Basic Invariants of the Complex Reﬂection Group
No.34 Constructed by Conway and Sloane
Manabu Oura∗

Jiro Sekiguchi†

Abstract
This paper studies the basic invariants, constructed by Conway and Sloane, of
the complex reﬂection group numbered as 34 in the list of Shephard-Todd [14].

Keywords: complex reﬂection group, invariant theory.
MSC2020: Primary 20F55; Secondary 13A50.

1Introduction

In this paper, we treat the complex reﬂection group of number 34 in the list of Shephard
and Todd [14] (denote this group by ST34 in this paper) which has order 6 5 · 7!. The
degrees of its basic invariants are 6,12,18,24,30,42. If x = (x1 , x2 , · · · , x6 ) is a linear
coordinate of the 6 dimensional representation space of ST34, the basic invariants arepolynomials of x. Conway and Sloane [5] constructed such basic invariants which aredenoted by µj (j = 6, 12, 18, 24, 30, 42). It is hard to write them down as polynomialsof x because of their lengthy. The main purpose of this paper is to write them in areasonable size. We explain our idea to accomplish the purpose brieﬂy. Let G(3, 3, 6)
be the imprimitive complex reﬂection group of rank 6. Since G(3, 3, 6) is a subgroup of
ST34, each of µj (j = 6, 12, 18, 24, 30, 42) is written as a polynomial of the basic invariantsof G(3, 3, 6), say p3 , p6 , p9 , p12 , p15 , s6 . It is easy to write p3 , p6 , p9 , p12 , p15 , s6 down aspolynomials of x. Along this idea, we succeeded to write µj (j = 6, 12, 18, 24, 30, 42)
as polynomials of p3 , p6 , p9 , p12 , p15 , s6 in a reasonable size. Though it follows from thedeﬁnition that each µj is invariant by ST34, we shall give an alternative proof of theinvariance by use of symmetric polynomials of six letters (see Theorem 1).
The discriminant of ST34 is expressed as a polynomial of µj (j = 6, 12, 18, 24, 30, 42).
It is known (cf. [1]) that the discriminant of a complex reﬂection group G is expressed as
the determinant of the Saito matrix of G. In ST34 case, Terao and Enta [15] proposed analgorithm to compute the Saito matrix for the basic invariants µj (j = 6, 12, 18, 24, 30, 42)
and along this line, Bessis and Michel actually computed it explicitly (unpublished). On
the other hand, Kato, Mano and Sekiguchi ([7], [8]) formulated the notion of the ﬂat∗

Institute of Science and Engineering, Kanazawa University, Japan.
Email address: oura@se.kanazawa-u.ac.jp†
Emeritus Professor, Tokyo University of Agriculture and Technology, Japan.
Email address: jinomino@yahoo.co.jp

1structure which is a generalization of Frobenius manifold structure. Applying it to acomplex reﬂection group G, one can construct the Saito matrix for the ﬂat coordinateof G when G satisﬁes some conditions on invariants. In ST34 case, the Saito matrix
corresponding to a potential vector ﬁeld of 6 variables whose weights are the same as
the degrees of the basic invariants of ST34 is constructed in [7] and the determinant of
this Saito matrix is expected to be the discriminant of ST34 if we regard the variables
as appropriate basic invariants. Theorem 2 says that this is actually valid. Theorem 2
itself is the same as the result in [12] but its proof is diﬀerent from that given in [12] (seeRemark 2 in the main text).
In the rest of this paper, we shall treat two topics related with the group ST34. One
is the correspondence between the totality of minimal vectors of the Coxeter-Todd lattice[5] and that of pseudo-reﬂections of ST34. We shall describe a natural 6-1 correspondence
between them. The other is concerned with the restriction of the basic invariants µj (j =6, 12, 18, 24, 30, 42) to the representation space of the group ST33 which is the complex
reﬂection group No.33 in [14]. The basic invariants of ST33 were ﬁrst constructed by
Burkhardt (cf. [9]).
This paper is organized as follows. In section 2, we mention the Coxeter-Todd lattice, its minimal vectors and the basic invariants µj (j = 6, 12, 18, 24, 30, 42) of ST34
constructed by Conway and Sloane. In section 3, we ﬁrst introduce polynomials mj (j =1, 2, 3, 4, 5, 7) written as polynomials of basic invariants p3 , p6 , p9 , p12 , p15 , s6 of the complex
reﬂection group G(3, 3, 6). Then in Theorem 1 we shall show that mj coincides with µ6j
up to a constant. In section 4, we mention two results on invariants and a discriminant of
ST34 due to Terao, Enta, Bessis and Michel. In section 5, we describe the discriminantof ST34 in terms of the ﬂat coordinates of ST34 (cf. [7], [8], [12]) In section 6, we shalltreat two topics on ST34. One is concerned with the correspondence between the totality of minimal vectors of the Coxeter-Todd lattice and that of hyperplanes ﬁxed by thepseudo-reﬂections of ST34. The other is the relationship between the basic invariants ofthe group ST33 and those of ST34 constructed by Conway and Sloane. In the Appendix,
we give the explicit relationship between Burkhardt invariants and our invariants.
We ﬁnally mention that the softwares Mathematica, Maple and Magma [2] are used
to obtain the results in this paper.
Acknowledgments. The second-named author is supported by JSPS KAKENHI Grant
Number 17K05269. The authors would like to thank the referee for valuable comments.

2

The basic invariants by Conway-Sloane

By the classiﬁcation of complex reﬂection groups of Shephard-Todd [14], it is known that
there are three inﬁnite series of complex reﬂection groups and thirty-four sporadic ones
numbered as 1, 2, . . . , 37. In this paper we focus our attention on the group No.34 which
we denote by ST34. The basic invariants of ST34 are constructed in Theorem 10 of[5] and they are denoted by µj (j = 6, 12, 18, 24, 30, 42). We now explain one of theirconstructions brieﬂy. Let Λ(3) be the Coxeter-Todd lattice introduced at p.424 of [5].
Then its automorphism group √is nothing else but ST34. Throughout this paper, we writeω = e2πi/3 and θ = ω − ω = −3 without any comment. The minimal vectors of Λ(3)consist of the vectors2

±(ω a θ, −ω b θ, 0, 0, 0, 0), a, b ∈ {0, 1, 2},(ω a , ω b , ω c , ω d , ω e , ω f ), a, b, c, d, e, f ∈ {0, 1, 2}, a + b + c + d + e + f ≡ 0 (mod 3)()
where all possible coordinate changes are considered in the ﬁrst type. There are 2·32 · 62 =270 minimal vectors of the ﬁrst type, whereas 2 · 35 = 486 minimal vectors of the second
type. In total, we have 756 minimal vectors of Λ(3) . We set∑µk =(v1 x1 + · · · + v6 x6 )k , k = 0, 1, 2 . . . ,where v = (v1 , . . . , v6 ) runs through 756 minimal vectors of Λ(3) . Theorem 10 in [5] saysthat the invariant ring of ST34 is generated byµ6 , µ12 , µ18 , µ24 , µ30 , µ42 .
We collect the basic properties of ST34. There are 126 pseudo-reﬂections in ST34.
The hyperplanes ﬁxed by the pseudo-reﬂections of ST34 are
45
30
20
30
1

xi − ω a xj = 0, a ∈ {0, 1, 2},
x1 + x2 + x3 + x4 + ωx5 + ω 2 x6 = 0,x1 + x2 + x3 + ω(x4 + x5 + x6 ) = 0,x1 + x2 + ω(x3 + x4 ) + ω 2 (x5 + x6 ) = 0,x1 + x2 + x3 + x4 + x5 + x6 = 0

where the number on the left side denotes the cardinality of each type. ST34 can begenerated by transformations P1 , P2 , P3 , Q1 , R1 , R2 given at p.298 in [14]. The hyperplanes
ﬁxed by the generators are
P1 : x2 − x3 = 0,
P2 : x3 − x4 = 0,
P3 : x4 − x5 = 0,
Q1 : x1 − x2 = 0,
R1 : x1 − ωx2 = 0,
R2 : x1 + x2 + x3 + x4 + x5 + x6 = 0.
The 7-th power of the matrix R2 R1 Q1 P1 P2 P3 given at p.299 of [14] is −ωI6 . The center Z34
of ST34 is of order 6 (cf. Table VII of [14]) and −ωI6 centralizes ST34. As a consequence,−ωI6 generates Z34 .

