Sci. Kyoto Univ., 29 (1993), 535-579.

    12. K. Saito. Uniformization of orbifold of a finite reflection group, Frobenius Manifold, Quantam Cohomology and Singularities. A Publication of the Max-Planck-Institute, Mathematics, Bonn, 265-320.

    13. J. Sekiguchi. A classification of weighted homogeneous Saito free divisors, J. Math. Soc. Japan, 61 (2009), 1071-1095.

    14. J. Sekiguchi. Three dimensional Saito free divisors and deformations of singular curves, J. Siberian Federal Univ., Mathematicas and Physics, 1 (2008), 33-41.

    15. J. Sekiguchi. Systems of uniformization equations and hyperelliptic integrals, preprint.

    16. G. C. Shephard, A. J. Todd. Finite reflection groups, Canad. J. Math., 6 (1954), 274-304.

    17. T. Yano, J. Sekiguchi. The microlocal structure of weighted homogeneous polynomials associated with Coxeter systems. I, II, Tokyo J. Math. 2 (1979), 193-219, Tokyo J. Math.

    4 (1981), 1-34.

    ГРУПИ ВІДОБРАЖЕННІ РАНГУ ТРИ І СИСТЕМИ РІВНЯНЬ ДІФОРМАЛІЗАЦІІ Джиро Секигучи

    Токійський університет сільського господарства і технології, Коганом, Токіо, 184-8588, Японія, e-mail: Этот адрес электронной почты защищён от спам-ботов. У вас должен быть включен JavaScript для просмотра.

    Анотація. У роботі наведені результати, що описують дискриминант непріводімих комплексних груп відображення рангу три.

    Ключові слова: Непріводімие рівняння, дискримінант комплексних груп відображення.


    Ключові слова: uniformization equations / discriminants of complex reflection groups / saito free divisors

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