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Complex Reflection Subgroups of Real Reflection Groups
Journal of Algebra 232, 310᎐330 Ž2000. doi:10.1006rjabr.2000.8404, available online at http:rrwww.idealibrary.com on
Complex Reflection Subgroups of Real Reflection Groups J. Patrick Brewer Lebanon Valley College, Ann¨ ille, Pennsyl¨ ania 17003 E-mail: [email protected] Communicated by Michel Broue´ Received July 8, 1999
We find all irreducible rank n complex reflection subgroups of finite irreducible rank 2 n real reflection groups. 䊚 2000 Academic Press Key Words: real reflection groups; complex reflection groups.
1. INTRODUCTION In this paper a triple Ž V, G, W . denotes an n-dimesnional Ž n ) 1. complex vector space V, an irreducible rank n truly complex Žnot complexified. reflection group G in V, and a rank 2 n finite real reflection group W in R V, such that G ; W and W is minimal for G; i.e., there is no real reflection group W X such that G - W X - W. We classify such triples up to isomorphism Žsee Table 4., where two triples Ž V1 , G1 , W1 . and Ž V2 , G 2 , W2 . are isomorphic if there exists a complex linear isomorphism : V1 ª V2 such that G1 y1 s G 2 and W1 y1 s W2 . Hereafter, we will suppress the vector space and write simply G - W when referring to a triple Ž V, G, W .. If W is a real reflection group, ⌽ W denotes its root system and AutŽ ⌽ W . the orthogonal root system automorphisms. In w11x Springer defined regular elements of complex reflection groups and proved that their centralizers are again complex reflection groups. Using that result we show that if V is a finite dimensional real vector space, W is an irreducible reflection group in V , and g AutŽ ⌽ W . with characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n, then CW Ž . is a complex reflection group in V, where 2 d s e i r d and V is the complex vector space defined via Ž1. R V s V and Ž2. 2 d ¨ s Ž ¨ ., ¨ g V ŽTheorem 3.2.. 310 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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On the other hand, if G - W, then W is irreducible ŽProposition 3.1., roots of unity act as orthogonal transformations ŽProposition 3.2., and there exists a d g 2, 3, 4, 54 such that 2 d is an automorphism of ⌽ W ŽCorollary 3.1.. Thus CW Ž 2 d . is a complex reflection group that contains G, so the problem reduces to identifying such centralizers. To do that we rely heavily on the fact that if G - W has reflections of orders d and e, d - e, then d s 2 and e s 4 ŽTheorem 3.1.. Throughout this paper, both real and complex reflections are discussed. To distinguish them, real reflections are denoted by subscripted lowercase esses: s␣ , s , etc., while complex reflections are denoted with greek letters: , .
2. PRELIMINARIES 2.1. Real Reflection Groups w2, 7x are common references for real reflection groups. Here we gather together in tabular form a summary of facts needed later. Table 1 lists the finite irreducible real reflection groups encountered later together with their root systems and orthogonal root system automorphisms. The root systems are described with respect to an orthonormal basis. We will need to know which conjugacy classes in AutŽ ⌽ W . have characteristic polynomials of the form Ž x 2 y 2 cosŽ 2d . x q 1. n, where rankŽW . s 2 n. Table 2 lists those polynomials and the notation used by different authors to refer to the corresponding conjugacy classes. The notations yA32 and yA42 do not appear in w4x, but they refer to the conjugacy classes with representatives yg and yh, where g and h are representatives of A32 and A42 , respectively. 2.2. Complex Reflection Groups Let G be a complex reflection group in a complex vector space V. Then G is called complexified if there is a G-invariant R-subspace V0 of V such that the canonical map C m V0 ª V is bijective. We are concerned only with complex reflection groups that are truly complex, not complexified. The group G is called imprimiti¨ e if V is a direct sum V s V1 [ V2 [ ⭈⭈⭈ [ Vk of nontrivial proper linear subspaces Vi of V, 1 F i F k, such that V1 , V2 , . . . , Vk 4 is invariant under G; G is primiti¨ e if it is not imprimitive. Table 3 lists the truly complex irreducible primitive reflection groups as well as some imprimitive families encountered later. The notation of the
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TABLE 1 Some Irreducible Real Reflection Groups and Their Orthogonal Root System Automorphisms ⌽W
W B2 n
D2 n n) 2
D4
AutŽ ⌽ W .
"e i " e j
1 F i / j F 2n
'2 "e i "e i " e j
1 F i F 2n
'2 "e i " e j
'2 "e i " e j
E6
'2 "
E8
1
Ž e8 y e 7 y e6 q Ý5is1 i e i .
2'2 "e i " e j 1
1Fi/jF4
D4 i S3 s F4
1Fi-jF5
E6 i Z 2
i s "1, Ł 5is 1 i
s y1
1Fi/jF4
"e i 1 Ž . 2 "e1 " e 2 " e 3 " e 4 ŽŽ"1, 0, 0, 0..U
1FiF4
ŽŽ"b, " , " a, 0..
a s cos
ž /
Ž" 12 , " 12 , " 12 , " 12 .
b s cos
ž /
1 2
E8
Ł 8is 1 i s 1
'2
H4
B2 n
i s "1,
Ý8is 1 i e i
2'2 "e i " e j
F4
1 F i / j F 2n
1Fi-jF8
'2 "
B2 n
5 2
F4 i Z 2
H4 ,
5
* The double parentheses indicate all even permutations of the entries.
primitive groups is from w9x. The missing groups G 23 , G 28 , and G 30 are the complexified Coxeter groups H3 , F4 , and H4 , respectively. 3. CLASSIFICATION We begin with a series of simple results which ultimately restrict both G and W: W must be irreducible ŽProposition 3.1. and not isomorphic to A 2 n ŽProposition 3.7.; G may contain reflections of only one order, unless it has order 4 reflections Žmixing theorem.. The main tool used in this
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TABLE 2 Characteristic Polynomials of the Form Ž x 2 y 2 cosŽ 2d . x q 1. n W B2 n D4 D2 n E6 E8 F4
H4
Charpoly Ž x 2 q 1. n Ž x 2 q 1. 2 Ž x 2 q 1. n Ž x 2 y x q 1. 3 Ž x 2 q 1. 4 Ž x 2 y x q 1. 4 Ž x 2 q 1. 2 Ž x 2 q 1. 2 Ž x 2 y x q 1. 2 Ž x 2 y '2 x q 1. 2 Ž x 2 q 1. 2 Ž x 2 y x q 1. 2 Ž x 2 y 2 cosŽ 210 . x q 1. 2
Carter w4x
Grove w6x
Shinoda w10x
B2n D4 Ž a1 . B2n yA32 D4 Ž a1 . 2 yA42 D4 Ž a1 . ᎏ F4 Ž a1 .
ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ
ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ w8 ᎏ
ᎏ ᎏ ᎏ ᎏ
ᎏ C1 = C1 C1 = C4 C1 = C 8
ᎏ ᎏ ᎏ ᎏ
context is the fact that roots of unity act as orthogonal transformations ŽProposition 3.2.. We finish by finding all the complex reflection subgroups of F4 , B4 , D4 , H4 , E6 , E8 , and finally B2 n and D 2 n for n ) 2. 3.1. Tools Throughout, R denotes the set of real numbers and C the set of complex numbers. The symbol n refers to the set 1, 2, . . . , n4 , and G - W is an abbreviation for Ž V, G, W .. PROPOSITION 3.1.
If G - W, then W is irreducible.
Proof. It is enough to prove that if G - W, then G is a complexified Coxeter group or R V is RG-irreducible, so assume R V is RG-reducible and choose a simple RG-submodule V0 of R V. Since V is CG-irreducible, iV0 [ V0 s V and : C mR V0 ª iV0 [ V0 , [ ¨ ¬ ŽImŽ . ¨ , ReŽ . ¨ . is bijective, so G is complexified. PROPOSITION 3.2. mations.
If G - W, then roots of unit act as orthogonal transfor-
Proof. The result follows from the fact that End R G ŽR V . s C. Indeed, assume End R G ŽR V . s C and let Ž , . be a W-invariant inner product on ² , : be a G-invariant unitary inner product on V. Since Ž , . R V, and let and Re² , : are both G-invariant and non-degenerate, there exists a G-equivariant linear map such that Ž x, y . s Re² x, Ž y .:, and since End R G ŽR V . s C, Ž x, y . s Re² x, Ž y .: s ReŽ² x, y :. s ReŽ ² x, y :. for
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J. PATRICK BREWER
TABLE 3 Primitive Complex Reflection Groups and Some Imprimitive Families w5x Number of reflections of order G
RankŽ G .
< ZŽ G .<
2
3
4
5
G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G 20 G 21 G 22 G 24 G 25 G 26 G 27 G 29 G 31 G 32 G 33 G 34 GŽ4, 1, n. GŽ4, 2, n. GŽ m, m, n.
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 5 6 n n n
24 72 48 144 96 192 288 576 48 96 144 288 600 1200 1800 3600 360 720 240 336 648 1296 2160 7680 46080 155520 51,840 39,191,040 4 n n! 2 ⭈ 4 ny 1 n! m ny 1 n!
2 6 4 12 4 8 12 24 2 4 6 12 10 20 30 60 6 12 4 2 3 6 6 4 4 6 2 6 4 2 gcdŽ m, n.
0 0 6 6 6 18 6 18 12 18 12 18 0 30 0 30 0 30 30 21 0 9 45 40 60 0 45 126 nŽ2 n y 1. nŽ2 n y 1.
8 16 8 16 0 0 16 16 0 0 16 16 0 0 40 40 40 40 0 0 24 24 0 0 0 80 0 0 0 0 0
0 0 0 0 12 12 12 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2n 0 0
0 0 0 0 0 0 0 0 0 0 0 0 48 48 48 48 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m Ž n y 1. n
2
some g C. In particular, if is a root of unity, then Ž x, x . s ReŽ ² x, x :. s ReŽ ² x, x :. s ReŽ ² x, x :. s Ž x, x .. To prove End R G ŽR V . s C, put A s End R G ŽR V .. Since G and V are complex, C ; A, so dim C Ž A. G 1. Let g be a reflection and let V s V1 [ V be the decomposition of V into g-eigenspaces. Since V s ¨ ¬ g 2 ¨ y Ž q . g¨ q ¨ s 04 and q g R, AV s V . But the proof of Proposition 3.1 implies that A is an R-division ring, so C ¨ A ¨ End R Ž V . , M2 ŽR..
COMPLEX REFLECTION SUBGROUPS
315
If dim C Ž A. s 2 then A , M2 ŽR.. Since M2 ŽR. is not a division ring, dim C Ž A. s 1. Remark 3.1. Proposition 3.2 strongly links the reflections of G to certain rank 2 root subsystems of ⌽ W : the eigenspaces of nontrivial eigenvalues contain rank 2 root subsystems of ⌽ W . More specifically, let g G - W be an order d reflection and 2 d s e i r d. Choose ␣ ,  g ⌽ W such that s s␣ s w4, Lemma 2x. Then the angles between ␣ and " are d ⭈ n and y d ⭈ n, where n is an integer relatively prime to d, so the possible rank two root subsystems containing ␣ and  are limited. Indeed, L Žthe -eigenspace corresponding to the nontrivial eigenvalue. is a complex subspace, so ␣⌽ [ ␣ , 2 d ␣ , . . . , 2Ž2ddy1.␣ 4 ; L , and since 2 d acts orthogonally, ␣⌽ ; ⌽ W , so ␣ ⌽ is a rank 2 root subsystem of ⌽ W . Thus, since dimŽ V . ) 1, dimŽW . ) 2, so d g 2, 3, 4, 54 . When d / 2, ␣⌽ s ⌽ W l L and is determined by : ␣ ⌽
s ⌽ W l L
¡⌽ ¢⌽
A2 ,
if d s 3
B2 ,
if d s 4
H2 ,
if d s 5.