3

From G(3, 3, 6) to ST34

Since the group G(3, 3, 6) is a subgroup1 of ST34 and since these two groups are of thesame rank, every invariant by ST34 is a polynomial of the basic invariants of G(3, 3, 6).
1

The group G(3, 3, 6) is generated by the transformations on (x1 , . . . , x6 ): (1) the permutations on
x1 , . . . , x6 , (2) xi 7→ ϵi xi (i = 1, . . . , 6), ϵ3i = 1, ϵ1 . . . ϵ6 = 1. The reﬂection corresponding to x5 − x6 = 0
is given as the product of (R2 R1 Q1 P1 P2 )9 and −I6 . The resulting matrix is an element of ST34.

3

It is known (p.284 of [14]) that as the basic invariants of G(3, 3, 6), we may takep3j (j = 1, 2, 3, 4, 5) and s6 deﬁned by
3j
3j
3j
3j
3j
p3j = x3j
1 + x2 + x3 + x4 + x5 + x6 (j = 1, 2, 3, 4, 5), s6 = x1 x2 x3 x4 x5 x6 .

Let mj (j = 1, 2, 3, 4, 5, 7) be the polynomials of pj (j = 3, 6, 9, 12, 15) and s6 deﬁned by
the following identities. These mj ’s are indeed invariants of ST34 as we will show below.
m1
m2
m3

=
=
=

m4

=

m5

=

−5p23 + 6p6 − 180s6 ,
1
4
2
2
2
2
17 (−10125p12 + 1925p3 − 10395p3 p6 + 6237p6 + 12375p3 p9 + 51975p3 s6 − 40095p6 s6 + 935550s6 ),(12
643644 (−1088916048p15 p3 + 1401597270p12 p3 − 52286707p3 − 3171150p12 p6 + 498415005p3 p6−723353895p23 p26 + 9550629p36 − 961539480p33 p9 + 936512280p3 p6 p9 − 16804260p29 + 2817424620p12 s6−804053250p43 s6 + 3721617900p23 p6 s6 − 1435481190p26 s6 − 4300536240p3 p9 s6 − 13894040160p23 s26+8931882960p6 s26 − 99578658960s36 ),
1
2
3
4
8
984160 (616868762940p12 + 13316119811280p15 p3 − 16940634571080p12 p3 + 594592089355p3
2
6−21567358775952p15 p3 p6 + 28072219631940p12 p3 p6 − 6811339217958p3 p6 − 368171695980p12 p26+18219805403580p43 p26 − 14058186924870p23 p36 + 98666056665p46 + 9579454289856p15 p9−13482219282840p12 p3 p9 + 11937182241204p53 p9 − 34871337186840p33 p6 p9 + 24742879985340p3 p26 p9+9126295239600p23 p29 − 8204834872080p6 p29 + 417113860322016p15 p3 s6 − 559068289484760p12 p23 s6+24970204832364p63 s6 + 17333883551160p12 p6 s6 − 219949713575460p43 p6 s6 + 302052795820380p23 p26 s6−8679490869060p36 s6 + 401008538842560p33 p9 s6 − 376118508300480p3 p6 p9 s6 + 1336769866080p29 s6−732306533056560p12 s26 + 311680371562560p43 s26 − 1202195718884160p23 p6 s26 + 386420052498480p26 s26+1238625892183680p3 p9 s26 + 3635549644541760p23 s36 − 1960702790302080p6 s36 + 14488157891797920s46 ),12 2
2106288160 (−855829791445492800p12 p3 + 2089201018632300p12 p6 + 1165675074483701520p12 p15 p3
64+138192145506496980p12 p3 − 877484405515505400p12 p3 p6 + 1379987318783043000p12 p3 3 p9+903656701052469000p12 p3 2 p6 2 − 1000122875204029200p12 p3 p6 p9 − 906698664553500p12 p6 3+580661494716600p12 p9 2 − 386759386157508288p15 2 − 103164357358799784p15 p3 5+631029397643881200p15 p3 3 p6 − 965383545286463760p15 p3 2 p9 − 618935794330053240p15 p3 p6 2+662798218317617520p15 p6 p9 − 3721032235241428p3 10 + 56929735554738525p3 8 p6−100670803599215340p3 7 p9 − 269791048596035610p3 6 p6 2 + 754513827309400680p3 5 p6 p9+455456084098977900p3 4 p6 3 − 540975355278457500p3 4 p9 2 − 1258582004466149700p3 3 p6 2 p9−237876161937857550p3 2 p6 4 + 824140299141199800p3 2 p6 p9 2 + 530150266143331200p3 p6 3 p9−157057655983200p3 p9 3 + 168811773953535p6 5 − 285007424484525300p6 2 p9 2 )2349(3336381283590p12 2 − 90991170059715p12 p3 4 + 132473392247970p12 p3 2 p6
− 5314408−60473056848750p12 p3 p9 − 3642116633685p12 p6 2 + 69747109280496p15 p3 3−96145773383952p15 p3 p6 + 39062903267592p15 p9 + 3483778065289p3 8 − 38013477459228p3 6 p6+65401030889973p3 5 p9 + 94878525690540p3 4 p6 2 − 173998725667650p3 3 p6 p9−66633212619360p3 2 p6 3 + 42238214518500p3 2 p9 2 + 110872962731835p3 p6 2 p9 + 987000196815p6 4−32583765485700p6 p9 2 )s6
26− 221240565
5314408 (−14731974630p12 p3 + 693432090p12 p6 + 10397417904p15 p3 + 842268971p3
43
2 2−6659025525p3 p6 + 11239209720p3 p9 + 8453844855p3 p6 − 9972788040p3 p6 p9 − 348684345p6 3+86329800p9 2 )s26
98415
(7805322022518p12 − 4961331496841p3 4 + 16060185954756p3 2 p6 − 14673183959034p3 p9+ 1328602−4333560899931p6 2 )s36
24− 231176835
664301 (9206527958p3 − 4219165557p6 )s6
5437195842606312330 5−s6 ,664301

4

m7

=

1
14
+ 1432379563699839285857067p6 p3 12619872782080 (−68155702606842557785745p3
11−2557436825642562030969900p9 p3 − 10698541080341814802394019p6 2 p3 10+3582500028944048652957174p12 p3 10 − 2730902888436717092575680p15 p3 9+32708776402929577905944640p6 p9 p3 9 + 34924283387125205511721185p6 3 p3 8−25654418121315705916250700p9 2 p3 8 − 41680220280677434506011850p12 p6 p3 8+31369465880845892613346176p15 p6 p3 7 − 123740484999420235384965240p6 2 p9 p3 7+69769318329654728499822240p12 p9 p3 7 − 52762146031720621233390555p6 4 p3 6−47257748766395678591117100p12 2 p3 6 + 126780633297174166548934500p12 p6 2 p3 6+130162353083938619632641600p6 p9 2 p3 6 − 52307455008394026247756320p15 p9 p3 6−34758789936079286842349760p9 3 p3 5 − 92627028581352964521141312p15 p6 2 p3 5+70714813144026703680064512p12 p15 p3 5 + 186354004242240630334859040p6 3 p9 p3 5−268191568191852515860009920p12 p6 p9 p3 5 + 33299590235273255328597225p6 5 p3 4−127770635569767758534767500p12 p6 3 p3 4 − 26421815125729745638761600p15 2 p3 4
−201898474896088367649617400p6 2 p9 2 p3 4 + 113298473516080679802360000p12 p9 2 p3 4+122348893127056155271952100p12 2 p6 p3 4 + 190543330015178442011760960p15 p6 p9 p3 4+87812086740565495801127040p15 p6 3 p3 3 + 62665614050169799982476800p6 p9 3 p3 3−74023416302855483255097600p15 p9 2 p3 3 − 167964296041394365510972800p12 p15 p6 p3 3−106338866778530075623833900p6 4 p9 p3 3 − 111554371285879313232176400p12 2 p9 p3 3+262063973509069744833861600p12 p6 2 p9 p3 3 − 5126918175165039183578025p6 6 p3 2+27931623849212462954949150p12 p6 4 p3 2 − 23152418725131358790400p9 4 p3 2+29692604870039550095987400p12 3 p3 2 − 50201877703761207864827100p12 2 p6 2 p3 2+103065371624630753152920000p6 3 p9 2 p3 2 − 115014969795605791696092000p12 p6 p9 2 p3 2+56973925496711539138136832p15 2 p6 p3 2 − 169930072089259625905241760p15 p6 2 p9 p3 2+135574470385751288032364160p12 p15 p9 p3 2 − 14845481613081683657896320p15 p6 4 p3−27945506509393387416667200p6 2 p9 3 p3 + 34586384532087307046400p12 p9 3 p3+51824289113657057141222400p12 p15 p6 2 p3 + 65877259573354459565260800p15 p6 p9 2 p3−44231372298721494546912000p12 2 p15 p3 + 12594866818006221239484000p6 5 p9 p3−44075507458043013535144800p12 p6 3 p9 p3 − 38836159574544740438311680p15 2 p9 p3+37742148440266625813304000p12 2 p6 p9 p3 + 841205579632152292755p6 7 − 6381502411588805286450p12 p6 5+20837200939432768800000p6 p9 4 + 12591883060088162966100p12 2 p6 3 − 18536934715459277725440p15 p9 3+16391367254585859259256064p12 p15 2 − 10659165988259339409769344p15 2 p6 2−7683466994926210007484300p6 4 p9 2 − 8464031877608794242000p12 2 p9 2+11976082225174674297633600p12 p6 2 p9 2 + 4971887755839338531400p12 3 p6
+18094988745615635242320960p15 p6 3 p9 − 28029510009872734558063680p12 p15 p6 p9 )2583(−826650327935043055861p3 12 + 14425925885422918422282p6 p3 10+ 309936391040−25452001272005719299540p9 p3 9 − 83351888440402255303005p6 2 p3 8 + 35197343052145422459210p12 p3 8−26417038086987608202768p15 p3 7 + 242911215262797035732640p6 p9 p3 7 + 186506835419875414158900p6 3 p3 6
−183741456439968965505360p9 2 p3 6 − 298429989407564825623320p12 p6 p3 6+220420573635965479389936p15 p6 p3 5 − 563000045054149040896440p6 2 p9 p3 5+493130102035689915584400p12 p9 p3 5 − 161110222055187816091275p6 4 p3 4−328749745772938424739300p12 2 p3 4 + 485097352985170111227300p12 p6 2 p3 4+429094779958543624758000p6 p9 2 p3 4 − 361895253749323218243360p15 p9 p3 4−20582878620895939008000p9 3 p3 3 − 347376551409240131304720p15 p6 2 p3 3+479349562682466181945440p12 p15 p3 3 + 428618573732090041704000p6 3 p9 p3 3−673892039580382647561600p12 p6 p9 p3 3 + 36297779413268324630250p6 5 p3 2−143166572523976223269800p12 p6 3 p3 2 − 173568570666357873223296p15 2 p3 2−293344069145812931192400p6 2 p9 2 p3 2 + 43066790405325599148000p12 p9 2 p3 2+141143057326507215360600p12 2 p6 p3 2 + 469676819583053026119360p15 p6 p9 p3 2+90703842698655568839600p15 p6 3 p3 + 19163765066700626496000p6 p9 3 p3−22962898216765380119040p15 p9 2 p3 − 178484620941100228128480p12 p15 p6 p3

5

−81029269710700131377700p6 4 p9 p3 − 19161435544300302037200p12 2 p9 p3+169202690869163631008400p12 p6 2 p9 p3 − 44454150584215675875p6 6 + 236755439504502230250p12 p6 4−95534751065126400p9 4 + 236549851238374857000p12 3 − 413969210349675356700p12 2 p6 2
+43860401994452195041200p6 3 p9 2 − 11869779935007266508000p12 p6 p9 2 + 53589954112307329756032p15 2 p6
−97278586484753853185760p15 p6 2 p9 + 14219410871098234843200p12 p15 p9 )s6
637362999− 15496819552(4857452670185866p3 10 − 65282378295761109p6 p3 8 + 108907344194688000p9 p3 7+260295982554842820p6 2 p3 6 − 144443551320429948p12 p3 6 + 105268487601706560p15 p3 5−644181269743573344p6 p9 p3 5 − 367603906096725030p6 3 p3 4 + 384498967408122240p9 2 p3 4+678555020839017240p12 p6 p3 4 − 463955787849692736p15 p6 p3 3 + 895430833203929760p6 2 p9 p3 3−880017911185364640p12 p9 p3 3 + 153022160508667650p6 4 p3 2 + 472031454180039120p12 2 p3 2−541859921711888940p12 p6 2 p3 2 − 505515465619816320p6 p9 2 p3 2 + 568858413963720960p15 p9 p3 2+1078387051311360p9 3 p3 + 333737005975634880p15 p6 2 p3 − 573606450773307072p12 p15 p3−298809060268406400p6 3 p9 p3 + 519137518908140640p12 p6 p9 p3 − 740481396190785p6 5+2897609287598280p12 p6 3 + 163131844841144064p15 2 + 133116924334917600p6 2 p9 2−769650476421600p12 p9 2 − 2833333167373020p12 2 p6 − 295206239616621696p15 p6 p9 )s26
137781− 7748409776(212849412401280527467p3 8 − 2056737692333330079912p6 p3 6 + 3301675794233514997830p9 p3 5+4487246757920672916210p6 2 p3 4 − 4310703736169475684240p12 p3 4 + 3103075441442929625568p15 p3 3−7542458411263968569940p6 p9 p3 3 − 2624550943988148988320p6 3 p3 2 + 1648635226201837343160p9 2 p3 2+5112301708617615116280p12 p6 p3 2 − 3259289326066664689728p15 p6 p3 + 3870577632979310328810p6 2 p9 p3−2142794233908947745900p12 p9 p3 + 66893663401936554375p6 4 + 193991306712427220580p12 2−230654127137064281640p12 p6 2 − 903663040115630763720p6 p9 2 + 1073610002137218186960p15 p9 )s36
457570701− 3874204888(393541072937010113p3 6 − 2483580961309967805p6 p3 4 + 3573657602659051360p9 p3 3+2671680990475403505p6 2 p3 2 − 4136783742478404450p12 p3 2 + 2632941690092063712p15 p3−2833712226159225840p6 p9 p3 − 136991407212027225p6 3 + 52735937047941600p9 2 + 266860623034204470p12 p6 )s46(−1416246125627p3 4 + 3374374908123p6 p3 2 − 2450724434568p9 p3+ 125452159093170
484275611−673355559033p6 2 + 1069099330203p12 )s56
2
6
− 9039811410
484275611 (105868047331971014p3 − 36630825385046211p6 )s6
1266409465981399253335790610 7
s6 .
−484275611

The following theorem is the main result of this paper.
Theorem 1 The polynomials mj (j = 1, 2, 3, 4, 5, 7) are invariants of ST34.
Proof. The group ST34 can be generated by the pseudo-reﬂections P1 , P2 , P3 , Q1 , R1 , R2 .
It is clear that p3j (j = 1, 2, 3, 4, 5) and s6 are invariant under the action of P1 , P2 , P3 , Q1 , R1
and so are mj (j = 1, 2, 3, 4, 5, 7).
As a consequence, it is suﬃcient to show that each of mj is invariant by the action of
R2 . For this purpose, we introduce the polynomials
qj =

6
∑

xjk

(j = 1, 2, 3, 4, 5).

k=1

6

Then p3j (j = 1, 2, 3, 4, 5) are polynomials of qj (j = 1, 2, 3, 4, 5) and s6 . Indeed, we have
p3
p6
p9