s~⌽
If d s 2, then ␣⌽ s ⌽A 1=A 1, but ⌽ W l L may be ⌽A 1=A 1 of ⌽B 2 . If ⌽ W l L s ⌽A 1=A 1, then ␣ ⌽ again depends only on , but if ⌽ W l L s ⌽B 2 , it is possible that s␣ s s s s␥ s␦ and ␣ ⌽ , ␥ ⌽ ; ⌽ W l L , but ␣⌽ l␥ ⌽ s ⭋. PROPOSITION 3.3. Let G - W, 2 d s e i r d, ⌽d s ␣ g ⌽ W ¬ 2 d ␣ g ⌽ W 4 , and Wd s ² s␣ ¬ ␣ g ⌽d :. Then ⌽d is a root subsystem of ⌽w . Proof. Put ⌽d s w Ž  . ¬ w g Wd ,  g ⌽d 4 . Then ⌽d is a root system for Wd that contains ⌽d . In fact ⌽d s ⌽d . To see this, let ␣ g ⌽d . Then 2 d ␣ and ␣ span a rank 2 root subsystem, ⌽ 2 . Since 2 d acts orthogonally, 2 d Ž 2 d ␣ . g ⌽ 2 , so 2 d ⌽d s ⌽d . Then 2 d Wd y1 2 d s Wd which implies 2 d ⌽d s ⌽d , so ⌽d ; ⌽d . The following theorem is vital. It implies that if G - W, then all the reflections in G have the same order, unless G has order 4 reflections, in which case G has order 4 and order 2 reflections. THEOREM 3.1 Žmixing theorem.. If G - W has reflections of orders d and e, d - e, then d s 2 and e s 4. Proof. The theorem follows from the rank 2 case. Indeed, assume the rank 2 case is true and that rankŽ G . ) 2. Let GŽ d . and GŽ e . denote the subgroups of G generated by its order d and order e reflections, respec-
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tively. Since GŽ d . and GŽ e . are normal, they have full rank in G, so not all the roots of reflections in GŽ d . can be orthogonal to all those of GŽ e .. Choose reflections g GŽ d ., g GŽ e . corresponding to non-orthogonal roots. If GŽ d . and GŽ e . commute, is a reflection whose order must be 2, 3, 4, or 5, so d s 2 and e s 4. If and do not commute, then ² , : is an irreducible rank 2 reflection group, so d s 2 and e s 4 by assumption. Assume G has rank 2. If G is imprimitive, then G s GŽ m, p, 2., so let V1 , V2 4 be an imprimitivity system and s s␥ si␥ a reflection which interchanges V1 and V2 . Put ⌽X s V1 l ⌽ W , ⌽Y s V2 l ⌽ W . Then ⌽X and ⌽Y are Žisomorphic. irreducible rank 2 root subsystems. Choose ␣ g ⌽X such that Ž ␣ , ␥ . / 0 / Ž i ␣ , ␥ . and put  s Ž ␣ .. Then B [ Ž ␣ , i ␣ ,  , i  . is a real ordered basis for V. Let x s Ž␥ , ␣ . and y s Ž␥ , i ␣ .. Since Ž ␣ . s  and Ž␥ . s y␥ , ␥ s Ž x, y, yx, yy . B , so x 2 q y 2 s 12 . Since ␣ , ␥ 4 is contained in a root system, x, y g " 12 , " 1r '2 , " cosŽ 5 ., "cosŽ 25 .4 . But <Ž ␣ , ␥ .< 2 q <Ž  , ␥ .< 2 s 12 , so x, y g " 12 4 . Let ␥ 1 be the projection of ␥ along V1. Then 5 ␥ 1 5 s 1r '2 , and for ␦ g ⌽X , Ž ␦ , ␥ . s Ž ␣ , ␥ 1 .. If Ž ␦ , ␥ . / 0 / Ž i ␦ , ␥ ., then <Ž ␦ , ␥ 1 .< s 12 , so
C ␣ l ⌽W
¡⌽ s~⌽ ¢⌽
A2 ,
if e s 3
B2 ,
if e s 4
H2 ,
if e s 5,
so the theorem follows. If ⌽e / ⌽ W , then 2 e g AutŽ ⌽e ._W. Moreover, ⌽e is an irreducible rank 4 root system that is properly contained in another irreducible rank 4 root system. The possible pairs Ž ⌽e , ⌽ W . are Ž ⌽D , ⌽F ., Ž ⌽D , ⌽B ., Ž ⌽D , ⌽H ., Ž ⌽B , ⌽F ., and Ž ⌽A , ⌽H .. Now, Ž ⌽D , 4 4 4 4 4 4 4 4 4 4 4 ⌽F4 ., Ž ⌽B4 , ⌽F4 ., and Ž ⌽A 4 , ⌽H 4 . are ruled out since AutŽ ⌽D 4 ., AutŽ ⌽B4 . ; F4 , and AutŽ ⌽A 4 . ; H4 . Before considering Ž ⌽D 4 , ⌽B4 . and Ž ⌽D 4 , ⌽H 4 . individually, note that neither 4 nor 5 is an angle between roots of ⌽D 4 , so if ⌽e s ⌽D 4 , then e s 3 and d s 2.
COMPLEX REFLECTION SUBGROUPS
317
Assume Ž ⌽ 3 , ⌽ W . s Ž ⌽D 4 , ⌽B4 .. Since ⌽ 2 has rank 4 and ⌽ 2 l ⌽ 3 s ⭋, ⌽ 2 s ⌽A 1=A 1=A 1=A , so GŽ2. ; W2 s A1 = A1 = A1 = A1 is Abelian and therefore reducible. But GŽ2. reducible implies that G is imprimitive, a contradiction. Assume Ž ⌽ 3 , ⌽ W . s Ž ⌽D 4 , ⌽H 4 .. Let k be the order of 6 g AutŽ ⌽D 4 .rD4 , S3 . Then 6k g D4 and has characteristic polynomial Ž x 2 y2 cosŽ 26 k . x q 1. 2 . For k s 1 or k s 2, D4 does not have an element with the required characteristic polynomial, so k s 3. But order 3 outer automorphisms of ⌽D 4 ; ⌽H 4 are contained in a subgroup H3 F H4 , a contradiction. To see this let ⌫ be the Coxeter diagram ⌽D 4 ; ⌽H 4 . Since ⌽H 4 is a single root orbit, we may choose the middle vertex of ⌫ to be e4 , so an order 3 graph automorphism must fix e4 and permute "e1 , " e2 , " e34 . Now, ␣ g ⌽H 4 ¬ Ž ␣ , e4 . s 04 is a root system of type H3 , and the corresponding reflection group, H3 , contains all order 3 permutations of "e1 , " e2 , " e34 . Indeed, since s e1, s e 2 , and s e 3 are in H3 , it is enough to show that e1 ª e2 ª e3 ª e1 is in H3 . There are four elements in H3 which send e1 to e2 . Since these elements fix e4 , they must permute "e1 , " e2 , " e34 , so multiplying by members of s e1, s e 2 , s e 3 4 as necessary, there are two possibilities: e1 ª e 2 ª e1 ,
e3 ª e3
and
e1 ª e 2 ª e 3 ª e1 .