=
=
=

p12

=

p15

=

q3 ,

1
6
4
2 2
3
3
2
2
120 (q1 − 15q1 q2 + 45q1 q2 − 15q2 + 40q1 q3 − 120q1 q2 q3 + 40q3 − 90q1 q4 + 90q2 q4 + 144q1 q5 − 720s6 ),
1
9
7
5 2
3 3
6
4
2 2
3
3 2
2
720 (q1 − 9q1 q2 − 27q1 q2 + 135q1 q2 + 33q1 q3 + 45q1 q2 q3 − 135q1 q2 q3 − 135q2 q3 + 120q1 q3 − 360q1 q2 q3
3
5
3
2
4
2
2
+80q3 − 54q1 q4 − 270q1 q2 q4 − 270q1 q3 q4 + 270q2 q3 q4 + 54q1 q5 + 324q1 q2 q5 + 162q2 q5 + 432q1 q3 q5+324q4 q5 − 1080q13 s6 − 3240q1 q2 s6 − 2160q3 s6 ),
1
12
10
8 2
6 3
4 4
9
7
5 2
3 3
43200 (q1 − 12q1 q2 + 90q1 q2 − 900q1 q2 + 2025q1 q2 + 80q1 q3 − 720q1 q2 q3 + 3600q1 q2 q3 − 3600q1 q2 q3
6 2
4
2
2 2 2
3 2
3 3
3
4
8
+1680q1 q3 − 3600q1 q2 q3 + 3600q1 q2 q3 − 3600q2 q3 + 3200q1 q3 − 9600q1 q2 q3 + 1600q3 − 45q1 q4 + 180q16 q2 q4−6750q14 q22 q4 + 8100q12 q23 q4 − 2025q24 q4 − 2880q15 q3 q4 + 7200q13 q2 q3 q4 − 21600q1 q22 q3 q4 − 7200q12 q32 q4+7200q2 q32 q4 − 16200q12 q2 q42 + 8100q22 q42 + 2700q43 + 288q17 q5 − 3456q15 q2 q5 + 12960q13 q22 q5 + 11520q14 q3 q5−17280q12 q2 q3 q5 + 11520q1 q32 q5 − 17280q13 q4 q5 + 25920q1 q2 q4 q5 + 17280q3 q4 q5 + 20736q12 q52 + 10368q2 q52−1440q16 s6 − 64800q12 q22 s6 − 57600q13 q3 s6 − 57600q32 s6 − 129600q2 q4 s6 − 207360q1 q5 s6 + 259200s26 ),
1
13
11 2
9 3
7 4
5 5
3 6
12
15
1036800 (13q1 − 330q1 q2 + 2565q1 q2 − 4500q1 q2 − 15525q1 q2 + 44550q1 q2 − 10125q1 q2 + 895q1 q3
10
8 2
6 3
4 4
2 5
6
−12090q1 q2 q3 + 22275q1 q2 q3 + 78300q1 q2 q3 − 165375q1 q2 q3 − 20250q1 q2 q3 + 10125q2 q3 + 16000q19 q32−36000q17 q2 q32 − 64800q15 q22 q32 + 108000q13 q23 q32 + 108000q1 q24 q32 + 24800q16 q33 − 84000q14 q2 q33 + 108000q12 q22 q33−36000q23 q33 + 32000q13 q34 − 96000q1 q2 q34 + 12800q35 − 1710q111 q4 + 19800q19 q2 q4 − 2700q17 q22 q4−178200q15 q23 q4 + 101250q13 q24 q4 − 58050q18 q3 q4 − 16200q16 q2 q3 q4 + 337500q14 q22 q3 q4 + 81000q12 q23 q3 q4−101250q24 q3 q4 − 43200q15 q32 q4 + 216000q13 q2 q32 q4 − 432000q1 q22 q32 q4 − 72000q12 q33 q4 + 72000q2 q33 q4+56700q17 q42 + 113400q15 q2 q42 − 121500q13 q22 q42 + 40500q14 q3 q42 − 405000q12 q2 q3 q42 + 121500q22 q3 q42 − 81000q13 q43+81000q3 q43 + 2196q110 q5 − 22140q18 q2 q5 − 33480q16 q22 q5 + 243000q14 q23 q5 + 72900q12 q24 q5 − 24300q25 q5+77760q17 q3 q5 + 60480q15 q2 q3 q5 − 302400q13 q22 q3 q5 − 388800q1 q23 q3 q5 + 216000q14 q32 q5 − 432000q12 q2 q32 q5−43200q22 q32 q5 + 115200q1 q33 q5 − 136080q16 q4 q5 − 550800q14 q2 q4 q5 + 97200q12 q22 q4 q5 + 97200q23 q4 q5−259200q13 q3 q4 q5 + 259200q1 q2 q3 q4 q5 + 259200q32 q4 q5 − 97200q12 q42 q5 + 291600q2 q42 q5 + 82944q15 q52+414720q13 q2 q52 + 311040q1 q22 q52 + 414720q12 q3 q52 + 207360q2 q3 q52 + 311040q1 q4 q52 + 41472q53 − 23400q19 s6+280800q17 q2 s6 − 226800q15 q22 s6 − 1620000q13 q23 s6 + 243000q1 q24 s6 − 820800q16 q3 s6 + 216000q14 q2 q3 s6+3240000q12 q22 q3 s6 + 648000q23 q3 s6 − 864000q13 q32 s6 + 864000q1 q2 q32 s6 − 576000q33 s6 + 1555200q15 q4 s6+2592000q13 q2 q4 s6 − 1944000q1 q22 q4 s6 − 2592000q2 q3 q4 s6 − 972000q1 q42 s6 − 2073600q14 q5 s6−6220800q12 q2 q5 s6 − 1555200q22 q5 s6 − 4147200q1 q3 q5 s6 − 1555200q4 q5 s6 + 10368000q13 s26 + 15552000q1 q2 s26+5184000q3 s26 ).

The reﬂection R2 acts on (x1 , x2 , x3 , x4 , x5 , x6 ) by
1∑
xk
R2 : xj 7→ xj −3 k=1
6

(j = 1, 2, · · · , 6).

If f = f (x1 , · · · , x6 ) is a polynomial of (x1 , · · · , x6 ), we write f ◦ R2 for f (R2 (x1 , · · · , x6 )).

7Then by a direct computation, we havep1 ◦ R 2
p2 ◦ R 2
p3 ◦ R 2

p4 ◦ R 2

p5 ◦ R 2s 6 ◦ R2==

1 3
9 (q1 − 9q1 q2 + 9q3 ),
1
6
4
2 2
3
3
2
2
9720 (−79q1 + 585q1 q2 + 3645q1 q2 − 1215q2 − 3960q1 q3 − 9720q1 q2 q3 + 3240q3 + 8910q1 q4 + 7290q2 q4−7776q1 q5 − 58320s6 ),
1
9
7
5 2
3 3
4
6
4
524880 (−1384q1 + 22707q1 q2 − 85293q1 q2 − 91125q1 q2 + 98415q1 q2 − 57375q1 q3 + 441045q1 q2 q3
2 2
3
3 2
2
3
5
+338985q1 q2 q3 − 98415q2 q3 − 456840q1 q3 − 262440q1 q2 q3 + 58320q3 + 82134q1 q4 − 634230q13 q2 q4−393660q1 q22 q4 + 1027890q12 q3 q4 + 196830q2 q3 q4 − 393660q1 q42 − 53946q14 q5 + 446148q12 q2 q5 + 118098q22 q5−524880q1 q3 q5 + 236196q4 q5 + 612360q13 s6 + 3936600q1 q2 s6 − 1574640q3 s6 ),1= 283435200(−61441q112 + 1761678q110 q2 − 17211690q18 q22 + 57445200q16 q23 + 2460375q14 q24 − 32476950q12 q25−2696580q19 q3 + 59428080q17 q2 q3 − 327612600q15 q22 q3 − 23619600q13 q23 q3 + 64953900q1 q24 q3 − 34739280q16 q32+553748400q14 q2 q32 + 81356400q12 q22 q32 − 23619600q23 q32 − 235612800q13 q33 − 62985600q1 q2 q33 + 10497600q34+1843155q18 q4 − 58480380q16 q2 q4 + 388739250q14 q22 q4 + 53144100q12 q23 q4 − 13286025q24 q4 + 19595520q15 q3 q4−1165233600q13 q2 q3 q4 − 141717600q1 q22 q3 q4 + 530128800q12 q32 q4 + 47239200q2 q32 q4 + 43302600q14 q42+543250800q12 q2 q42 + 53144100q22 q42 − 259815600q1 q3 q42 + 17714700q43 + 93312q17 q5 + 23514624q15 q2 q5−226748160q13 q22 q5 + 29393280q14 q3 q5 + 579467520q12 q2 q3 q5 − 201553920q1 q32 q5 − 113374080q13 q4 q5−453496320q1 q2 q4 q5 + 113374080q3 q4 q5 + 52907904q12 q52 + 68024448q2 q52 + 75232800q16 s6−692841600q14 q2 s6 − 1464415200q12 q22 s6 + 1700611200q13 q3 s6 + 2078524800q1 q2 q3 s6 − 377913600q32 s6−2078524800q12 q4 s6 − 850305600q2 q4 s6 + 1133740800q1 q5 s6 + 1700611200s26 ),
1(190052q115 + 7543305q113 q2 − 314760330q111 q22 + 3315388725q19 q23 − 11699575200q17 q24= 61222003200+4687014375q15 q25 + 3587226750q13 q26 − 597871125q1 q27 + 41009175q112 q3 + 135576990q110 q2 q3−12521395125q18 q22 q3 + 74175385500q16 q23 q3 − 20512146375q14 q24 q3 − 12356003250q12 q25 q3 + 597871125q26 q3+1263778200q19 q32 + 7463793600q17 q2 q32 − 163560481200q15 q22 q32 + 17950896000q13 q23 q32 + 10097379000q1 q24 q32+6796029600q16 q33 + 138332124000q14 q2 q33 + 3070548000q12 q22 q33 − 2125764000q23 q33 − 36321696000q13 q34−5668704000q1 q2 q34 + 755827200q35 − 124608780q111 q4 + 1213748550q19 q2 q4 + 7755102000q17 q22 q4−79344141300q15 q23 q4 + 12488863500q13 q24 q4 + 8370195750q1 q25 q4 − 6226279650q18 q3 q4 + 17065161000q16 q2 q3 q4+320016055500q14 q22 q3 q4 − 2657205000q12 q23 q3 q4 − 5978711250q24 q3 q4 − 55642528800q15 q32 q4−327840048000q13 q2 q32 q4 − 10628820000q1 q22 q32 q4 + 81723816000q12 q33 q4 + 4251528000q2 q33 q4+5558479200q17 q42 − 30711384900q15 q2 q42 − 150397803000q13 q22 q42 − 16740391500q1 q23 q42 + 90020200500q14 q3 q42+214170723000q12 q2 q3 q42 + 7174453500q22 q3 q42 − 44641044000q1 q32 q42 − 35783694000q13 q43−33480783000q1 q2 q43 + 4782969000q3 q43 + 122974524q110 q5 − 1648516860q18 q2 q5 + 569232360q16 q22 q5+41133533400q14 q23 q5 − 11319693300q12 q24 q5 − 1434890700q25 q5 + 5427259200q17 q3 q5 − 30377954880q15 q2 q3 q5−148803480000q13 q22 q3 q5 + 12754584000q1 q23 q3 q5 + 45381124800q14 q32 q5 + 125278358400q12 q2 q32 q5−2550916800q22 q32 q5 − 24942297600q1 q33 q5 − 8299927440q16 q4 q5 + 58741945200q14 q2 q4 q5+130734486000q12 q22 q4 q5 + 5739562800q23 q4 q5 − 118475913600q13 q3 q4 q5 − 127545840000q1 q2 q3 q4 q5+15305500800q32 q4 q5 + 56757898800q12 q42 q5 + 17218688400q2 q42 q5 + 2781444096q15 q52 − 23128312320q13 q2 q52−24488801280q1 q22 q52 + 34012224000q12 q3 q52 + 12244400640q2 q3 q52 − 24488801280q1 q4 q52 + 2448880128q53+1452897000q19 s6 − 35980524000q17 q2 s6 + 213119650800q15 q22 s6 + 172186884000q13 q23 s6−52612659000q1 q24 s6 + 45885009600q16 q3 s6 − 807317928000q14 q2 q3 s6 − 522937944000q12 q22 q3 s6+38263752000q23 q3 s6 + 530968608000q13 q32 s6 + 289103904000q1 q2 q32 s6 − 34012224000q33 s6−20596291200q15 q4 s6 + 926833104000q13 q2 q4 s6 + 420901272000q1 q22 q4 s6 − 833299488000q12 q3 q4 s6−153055008000q2 q3 q4 s6 + 210450636000q1 q42 s6 − 11337408000q14 q5 s6 − 510183360000q12 q2 q5 s6−91833004800q22 q5 s6 + 326517350400q1 q3 q5 s6 − 91833004800q4 q5 s6 − 578207808000q13 s26−1224440064000q1 q2 s26 + 306110016000q3 s26 ),= (−13q16 + 180q14 q2 − 405q12 q22 − 450q13 q3 + 810q1 q2 q3 + 810q12 q4 − 972q1 q5 + 14580s6 )/14580.
=

We now take one of m1 , m2 , m3 , m4 , m5 , m7 and write it, say mj . It is possible to show theexpressions of both mj and mj ◦ R2 as polynomials of q1 , q2 , q3 , q4 , q5 and s6 . Comparing
these two expressions, we ﬁnd that mj ◦R2 = mj . This completes the proof of the theorem.
Remark 1 The normalization of the choice of mj ’s is that as a polynomial of x, each mj
contains the term x6j
1 with coeﬃcient 1. This implies that as a polynomial of p3 , p6 , p9 , p12 , p15 , s6 ,the value of mj at p3 = p6 = p9 = p12 = p15 = 1, s6 = 0 is 1.

8

Corollary 1 The relations between µ6 , µ12 , · · · and m1 , m2 , · · · are given by
µ6
µ12
µ18
µ24
µ30
µ42

=
=
=
=
=
=

−1944m1 ,66096m2 ,−1770984m3 ,47830176m4 ,−1291401144m5 ,−941431787784m7 .

Proof. The polynomials mj (j = 1, 2, 3, 4, 5, 7) are obtained by the same idea as theconstruction of µ6j (j = 1, 2, 3, 4, 5, 7) given in [5]. Actually mj coincides with µ6j up to a
constant factor. The constants are speciﬁed by evaluating polynomials at special values
and the result follows.

4

Two results on invariants and a discriminant of
ST34

We give two results which play important roles in our consideration.

4.1

Invariants by Terao and Enta

We recall the invariants f1 , f2 , f3 , f4 , f5 , f6 given in Terao and Enta [15] (see also [11]). By
a direct computation we have
f1
f2
f3
f4
f5
f6

=
=
=
=
=
=

1
µ (= −m1 ),1944 61µ (= 17m2 ),3888 121µ (= −911m3 ),1944 1831670896436
233872961
63038467
6151
1937160963
m4 − 19773045015625
m21 m2 + 754856437500
m22 + 258156875724
m1 m3 − 205320951
m4 ,
1797549546875 1
1µ(=−664301m),
5
1944 301µ(=−484275611m
42
7 ),
1944

where f4 is the determinant of the Hessian matrix of f1 up to a constant factor.

4.2

Discriminant by Bessis and Michel

Concerning the discriminant of ST34, there are studies by Terao and Enta [15] (see also[11]), Bessis and Michel (unpublished). Bessis and Michel [1] constructed Saito matrices
of some complex reﬂection groups. They commented that the case ST34 is not treated in[1] since the Saito matrix of ST34 is “too large to be printed” (p.261 of [1]). Moreoverbased on the idea of [15], they constructed the Saito matrix of the invariants by Conway
and Sloane explicitly (unpublished).
Prof. J. Michel kindly sent the data of the Saito matrix of ST34 for the basic invariants{f1 , f2 , · · · , f6 } on the request of the second author (J. S.). We call this matrix M34 whose
determinant is the discriminant of ST34 up to a non-zero constant factor. Each matrix
entry of M34 is a polynomial of x, y, z, t, u, v which are the same as f1 , f2 , f3 , f4 , f5 , f6 of
Terao and Enta , respectively.
95Construction of the Saito matrix in terms of a potential vector ﬁeld