The first case would imply that s e1ye 2 g H4 , a contradiction, so e1 ª e2 ª e3 ª e1 is in H3 . COROLLARY 3.1. If G - W, then there exists d g 2, 3, 4, 54 such that ⌽ W s ⌽d . Therefore, if 2 d denotes the R-linear map ¨ ¬ e i r d ¨ , then 2 d g AutŽ ⌽ W .. Proof. Let G - W. If G has solely order d reflections, then G - Wd , so since W is minimal, W s Wd , and ⌽ W s ⌽d . If G has an order 4 reflection, then ⌽4 : ⌽ 2 , so G - W2 and ⌽ W s ⌽ 2 . Remark 3.2. Springer defined regular elements in complex reflection groups and proved that their centralizers are again complex reflection groups w11, Theorem 4.2x. Below, the fact that 2 d g AutŽ ⌽ W . Ž d g 2, 3, 4, 54 , as above. is used to prove that 1 m 2 d is a regular element in the complex reflection group 1 m W so that C1mW Ž1 m 2 d . is a complex reflection group. It then follows that CW Ž 2 d . is a complex reflection group, which reduces the problem to identifying such centralizers and determining their subgroups. LEMMA 3.1. Let W be an irreducible rank 2 n Ž n ) 1. real reflection group and d g 2, 3, 4, 54 . If g AutŽ ⌽ W . has characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n, then is regular.
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J. PATRICK BREWER
Proof. If W s E6 , E8 , F4 , or H4 , the result follows from w11, Sects. 5 and 6x via case-by-case inspection. If W s A 2 n , the result follows from the proof of Proposition 3.7 below. For W s B2 n , note that its elements have characteristic polynomials of the form ŁŽ x n i " 1., where ⌺n i s 2 n, so g AutŽ B2 n . s B2 n implies d s 2, in which case is regular w11, Sects. 5 and 6x. Finally, the case W s D 2 n follows from the above arguments since AutŽ ⌽D 4 . s F4 and AutŽ ⌽D 2 n . s B2 n for n ) 2. THEOREM 3.2. Let V be a 2 n-dimensional real ¨ ector space Ž n ) 1., let W be an irreducible rank 2 n reflection group in V , and let g AutŽ ⌽ W . with characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n. Define a complex ¨ ector space V ¨ ia Ž1. R V s V and Ž2. 2 d ¨ s Ž ¨ ., where 2 d s e i r d. Then CW Ž . is a complex reflection group in V. Proof. Apply Springer’s theorem w11, Theorem 4.2x to the complex reflection group 1 m W in U [ C mR V. By the above lemma, 1 m is regular, so C1mW Ž1 m . is a complex reflection group in U , where U s U m U is the decomposition of U into 1 m eigenspaces w11, Theorem 4.2x. If 1 m g C1mW Ž1 m . is a complex reflection in U , then charpolyR V Ž . s charpolyU Ž1 m . s charpolyU˜Ž1 m .charpolyUŽ1 m . s Ž x y 1. ny 1 Ž x y .Ž x y 1. ny1 Ž x y ., where 2 n s dim R V and is the non-trivial eigenvalue of 1 m . Thus, fixes a complex hyperplane, and so is a complex reflection. The next four propositions are more specific than some previous results but help streamline the procedure for finding complex reflection subgroups. The first two concern the case when G is imprimitive, the third restricts W when G is primitive and has order four reflections, and the fourth states that A 2 n has no complex reflection subgroups of rank n. PROPOSITION 3.4. If GŽ m, p, n. - W, then W has an element with characteristic polynomial Ž o¨ er R. equal to Ž x 2 y 2 cosŽ 2m . x q 1. 2 Ž x y 1. 2 ny4 , and if m / p, then m g 4, 84 . Proof. The group G [ GŽ m, p, n. is subgroup of both GLnŽC. and W, so if g G is an order d reflection, it may be written as a diagonal matrix, s dŽ a1 , a2 , . . . , a n ., where one a i is a primitive dth root of unity and the others are 1, but since g W, it is also a product of real reflections in W, s s␣ s w4, Lemma 2x, where ␣ ,  g ⌽ W and Ž ␣ ,  . s "cosŽ d .. Let Vi ¬ i g n x be an imprimitivity system for G, and let m be a . primitive mth root of unity. Then [ dŽ m , y1 m , 1, . . . , 1 g G, and 2 2 2 2 ny4 charpoly R Ž . s Ž x y 2 cosŽ m . x q 1. Ž x y 1. , which proves the first statement. For the second, assume m / p and note that since G has order 2 reflections, the mixing theorem implies ⌽ W s ⌽ 2 . But G also has
COMPLEX REFLECTION SUBGROUPS
319
reflections of order mp , so mp s 2 if G has only order 2 reflections, while m p s 4 if G has order 4 reflections. In any case, m is even, so [ dŽy1, 1, . . . , 1. g G. Choose ␣ ,  g ⌽ W such that s s␣ s . Then Ž ␣ ,  . s 0, so ⌽ W l V1 s ⌽A 1=A 1 or ⌽ W l V1 s ⌽B 2 . Since g W, Ž ⌽ W . s ⌽ W , and Ž V1 . s mV1 s V1 , so Ž ⌽ W l V1 . s mŽ ⌽ W l V1 . s ⌽ W l V1. Therefore, since m acts orthogonally, m g 2, 4, 84 . But GŽ2, 1, n. and GŽ2, 2, n. are complexified Coxeter groups, so m / 2. Thus, m g 4, 84 . Proposition 3.5 is used below to limit the possible complex subgroups of H4 , E6 , and E8 . PROPOSITION 3.5. Let G s GŽ4, 2, n. and assume W is a real reflection group whose Coxeter diagram has no double bond. If G - W, then W s D 2 n . Proof. Remark 3.1 describes how roots are associated with complex reflections. The set of all such roots corresponding to the reflections in G forms the root system ⌽D 2 n, so the proposition follows from the minimality of W. Indeed, let Vk ¬ k g n x be an imprimitivity system for G. There are two types of reflections in Gᎏthose that act diagonally on Vk ¬ k g n4 and those that interchange a pair of subspaces in Vk ¬ k g n4 . But since the Coxeter diagram of W has no double bond and all the reflections in G have order 2, the root subsystems associated with them are type A1 = A1. To identify these roots choose an R-basis from the root system as follows: For each k g n, put ⌽ k s Vk l ⌽ W , and choose ␣ k g ⌽ k ; then ⌽ k s "␣ k , " i ␣ k 4 . Fix i, j g n and let g G be a reflection that interchanges Vi and Vj . Choose ␥ g ⌽ W such that s s␥ si␥ , and let Ž a1 , a2 , . . . , a2 n . be the coordinates of ␥ with respect to the ordered R-basis Ž ␣ 1 , i ␣ 1 , ␣ 2 , i ␣ 2 , . . . , ␣ n , i ␣ n .. The coordinates of ␥ are inner products of roots, for example, a1 s Ž␥ , ␣ ., and the Coxeter diagram of W has no double bond, so the a i are in 0, " 12 4 . Since fixes Vk , k / i, j, there are at most four non-zero coordinates for ␥ , which must be " 21 , so there are at most 16 such roots. In fact there are exactly 16, since there are four reflections that interchange Vi and Vj , each corresponding to four roots. Thus, the coordinates of the roots corresponding to reflections that interchange a pair of subspaces in Vk ¬ k g n4 have exactly four nonzero entries taken from " 12 4 . For example, the coordinates for the roots corresponding to the reflections that interchange V1 and V2 are Ž" 21 , " 21 , " 21 , " 21 , 0, . . . , 0.. The coordinates of the roots corresponding to the diagonal reflections are Žy1, y1, 0, . . . , 0., Ž0, 0, y1, y1, 0, . . . , 0., . . . , Ž0, . . . , 0, y1, y1.. Define a new ordered R-basis e i : i g 2 n4 via e2 ky1 s Ž ␣ k q i ␣ k .r2 and e2 k s Ž ␣ k y i ␣ k .r2, for k g n. With respect to this basis the roots corresponding to the reflections in G are "e i " e j : 1 F i, j F 2 n4 .