The Saito matrix of a reﬂection group depends on the choice of basic invariants. Thereis an alternative way to construct the Saito matrix of ST34. We begin this section with
explaining the construction of the Saito matrix for the ﬂat coordinate. For the details,see [12], [7], [13].
The Saito matrix of ST34 can be expressed by a potential vector ﬁeld given in[7]. To show the result, we deﬁne the following polynomials hj (j = 1, 2, 3, 4, 5, 6) of
u1 , u 2 , u 3 , u 4 , u 5 , u 6 .
h1 = (20u41 u22 + 120u21 u32 − 60u42 − 12u51 u3 + 60u31 u2 u3 − 180u1 u22 u3 + 135u21 u23 + 135u2 u23 + 120u41 u4−180u21 u2 u4 + 540u22 u4 + 270u1 u3 u4 + 405u24 + 180u31 u5 + 540u1 u2 u5 + 405u3 u5 + 405u1 u6 )/405,
h2 = (64u91 − 288u71 u2 − 1728u51 u22 + 1728u31 u32 − 2592u1 u42 + 432u61 u3 + 3888u41 u2 u3 + 3888u21 u22 u3+1944u32 u3 + 972u31 u23 + 729u33 − 1296u51 u4 + 7776u31 u2 u4 + 8748u21 u3 u4 − 2916u2 u3 u4 − 5832u1 u24+3888u41 u5 − 2916u21 u2 u5 − 2916u22 u5 + 4374u1 u3 u5 + 4374u4 u5 + 4374u2 u6 )/4374,
3 2
5
7
5
6 2
4 3
8
h3 = (64u10
1 + 3456u1 u2 + 6624u1 u2 + 2160u1 u2 + 3888u2 + 7776u1 u3 + 19440u1 u2 u3 + 32400u1 u2 u3
4
3
6
2 2
2
2
4 2
3
+7776u1 u2 u3 + 19440u1 u3 + 26244u1 u2 u3 + 2916u2 u3 + 7290u1 u3 + 9504u1 u4 + 38880u1 u2 u4−11664u21 u22 u4 − 3888u32 u4 + 25272u31 u3 u4 + 34992u1 u2 u3 u4 + 8748u23 u4 + 5832u21 u24 + 8748u2 u24+11664u51 u5 + 25272u31 u2 u5 + 17496u1 u22 u5 + 26244u21 u3 u5 + 8748u2 u3 u5 + 26244u1 u4 u5+6561u25 + 6561u3 u6 )/6561,
9
7 2
5 3
3 4
8
6
4 2
h4 = (1152u11
1 + 832u1 u2 + 23616u1 u2 − 13824u1 u2 + 34560u1 u2 + 7776u1 u3 + 42768u1 u2 u3 + 16200u1 u2 u3
2 3
4
5 2
3
2
2 2
2 3
3
+7776u1 u2 u3 − 5832u2 u3 + 17496u1 u3 + 27216u1 u2 u3 + 23328u1 u2 u3 + 13122u1 u3 + 2916u2 u3+12960u71 u4 − 25920u51 u2 u4 + 103680u31 u22 u4 − 31104u1 u32 u4 + 19440u41 u3 u4 + 69984u21 u2 u3 u4−5832u22 u3 u4 + 17496u1 u23 u4 + 38880u31 u24 − 46656u1 u2 u24 − 4374u3 u24 + 2592u61 u5 + 21384u41 u2 u5−5832u21 u22 u5 + 11664u32 u5 + 26244u31 u3 u5 + 17496u1 u2 u3 u5 + 6561u23 u5 − 8748u21 u4 u5 + 17496u2 u4 u5+13122u1 u25 + 13122u4 u6 )/13122,
10
8 2
6 3
4 4
6
9
h5 = (10496u12
1 + 70656u1 u2 + 86976u1 u2 + 233856u1 u2 − 25920u1 u2 − 20736u2 + 71808u1 u3
7
5 2
3 3
4
6 2
+264384u1 u2 u3 + 393984u1 u2 u3 + 129600u1 u2 u3 + 93312u1 u2 u3 + 165888u1 u3 + 408240u41 u2 u23+221616u21 u22 u23 + 134136u31 u33 + 87480u1 u2 u33 + 10935u43 + 22464u81 u4 + 209088u61 u2 u4 + 38880u41 u22 u4+233280u21 u32 u4 + 23328u42 u4 + 241056u51 u3 u4 + 272160u31 u2 u3 u4 + 139968u1 u22 u3 u4 + 209952u21 u23 u4+87480u2 u23 u4 + 19440u41 u24 + 349920u21 u2 u24 − 69984u22 u24 + 139968u1 u3 u24 − 17496u34 + 10368u71 u5+38880u51 u2 u5 + 93312u31 u22 u5 − 23328u1 u32 u5 + 134136u41 u3 u5 + 227448u21 u2 u3 u5 + 52488u22 u3 u5+104976u1 u23 u5 + 198288u31 u4 u5 + 104976u1 u2 u4 u5 + 78732u3 u4 u5 + 78732u21 u25 + 26244u2 u25+39366u5 u6 )/39366,
12
10 2
8 3
6 4
4 5
h6 = (109056u14
1 + 433664u1 u2 + 1983744u1 u2 − 400512u1 u2 + 2784768u1 u2 − 282240u1 u2
2 6
7
11
9
7 2
+967680u1 u2 + 207360u2 + 403200u1 u3 + 2395008u1 u2 u3 + 3709440u1 u2 u3 + 6096384u51 u32 u3−846720u31 u42 u3 − 725760u1 u52 u3 + 1611792u81 u23 + 4445280u61 u2 u23 + 3492720u41 u22 u23 + 1360800u21 u32 u23+462672u42 u23 + 1496880u51 u33 + 2857680u31 u2 u33 + 734832u1 u22 u33 + 489888u21 u43 + 91854u2 u43
8
6 2
4 3
2 4
5
+177408u10
1 u4 + 80640u1 u2 u4 + 3499776u1 u2 u4 − 3870720u1 u2 u4 + 2540160u1 u2 u4 − 653184u2 u4
7
5
3 2
3
4 2
+1439424u1 u3 u4 + 7039872u1 u2 u3 u4 + 3991680u1 u2 u3 u4 + 1741824u1 u2 u3 u4 + 2721600u1 u3 u4+1388016u21 u2 u23 u4 + 244944u22 u23 u4 + 857304u1 u33 u4 + 1874880u61 u24 − 3175200u41 u2 u24+5225472u21 u22 u24 + 925344u31 u3 u24 + 2939328u1 u2 u3 u24 + 306180u23 u24 + 1469664u21 u34 − 816480u2 u34−8064u91 u5 + 254016u71 u2 u5 + 217728u51 u22 u5 + 362880u31 u32 u5 + 544320u1 u42 u5 + 707616u61 u3 u5+1632960u41 u2 u3 u5 + 2122848u21 u22 u3 u5 − 326592u32 u3 u5 + 1714608u31 u23 u5 + 1224720u1 u2 u23 u5+183708u33 u5 + 816480u51 u4 u5 + 4572288u31 u2 u4 u5 − 1959552u1 u22 u4 u5 + 2694384u21 u3 u4 u5+979776u2 u3 u4 u5 − 244944u1 u24 u5 + 1102248u41 u25 + 979776u21 u2 u25 + 489888u22 u25 + 734832u1 u3 u25+367416u4 u25 + 137781u26 )/275562.

Let wj = j/7 for j = 1, 2, 3, 4, 5 and w6 = 1. Then each hj is a weighted homogeneouspolynomial of u1 , . . . , u6 if wk is the weight of uk (k = 1, 2, · · · , 6). We shall deﬁne
6 × 6 matrices C and T in the following way. The matrix C = (Cij ) is deﬁned by
10

Cij = ∂ui hj (i, j = 1, 2, · · · 6). At this moment, we remark one of the important properties
of C. Namely the matrices ∂uj C (j = 1, . . . , 6) are commutative each other. For E =∑6
j=1 wj uj ∂uj , we set T = EC. Then T is regarded as a Saito matrix of the polynomial
F = det(T ), where F is a polynomial of u1 , u2 , u3 , u4 , u5 , u6 . It is expected that T is
the Saito matrix of ST34 if we regard {u1 , u2 , u3 , u4 , u5 , u6 } as a set of appropriate basic
invariants of ST34. Actually this is the case as is shown in the theorem below:
Theorem 2 We regard u1 , u2 , · · · , u6 as basic invariants of ST34 which are polynomialsof f1 , f2 , · · · , f6 deﬁned by the following identities:

f1




f2




f

3



f4




 f5


f6
















= k 1 u1 ,= 3k12 (484u21 + 125u2 )/56,= 3k13 (3840232u31 + 1647300u1 u2 + 466785u3 )/3136,= −k14 (10u21 u2 − 6u22 − 3u1 u3 + 3u4 )/175616,= 27k15 (8037231129928u51 + 7056011495960u31 u2 + 923992672120u1 u22 + 4020065168070u21 u3+320350835370u2 u3 + 1029935376420u1 u4 + 170567816805u5 )/9834496,= (41535291386925640512u71 + 55402268263969532360u51 u2 + 17134764422423982880u31 u22+1037814693245737680u1 u32 + 38364019383116279580u41 u3 + 13847169906634165920u21 u2 u3+402156952864771860u22 u3 + 1820695718372802075u1 u23 + 14361697034479016040u31 u4+2535507816645416760u1 u2 u4 + 322231741664151375u3 u4 + 5379672850162662240u21 u5+306822518227402695u2 u5 + 60643659460340565u6 )/120472576,(1)

where k1 = (64/27)1/7 . Then det(T ) is the discriminant of ST34 for the basic invariants
u1 , u 2 , · · · , u 6 .
Proof. We ﬁrst note that by the identities (1), {u1 , u2 , · · · , u6 } is a set of basic invariants of ST34. Hence we see that det(T ) is an invariant of ST34 of degree 252. Since
the action of ST34 on the set of hyperplanes ﬁxed by the pseudo-reﬂections of ST34 aretransitive, det(T ) is irreducible. We denote by ℓi (i =∏1, 2, . . . , 126) the linear forms ﬁxed
2of ST34,by the pseudo-reﬂections of ST34.
the product 126
i=1 ℓi is the discriminant∏ Since∏2
the theorem follows if det(T ) is i ℓi up to a constant factor. Note that i ℓi is a relative
invariant of ST34 of degree 126. Suppose∏ that det(T ) is divisible by x5 − x6 , which is oneof degree 126.∏Because ofof the linear forms ℓi . Then det(T )/ i ℓi is a relative invariant∏ℓcoincideswith i ℓi up to a
Corollary 6.43 at p.231 of [11], we conclude that det(T)/∏ 2 i iconstant factor.
This leads to det(T ) = (constant) i ℓi . Therefore in order to show that∏det(T ) is i ℓ2i up to a constant factor, we only need to show that det(T ) is divisible byx5 − x6 . Moreover, since det(T ) is homogeneous, it is suﬃcient to show that det(T ) = 0
under the condition x5 = x6 = 1.
To accomplish our purpose, we need some preparation. We deﬁne symmetric polynomials rk in x1 , x2 , x3 , x4 by
3j
3j
3j
r3j = x3j
1 + x2 + x3 + x4 (j = 1, 2, 3, 4), r4 = x1 x2 x3 x4 ,andp̃3j = p3j |x5 =x6 =1 (j = 1, 2, 3, 4, 5),s̃6 = s6 |x5 =x6 =1 .

Thenp̃3j = r3j + 2 (j = 1, 2, 3),11s̃6 = r4 ,andp̃12 =
p̃15 =

1(12 + r34 − 6r32 r6 + 3r62 + 8r3 r9 − 24r43 ),6
1(12 + r35 − 5r33 r6 + 5r32 r9 + 5r6 r9 − 30r3 r43 ).
6

Therefore each matrix entry of T̃ = T |x5 =x6 =1 is a polynomial of r3j (j = 1, 2, 3) and r4 .
Using this expression, we conclude det(T̃ ) = 0 by a direct computation with the help ofcomputer softwares.
Remark 2 Kato, Mano and Sekiguchi [7], [8] formulated a generalization of Frobenius
manifold structure and among others they introduced the notion of ﬂat coordinates forwell-generated complex reﬂection groups. As to the group ST34 the second author (J.S.)
constructed the Saito matrix for the ﬂat coordinate. As a consequence, determinant expression of the discriminant of ST34 for the ﬂat coordinate was established (cf. [12]). In
the course of the identiﬁcation of u1 , u2 , · · · , u6 with basic invariants of ST34, we use the
Saito matrix constructed by Bessis and Michel. It is worthwhile to mention the procedureof this identiﬁcation. Noting that the weight of fj is supposed to be wj (= deg fi /42), we
determine the undetermined constants cij and c0 by the conditions
f1
f2
f3
f4
f5
f6

=
=
=
=
=
=

c11 u1 ,c21 u2 + c22 u21 ,c31 u3 + c32 u1 u2 + c33 u31 ,c41 u4 + c42 u3 u1 + c43 u22 + c44 u2 u21 + c45 u41 ,c51 u5 + c52 u4 u1 + c53 u3 u2 + · · · ,c61 u6 + c62 u5 u2 + c63 u5 u21 + · · ·and
F = c0 det(M34 ).
Solving these equations, we determine the constants cij and c0 . The answer is given above(1) which already appeared in [12]. It is underlined here that the proof of Theorem 2 ismainly indebted to (1) and is independent of the result of Bessis and Michel on the explicit
form of the Saito matrix M34 .

6

More about the group ST34

6.1

Minimal vectors of Λ(3) and the reﬂections of ST34

We consider the correspondence between the totality of minimal vectors of Λ (3) and that of
hyperplanes ﬁxed by pseudo-reﬂections of ST34. The group generated by P1 , P2 , Q1 , R1 , R2
is identiﬁed with ST33 (the group numbered as 33 in [14]). From Table VII of [14] theorder of the center of ST33 is 2. The 9-th power of the matrix R2 R1 Q1 P1 P2 given at p.298
of [14] is


−1 0
0
0
0
0
 0 −1 0
0
0
0


0

0
−1
0
0
0


0
0
0 −1 0
0


0
0
0
0
0 −1
0
0
0
0 −1 012

which commutes with all generators of ST33. Hence this matrix is a generator of thecenter of ST33, however, does not commute with the reﬂection P3 . We have that thecenter Z34 of ST34 centralizes ST33, Z34 ∩ ST33 = {1} and |Z34 | = 6. As a consequence
H = ST33 · Z34 is well-deﬁned as a group and |H/ST33| = 6.
There is a natural map between coset spaces
ST34/ST33 −→ ST34/H,
in other words,
6:1

756 minimal vectors of Λ(3) −→ 126 reﬂections of ST34.
We shall construct this map concretely. For an element √13 (θ, −ω b θ, 0, 0, 0, 0) up to Z34 , weassign x1 −ω b x2 = 0 to this vector. We take up √13 (1, ω b , ω c , ω d , ω e , ω f ), b+c+d+e+f ≡ 0(mod 3). Without loss of generality, we may assume 0 ≤ b ≤ c ≤ d ≤ e ≤ f ≤ 2. Under
this condition, we solve b + c + d + e + f ≡ 0 (mod 3). First we have a trivial solution(b, c, d, e, f ) = (0, 0, 0, 0, 0) which corresponds to x1 + x2 + x3 + x4 + x5 + x6 = 0.
Case (b, c, d, e, f ) = (0, 0, 0, 0, f ), f 6= 0: This has no solution.
Case (b, c, d, e, f ) = (0, 0, 0, e, f ), e 6= 0: If e = 2 then f = 2 and this is not the case.
Let e = 1. If f = 2 then this is the case corresponding to x1 +x2 +x3 +x4 +ωx5 +ω 2 x6 = 0.
The case f = 1 does not satisfy the condition.
Case (b, c, d, e, f ) = (0, 0, d, e, f ), d 6= 0: If d = 2, then (e, f ) = (2, 2) satisﬁes thecondition and this corresponds to x1 + x2 + x3 + ω 2 (x4 + x5 + x6 ) = 0. However, this is
the same type as the one we shall see below, that is, the case (b, c, d, e, f ) = (0, 0, 1, 1, 1).
Let d = 1. If e = 2, then f = 2 and this is not the case. Let e = 1. Then f = 2 is not thecase, but the case f = 1 works. This corresponds to x1 + x2 + x3 + ω(x4 + x5 + x6 ) = 0.
Case (b, c, d, e, f ) = (0, c, d, e, f ), c 6= 0: If c = 2, then (d, e, f ) = (2, 2, 2) and this is
not the case. Let c = 1. If d = 2, then (e, f ) = (2, 2) does not satisfy the condition.
(d, e, f ) = (1, 2, 2) is the case and this corresponds to x1 +x2 +ω(x3 +x3 )+ω 2 (x5 +x6 ) = 0.
The cases (e, f ) = (1, 2), (1, 1) do not satisfy the condition.
Case (b, c, d, e, f ), b 6= 0: If b = 2 then (c, d, e, f ) = (2, 2, 2, 2) does not satisfy thecondition. Let b = 1. If c = 2, then (d, e, f ) = (2, 2, 2) satisﬁes the condition and thiscorresponds to x1 + ωx2 + ω 2 (x3 + x4 + x5 + x6 ) = 0. However, this type already appearedin Case (b, c, d, e, f ) = (0, 0, 0, 0, 1, 2) as type x1 + x2 + x3 + x4 + ωx5 + ω 2 x6 = 0. Let
c = 1. If d = 2, then (e, f ) = (2, 2) does not satisfy the condition. Let d = 1. The
case (e, f ) = (2, 2) does not satisfy the condition. The case (e, f ) = (1, 2) satisﬁes thecondition corresponding to x1 + ω(x2 + x3 + x4 + x5 ) + ω 2 x6 = 0, but this case alreadyappeared in Case (b, c, d, e, f ) = (0, 0, 0, 1, 2) as type x1 + x2 + x3 + x4 + ωx5 + ω 2 x6 = 0.
The case (e, f ) = (1, 1) does not satisfy the condition.
We have thus obtained the following proposition.
Proposition 1 There is a natural 6 to 1 correspondence between the minimal vectorsof Λ(3) up to Z34 and the hyperplanes ﬁxed by the pseudo-reﬂections of ST34. Typical13

correspondences are given by(θ, −ω b θ, 0, 0, 0, 0) → x1 − ω b x2 = 0, b ∈ {0, 1, 2},(1, 1, 1, 1, 1, 1) → x1 + x2 + x3 + x4 + x5 + x6 = 0,(1, 1, 1, 1, ω, ω 2 ) → x1 + x2 + x3 + x4 + ωx5 + ω 2 x6 = 0,(1, 1, 1, ω, ω, ω) → x1 + x2 + x3 + ω(x4 + x5 + x6 ) = 0,(1, 1, ω, ω, ω 2 , ω 2 ) → x1 + x2 + ω(x3 + x4 ) + ω 2 (x5 + x6 ) = 0.