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J. PATRICK BREWER
Proposition 3.6 restricts possible primitive subgroups for it implies that if G - W is primitive, then both 4 and 8 are in AutŽ ⌽ W .. PROPOSITION 3.6. ⌽4 s ⌽ 2 s ⌽ W .
If G - W is primiti¨ e and has order 4 reflections, then
Proof. The last equality follows from the minimality of W. To prove the first, note that no primitive group of rank greater than 2 has an order 4 reflection, so assume G has rank 2. The mixing theorem implies that G has only order 2 and order 4 reflections; there are only two such groups: G 8 and G 9 ŽTable 3.. Since all its order 2 reflections are squares of order 4 reflections, G 8 satisfies the theorem. Now, G 9 has 18 order 2 reflections, and each corresponds to a root subsystem of type A1 = A1 or B2 , so ⌽ W must have at least 18 ⭈ 4 s 72 roots. But the largest rank 4 root system whose Coxeter diagram has a double bond is ⌽F4 , which has only 48 roots. PROPOSITION 3.7.
A 2 n has no complex reflection subgroups of rank n.
Proof. If G - A 2 n , then there exists a d g 2, 3, 4, 54 such that ⌽d s ⌽A 2 n, so 2 d g AutŽ ⌽A 2 n . s A 2 n i "14 . Elements of A 2 n have characteristic polynomials of the form Ž x 1 y 1.Ž x 2 y 1. ⭈⭈⭈ Ž x k y 1.rŽ x y 1., 1 q 2 q ⭈⭈⭈ q k s 2 n q 1, and charpoly R Ž 2 d . s Ž x 2 y 2 cosŽ d . x q 1. n. But Ž x y 1. divides all the x i y 1, while 1 is not a root of x 2 y 2 cosŽ d . x q 1, so k s 1; thus 2 d f A 2 n . The elements in AutŽ ⌽A 2 n ._ A 2 n can be expressed as y1V ⭈ , where y1V : V ª V, ¨ ¬ y¨ , and g A 2 n . Since 2 n is even, charpoly R Žy1V ⭈ . s pŽyx ., where pŽ x . s charpoly R Ž ., so the above argument applies again; thus 2 d f AutŽ ⌽A 2 n .. 3.2. Classification Now we describe a general method for finding complex and real reflection groups G and W that satisfy G - W. Remark 3.1 implies that the possible orders of reflections in any complex reflection subgroup of W are determined by the rank 2 root subsystems of ⌽ W : ⌽A 1=A 1 l order 2 ⌽A 2 l order 3 ⌽H 2 l order 5 ⌽B 2 l order 2 or 4.
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The mixing theorem states that if G - W, then all the reflections in G have the same order or that G has reflections of orders 2 and 4. Moreover, Corollary 3.1 says that if G has reflections of a single order d, then ⌽ W s ⌽d , while ⌽ W s ⌽ 2 otherwise, so if G - W, there exists d g 2, 3, 4, 54 such that 2 d g AutŽ ⌽ W . and G - CW Ž 2 d .. Since Theorem 3.2 states that centralizers of the form CW Ž 2 d . are complex reflection groups, to find the complex reflection subgroups of W it is enough to identify such centralizers and their reflection subgroups. To identify a centralizer CW Ž 2 d . we will usually try to decide: Ž1. whether CW Ž 2 d . is primitive or imprimitive, and Ž2. how many order d reflections it contains. For Ž1., note that if ⌽ W s ⌽d , d / 2, then CW Ž 2 d . contains only order d reflections so must be primitive, since imprimitive groups have order 2 reflections. If ⌽ W s ⌽ 2 , then CW Ž 2 d . has order 2 reflections but may or may not have order 4 reflections and may or may not be primitive. For Ž2., recall that an order d reflection s s␣ s corresponds to a rank 2 root subsystem ␣ ⌽ of ⌽ W . In fact , 2 , . . . , dy 1 all correspond to ␣⌽ . Moreover, any two order d reflections correspond to the same rank 2 root subsystem or to disjoint subsystems since the subsystems are contained in the eigenspaces for the nontrivial eigenvalues of the respective reflections. Therefore, Ž< ⌽ W
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1. 2 ŽTable 2., so the complex reflection group CF4Ž 6 . is unique up to isomorphism. Since CF4Ž 6 . has Ž< ⌽
The action of 4 partitions ⌽ into 12 A1 = A1 root subsystems of the form ␣ , 4 ␣ , 42 ␣ , 43 ␣ 4 , so CF4Ž 4 . has at most 12 order 2 reflections. There are less than 12 exactly when pairs of those subsystems together form B2 subsystems. Since orthogonal roots in a B2 root subsystem of F4 are in the same F4 orbit, C Ž 4o . has 12 order two reflections and C Ž 4i . has 6. Thus, C Ž 4o . , G12 ŽTable 3.. To identify C Ž 4i . note that the action of 8 is given by an element from the conjugacy class in AutŽ ⌽ . with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , and since AutŽ ⌽ . s F4 i Z 2 , we may and shall assume that 82 s 4i . Since CW Ž 8 . has 488 ⭈ 2 s 12 order 4 reflections, it is isomorphic to G 8 ŽTable 3., and CW Ž 8 . ; CW Ž 82 . s CW Ž 4i ., so CW Ž 4i . s CW Ž 8 . , G 8 . Neither G12 nor G 8 have primitive reflection subgroups ŽTable 3., and since F4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , Proposition 3.4 implies that the only possible imprimitive subgroups of G 8 or G12 are GŽ4, 1, 2. and GŽ4, 2, 2.. Since Ž 0i 0i . g GŽ4, 2, 2. F GŽ4, 1, 2. and < ZŽ G12 .< s 2, G12 has no imprimitive reflection subgroups. On the other hand, the 2-Sylow subgroup of G 8 is isomorphic to GŽ4, 1, 2., but F4 is not minimal for GŽ4, 1, 2.. To see this, let H be the subgroup of G 8 generated by its order 2 reflections. Then H is an irreducible rank 2 complex reflection group whose 6 reflections all have order 2, so H is imprimitive ŽTable 3., and since H has only order 2 reflections H s GŽ4, 2, 2.. Now H is a 2-group, so it is contained in a 2-Sylow subgroup P2 of G 8 , and since G 8 has 12 order four reflections and three 2-Sylow subgroups, each 2-Sylow subgroup must contain at least 4 order four reflections, so P2 , GŽ4, 1, 2.. Let i s s␣ i s i , i s 1, 2, . . . , 10,
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be the reflections in GŽ4, 1, 2., where 1 , 2 , 3 , 4 have order 4 and 5 , 6 are squares of order 4 reflections. Then 7 , 8 , 9 , 10 are conjugate in GŽ4, 1, 2., and since GŽ4, 1, 2. - CF4Ž 4 . , G 8 , where 4 acts via an inner automorphism of ⌽, ␣ i and i are in the same F4 root orbit for i G 7, so s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 . The same is true for 5 and 6 , so the roots ␣ 5 , 5 , . . . , ␣ 10 ,  10 can be chosen from the same F4 root orbit, and therefore, the roots ␣ i , i , i s 1, 2, . . . , 10, can be chosen so that D10 is1 i⌽ i is a root system of type B4 or C4 , so F4 is not minimal for GŽ4, 1, 2.. Remark 3.3. In the last paragraph we saw that GŽ4, 1, 2. ; B4 l C4 since i s s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 for i G 5. In particular, both B4 and C4 are minimal for GŽ4, 1, 2. - F4 . Something similar occurs for GŽ4, 2, 2. - F4 . Let 1 , . . . , 6 be the reflections in GŽ4, 2, 2. and 1 , 2 4 , 3 , 4 , 5 , 6 4 the conjugacy classes. Choose ␣ 1 ,  1 , . . . , ␣ 6 , 6 such that i s s␣ i s i . Since 3 , 4 , 5 , 6 4 are conjugate, we may assume that ␣ 3 ,  3 , . . . , ␣ 6 , 6 4 are in the same F4 root orbit; similarly, ␣ 1 ,  1 , ␣ 2 ,  2 4 are in the same orbit. Since each F4 root orbit is a D4 root subsystem, s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 , for i G 1. Thus, we may choose the ␣ i and i so that D6is1 ␣ i⌽ i is either of the F4 root orbits; so if D4 and D4X are the groups corresponding to the orbits, then GŽ4, 2, 2. ; D4 l D4X . B4 Let G - B4 . Possible orders of reflections in G are 2, 3, and 4, so ⌽ s ⌽ 2 or ⌽ s ⌽ 3 . ⌽ s ⌽ 3 . Here, 6 g AutŽ ⌽ ., but B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 2 , so there are no complex reflection subgroups in this case. 䢇
⌽ s ⌽ 2 . There is exactly one conjugacy class in B4 with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , so the complex reflection group CB4Ž 4 . is unique. Since D4 e B4 and has only one conjugacy class with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , 4 g D4 . 䢇
Suppose CB4Ž 4 . is primitive. If CB4Ž 4 . has an order 4 reflection, then Proposition 3.6 implies ⌽ s ⌽ 2 s ⌽4 , so 8 g AutŽ ⌽ .. But B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 . On the other hand, if CB4Ž 4 . has solely order 2 reflections, then CB4Ž 4 . has at most eight reflections since there are only 324 s 8 4-orbits on ⌽, but there is no such primitive group ŽTable 3..
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Therefore, CB4Ž 4 . is imprimitive, and since B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , CB4Ž 4 . s GŽ4, 1, 2. or CB4Ž 4 . s GŽ4, 2, 2.. Direct computation shows that < CB4Ž 4 .< s 32, so CB4Ž 4 . s GŽ4, 1, 2.. Now, GŽ4, 2, 2. - GŽ4, 1, 2., but GŽ4, 2, 2. D4 . To see this let i s s␣ i s i be the six order 2 reflections in GŽ4, 2, 2. divided into conjugacy classes 1 , 2 , 3 , 4 4 and 5 , 6 4 . Since the corresponding reflections are conjugate, the 16 roots in D4is1 ␣ i⌽ i can be chosen from a single Žlong. B4-orbit. Assume i ) 4. If ␣ i and i are short, then i s s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 . Therefore, the ␣ i and i can be chosen so that the 24 roots of D6is1 ␣ i⌽ i are contained in a single B4-orbit and hence form a D4 root subsystem. D4 Possible orders of reflections in G - D4 are 2 and 3, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . There is only one conjugacy class in AutŽ ⌽ . s F4 with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 3 , so the complex reflection group CD 4Ž 6 . is unique. Since CD 4Ž 6 . has 246 ⭈ 2 s 8 order 3 reflections, CD 4Ž 6 . , G4 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . The complex reflection group CD 4Ž 4 . is unique since there is only one conjugacy class in F4 with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 . Since there are 244 s 6 4-orbits on ⌽, there are at most six order 2 reflections, so CD 4Ž 4 . is not primitive ŽTable 3.. Therefore, CD 4Ž 4 . is imprimitive, and since there can be no order 4 reflections, CD 4Ž 4 . s GŽ4, 2, 2.. The only reflection subgroup of GŽ4, 2, 2. is GŽ4, 4, 2., and it is unique since it is the subgroup of GŽ4, 2, 2. generated by the non-diagonal reflections. 䢇
䢇