6.2

Basic invariants of ST33 and those of ST34

We now mention the relationship between the basic invariants of the group ST33 andthose of ST34. For this purpose, we recall the symmetric polynomials rk in x1 , x2 , x3 , x4
introduced in the proof of Theorem 2. If we let x5 = x6 = y in the deﬁnition of mj (j =1, 2, 3, 4, 5, 7) at §1, then we get polynomials in x1 , x2 , x3 , x4 , y. Since we havep3j |x5 =x6 =y = r3j + 2y 3j (j = 1, 2, 3, 4, 5, 6),s6 |x5 =x6 =y = r4 y 2 ,we can regard mj |x5 =x6 =y as a polynomial in r3 , r4 , r6 , r9 , y. This procedure gives us theinvariants of ST33. First we set
K4 = 3r4 − r3 y + y 4 ,K10 = 21 (r32 r4 − 3r6 r4 + 2r33 y − 8r3 r6 y + 6r9 y + 36r42 y 2 − 8r3 r4 y 3 − 8r32 y 4 + 8r6 y 4 + 64r4 y 6 )andK6k = mk |x5 =x6 =y (k = 1, 2, 3, 4, 5, 7).
Then we get
K6
K12

=
=

K18

=

−5r32 + 6r6 − 180r4 y 2 − 20r3 y 3 − 8y 6 ,
1
4
2
2
3
2
2
2
34 (475r3 − 540r3 r6 + 2349r6 − 2250r3 r9 + 81000r4 + 103950r3 r4 y − 80190r6 r4 y
3 3
3
3
2 4
5
2 6
+30800r3 y − 83160r3 r6 y + 49500r9 y + 1871100r4 y + 415800r3 r4 y + 50820r3 y−33264r6 y 6 + 255420r4 y 8 + 6380r3 y 9 + 3676y 12 ),
1
6
4
2 2
3
3
1822 (−86585r3 + 1859625r3 r6 − 9692055r3 r6 + 3982527r6 − 86580r3 r92
2 3
3+12427020r3 r6 r9 − 8402130r9 − 80904420r3 r4 + 6342300r6 r4 − 167241240r34 r4 y 2+452096640r32 r6 r4 y 2 − 13384440r62 r4 y 2 − 271985040r3 r9 r4 y 2 − 5634849240r44 y 2−28007160r35 y 3 + 97895520r33 r6 y 3 − 45110520r3 r62 y 3 − 54455760r32 r9 y 3 + 29082240r6 r9 y 3−5768197920r3 r43 y 3 − 6947020080r32 r42 y 4 + 4465941480r6 r42 y 4 − 3216213000r33 r4 y 5+7443235800r3 r6 r4 y 5 − 4300536240r9 r4 y 5 − 603515640r34 y 6 + 1734248880r32 r6 y 6−18044208r62 y 6 − 1099360080r3 r9 y 6 − 60989423040r43 y 6 − 27788080320r3 r42 y 7−5927021100r32 r4 y 8 + 4572273420r6 r4 y 8 − 1157156000r33 y 9 + 3124321200r3 r6 y 9−2006741880r9 y 9 − 18856197360r42 y 10 − 2278916640r3 r4 y 11−126792120r32 y 12 + 127646064r6 y 12 − 200564640r4 y 14 − 5051040r3 y 15 − 4823744y 18 ).

We can show that K4 , K6 , K10 , K12 , K18 generate the invariant ring of ST33. It is
underlined here that a set of the generators of the invariant ring of ST33 was determinedby Burkhardt [3]. The invariants of Burkardt are also given in [9] or see the Appendix
of this paper. Moreover, we can see that K24 , K30 , K42 are polynomials of K4 , K6 , K10 ,
K12 , K18 as follows.

14

K24

=

K30

=

K42

=

1
2
2
3
362524562500 (3749217437791K12 − 72606621297375000K10 K4 − 548583360913500K12 K4
62+24202207099125000K4 + 2954928140625K18 K6 − 12101103549562500K10 K4 K6 − 19382588618832K12 K62+548583360913500K43 K62 + 13040967602916K64 ),1
3286134928412500 (398092491836609625000K10 + 1638875315504600K12 K18 − 9023429814963151500K10 K12 K43
52−129489400924462500K18 K4 + 796184983673219250000K10 K4 − 3829737121534827K12 K6
2−100339389337262625000K10
K4 K6 + 25366379163880500K12 K43 K6 + 33446463112420875000K46 K6
2+1580814161617275K18 K6 − 7699801741247286000K10 K42 K62 − 10484943427856196K12 K63+104123021760582000K43 K63 + 11181126000681648K65 ),1327401652548230514062500 (417391800833624595374824312500K10 K12 + 1797658480288030291174075K12 K182
2
2−15459093584438022965939062500K10 K18 K4 − 9460880818895490828496017750K10 K12 K4
3+22245016831722459672845062500000K10
K43 − 252569017560632216590387500K12 K18 K43+1339003983186291610000803375000K10 K12 K45 + 8770901365894143085607812500K18 K463
K6−44490033663444919345690125000000K10 K48 − 6443664189964048153587159K122
2
K12 K4 K6
K6 − 2721541301555381788577062500K10+504384671966073105703125K18
2
K43 K6−2576515597406337160989843750K10 K18 K42 K6 + 800906441502367667511497250K12
2+5561254207930614918211265625000K10
K44 K6 − 20983038899424542341187062500K12 K46 K6
3+1279530628386314942683324125000K10
K62 − 370850790506796659157525K12 K18 K62−19995403641453544837088839500K10 K12 K42 K62 − 299396852886277645255706250K18 K43 K622
K63+2054840875253587466115493500000K10 K45 K62 − 14237902960750897238629098K12
2
3−69323880439964411839202625000K10 K4 K6 + 1329325545599266972946080500K12 K43 K63+41380309308849671453485500000K46 K63 + 63445090000090696420950K18 K64+17448714170095736727621576000K10 K42 K64 + 30208710160434535723891548K12 K65−1578266116654724778611484000K43 K65 − 11514378808918757251753416K67 ).

Appendix
In this Appendix, we give the explicit relationship between the invariants of Burkhardt andour invariants at §6.2. We note that the variables used by Burkhardt are Y0 , Y1 , Y2 , Y3 , Y4
which relate with our variables by xj = −2Yj (j = 1, 2, 3, 4) and y = Y0 . Burkhardt
invariants Jj (Y0 , Y1 , Y2 , Y3 , Y4 ) (j = 4, 6, 10, 12, 18) can be expressed by our Kj ’s as
J4 (Y0 , Y1 , Y2 , Y3 , Y4 ) = K4 ,
1
J6 (Y0 , Y1 , Y2 , Y3 , Y4 ) = − K6 ,
81
J10 (Y0 , Y1 , Y2 , Y3 , Y4 ) =
K10 ,
512
1
(−750K43 − 17K62 + 17K12 ),J12 (Y0 , Y1 , Y2 , Y3 , Y4 ) =3840001
J18 (Y0 , Y1 , Y2 , Y3 , Y4 ) =(140035500K42 K10 − 115027K63 + 137802K6 K12 − 22775K18 ),
9177366528000

respectively.
Remark 3 There are some errors in the expressions of Burkhardt invariants in [3], as
pointed out by, for example, Coble [4, p.350], Hunt [6, p.154], Orlik [10, p.224], NaganoShiga [9, Remark 5.3]. The readers should be careful if they compare the invariants in [3]with the above expressions of the invariants in terms of K4 , K6 , K10 , K12 , K18 .

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15

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