H4 Let G - H4 . Possible orders of reflections in G are 2, 3, and 5, so ⌽ s ⌽ 2 , ⌽ s ⌽ 3 , or ⌽ s ⌽5 . Moreover, AutŽ ⌽H 4 . s H4 w1, Appendixx and each conjugacy class in H4 has a distinct characteristic polynomial, so the complex reflection groups CH 4Ž 2 d ., d g 2, 3, 54 are unique. ⌽ s ⌽5 . Since CH 4Ž 10 . has 120 10 ⭈ 4 s 48 order 5 reflections, it is isomorphic to G16 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 3 . Since CH 4Ž 6 . has 120 6 ⭈ 2 s 40 order 3 reflections, it is isomorphic to G 20 ; the only possible subgroups are G4 and G5 . To find 䢇
䢇
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them note that Z [ ZŽ G 20 . has order 6 and that G 20rA , A5 Žthe simple group of order 60. w5x. Moreover, : G 20 ª A5 induces a bijective correspondence between the 3-Sylow subgroups of G 20 and A5 . Choose P1 and P2 , 3-Sylow subgroups of A5 , so that y1 Ž² P1 , P2 :. , G5 . Since G4 - G5 , existence of both G4 and G5 in CH 4Ž 6 . is proven; next consider uniqueness. Let 3 be an order 3 element of Z. Then G5 s G4 = ² 3 :, so G5 and G4 have the same 2-Sylow subgroups; a counting argument shows that G4 has only one. Let P2 be the 2-Sylow subgroup contained in G4 l G5 . Since w G 20 : G5 x s 5, NG Ž P2 . s G5 or NG Ž P2 . s G 20 , but P2 eG 20 would im20 20 ply that G 20 rP2 and hence G 20 is solvable, which is not possible since G 20 has a factor isomorphic to A5 . Therefore, G5 s NG 20Ž P2 ., so there is a unique isomorphism class of G5 in G 20 . More about G4 follows in Remark 3.4 below; for now note that G4 - D4 - H4 . ⌽ s ⌽ 2 . Since the Coxeter diagram of H4 has no double bond, CH 4Ž 4 . has 120 4 s 30 order 2 reflections and no order 4 reflections, so it is isomorphic to G 22 . The only other irreducible, rank 2, primitive, complex reflection groups generated by order 2 reflections are G12 and G13 , but < G13 < does not divide < G 22 <, so consider G12 . Suppose G12 - G 22 . Then they share at least one 2-Sylow subgroup, P2 . Since G 22 is irreducible, its center, Z, contains only scalars. But Z has order 4, so Z - P2 , which contradicts the fact that G12 has a center of order 2. Thus, G 22 has no primitive subgroups. In fact G 22 has no imprimitive subgroups for which H4 is minimal. 䢇
As above, let P2 be a 2-Sylow subgroup of G 22 . A counting argument Žusing the fact that G 22 rZŽ G 22 . , A5 . shows that there are at most five 2-Sylow subgroups of G 22 , so each must contain at least 305 s 6 order 2 reflections. Since the subgroup of P2 generated by its reflections must be imprimitive ŽTable 3., it follows that P2 , GŽ4, 2, 2., so by Proposition 3.5, it is contained in D4 . Remark 3.4. Ž1. The reflections in G5 account for 48 roots. Since < ⌽F < s 48, it is not surprising that G5 - F4 , but < ⌽H < s 120, and ⌽H has 4 4 4 no root subsystem containing 48 roots. Something similar happens with GŽ4, 4, n. in D 2 n and G 29 in E8 . Ž2. Assume G4 - H4 and let P2 be the unique 2-Sylow subgroup of G4 . Since G4 - NG 20Ž P2 . s G5 - G 20 , it is enough to consider G 20-conjugacy classes of G4 in G5 . From the discussion of G4 in F4 , there are at least two distinct copies of G4 in G5 . To see that there are at most two, let G4 and G4X be distinct subgroups of G5 ; then G4 l G4X s P2 , so they share
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no reflections. Since G4 has 8 reflections and there are only 16 in G5 , there can be at most two copies of G4 in G5 . Those two copies of G4 are not conjugate in G 20 . Let G4 , G4X - G5 with G4 / G4X . Choose a reflection g G4 _G4X . Since G4 = ² 3 : s G4 s G4X = ² 3 :, there exist g X g G4X and X3 g ² 3 : such that s g XX3 . Since g X has order 3, it is a reflection with detŽ g X . / detŽ .. Suppose G4 and G4X are conjugate by an element of G 20 . Since reflections in G4X of a given determinant are conjugate Žin G4 . there exists x g G 20 such that x xy1 s Ž g X . 2 . Multiplying x by a suitable element from ² 3 :, if necessary, detŽ x . s 1. Choose a basis so that is represented as
ž
3 0
0 . 1
/
Then corresponding to Ž g X . 2 and x are
ž
1 0
0 3
/
and
ž
0 ycy1
c . 0
/
In particular, x has order 4. Since xG4 xy1 s G4X , xP2 xy1 s P2 , so x g NG 20Ž P2 . s G5 . But G5 s G4 = ² 3 :, so x g G4 , a contradiction. E6 Possible orders of reflections in G - E6 are 2 or 3, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . There is only one conjugacy class in AutŽ ⌽ . with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 2 , so the complex reflection group CE 6Ž 6 . is unique. Since CE 6Ž 6 . has 726 ⭈ 2 s 24 order 3 reflections, CE 6Ž 6 . , G 25 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . There are no complex subgroups in this case because AutŽ ⌽ . does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 3. 䢇
䢇
E8 Possible orders of reflections in G - E8 are 2 and 3, so ⌽ s ⌽ 2 or ⌽ s ⌽ 3 . Moreover, AutŽ ⌽ . s E8 and there is only one conjugacy class in E8 corresponding to the characteristic polynomials Ž x 2 y 2 cosŽ 24 . x q 1. 4 and Ž x 2 y 2 cosŽ 26 . x q 1. 4 , so the complex reflection groups CE 8Ž 4 . and CE 8Ž 6 . are unique.
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⌽ s ⌽ 3 . The complex reflection group CE 8Ž 6 . has 240 6 ⭈ 2 s 80 order 3 reflections, so it is isomorphic to G 32 , and it has no reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . Since CE 8Ž 4 . has 240 4 s 60 order 2 reflections, it is isomorphic to G 31 , and G 29 is its only primitive subgroup w5, p. 409x. There are at most two isomorphism classes of G 29 in G 31. To see this choose generators 1 , 2 , 3 , 4 for G 29 corresponding to the presentation given in w3x. Choose ␣ i , i g ⌽ such that i s s␣ i s i , and construct a graph ⌫ with vertices in one-to-one correspondence with ␣ 1 ,  1 , ␣ 2 ,  2 , ␣ 3 ,  3 , ␣ 4 , 4 4 by joining two vertices with an edge if the corresponding roots are not orthogonal; label the edge with the inner product of the roots w4x. If the inner product is " 12 , the edge label is often omitted. Note that since i has order 2, Ž ␣ i , i . s 0. Since the Coxeter diagram of E8 has no double bond, the resulting graph is unique and is displayed in Fig. 1. 䢇
䢇
The numbered nodes correspond to a D6 root subsystem and there is only one up to conjugacy in E8 , so let e6 y e5 , e5 y e4 , e4 y e3 , e3 y e2 , e2 y e1 , e1 q e2 correspond to nodes 1, 2, 3, 4, 5, and 6, respectively. Then the constraints of the diagram require that up to sign change there are only four roots that can correspond to node a: ye4 " e 7 and ye4 " e8 . Similarly, node b may correspond to 12 Žye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., where s s "1. Since the roots corresponding to nodes a and b are orthogonal, there are four possibilities: Žye4 q e 7 , ye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., Žye4 q e8 , ye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., Žye4 y e 7 , ye1 y e2 q e3 q e4 y e5 y e6 y e 7 y e8 ., Žye4 y e 7 , ye1 y e2 q e3 q e4 y e5 y e6 y e 7 y e8 .4 . These pairs of roots are all conjugate in E4 by elements which fix the chosen D6 root subsystem. Thus, there is a unique D 8-conjugacy class of G 29 in G 31. Assume G 29 , GX29 - G 31 Žs CE 8Ž 4 ... Put Z s ZŽ G 31 . s 1, 4 , 42 , 434 and choose g E8 such that G 29 y1 s GX29 . Since Z ; G 29 l GX29 , 4 y1 s 4 or 4 y1 s 43, there are at most two isomorphism classes of G 29 in G 31. In fact using a computer algebra system, it is possible to check that there is only one isomorphism class of G 29 in G 31. Finally, G 31 has no imprimitive subgroups for which E8 is minimal, for suppose GŽ m, p, 4. - G 31 - E8 . Then GŽ m, m, 4. - G 31 since GŽ m, m, 4. F GŽ m, p, 4.. But < GŽ m, m, 4.< s 3Ž2 m. 3, while < G 31 < s 2 10 ⭈ 3 ⭈ 5, so m F 4. Thus the only possibilities are GŽ4, 1, 4., GŽ4, 2, 4., and GŽ4, 4, 4.. Since
FIG. 1. Carter graph for G 29 .
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G 31 has no order 4 reflections, GŽ4, 1, 4. is eliminated, and by Proposition 3.5, GŽ4, 4, 4. - GŽ4, 2, 4. - D 8 . Remark 3.5. In w8x, Shephard described G 31 as the symmetry group of a four-dimensional complex polytope that has 240 vertices, and he listed the following coordinates for the vertices:
Ž 2 i a , 0, 0, 0 . Ž Ž 1 q i . i a , Ž 1 q i . i b , 0, 0 . Ž ia, ib, ic, id .
permuted, Ž a s 0, 1, 2, 3 . permuted, Ž a, b s 0, 1, 2, 3 .
Ž a q b q c q d s 0 Ž mod 2. . .
If Shephard’s complex basis is denoted e1 , e2 , e3 , and e4 , then with respect to the real ordered basis 1 s Ž1 q i . e1 , 2 s Ž1 y i . e1 , . . . , 7 s Ž1 q i . e4 , 8 s Ž1 y i . e4 , the coordinates are " i " j , i - j and 12 Ý " i Ževen number of plus signs., which is the usual realization of the E8 root system. B2 n Possible orders of reflections in G - B2 n are 2, 3, and 4, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . The elements in B2 n have characteristic polynomials of the form ŁŽ x n i " 1., where ⌺n i s 2 n, so 6 f AutŽ ⌽ . s B2 n . ⌽ s ⌽ 2 . There is one conjugacy class in B2 n with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , so the complex reflection group CB 2 nŽ 4 . is unique. 䢇
䢇
Since every order two reflection requires four roots from ⌽, the rank 3, 4, and 5 primitive reflection groups have too many order 2 reflections to be contained in B2 n . Thus, CB 2 nŽ 4 . is imprimitive, so from w11, Theorem 4.2x, it follows that CB 2 nŽ 4 . s GŽ4, 1, n.. Now GŽ4, 2, n. is the subgroup of GŽ4, 1, n. generated by the order 2 reflections and GŽ4, 4, n. is the subgroup generated by the non-diagonal order 2 reflections, so there is only one isomorphism class of each. The argument used to show that GŽ4, 2, 2. - D4 can also be used here to prove that GŽ4, 4, n. - GŽ4, 2, n. - D 2 n . D2 n Since there is one conjugacy class in D 2 n with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. n, the complex reflection group CD Ž 4 . is unique, 2n and it follows from the B2 n discussion that CD 2 nŽ 4 . s GŽ4, 2, n. and that the subgroup GŽ4, 4, n. is unique.
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TABLE 4 Complex Reflection Subgroups of Real Reflection Groups W F4
B4 D4
H4
G-W G12 , CF4Ž 4o . G 8 , CF4Ž 4i . G5 , CF4Ž 6o . GŽ4, 1, 2. , CB4Ž 4 . GŽ4, 2, 2. , CD 4Ž 4 . GŽ4, 4, 2. - GŽ4, 2, 2. G4 , CD 4Ž 6 . G 22 , CH 4Ž 4 . G 20 , CH Ž 6 . 4
E6 E8
B2 n D2 n
G5 - G 20 G16 , CH 4Ž 10 . G 25 , CE 6Ž 6 . G 31 , CE 8Ž 4 . G 29 - G 31 G 32 , CE 8Ž 6 . GŽ4, 1, n. , CB 2 nŽ 4 . GŽ4, 2, n. , CD 2 nŽ 6 . GŽ4, 4, n. - GŽ4, 2, n.
3.3. Summary Table 4 lists the rank 2 n Ž n ) 1. irreducible real reflection groups with their truly complex irreducible rank n reflection subgroups. When appropriate, the complex reflection groups are expressed as centralizers.
ACKNOWLEDGMENTS Much of this material first appeared in a dissertation written under the supervision of J. N. Spaltenstein at the University of Oregon. I gratefully acknowledge his support and encouragement. I also thank C. Curtis who suggested that w11x would be relevant, and I thank the referee, whose suggestions made this a better paper.
REFERENCES 1. E. Bannai, Automorphisms of Irreducible Weyl Groups, J. Fac. Sci. Uni¨ . Tokyo 16 Ž1969., 273᎐286.
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2. N. Bourbaki, ‘‘Groupes et Algebres de Lie,’’ Chaps. IV, V, VI, Hermann, Paris, 1968. ` 3. M. Broue, ´ G. Malle, and R. Roquier, On complex reflection groups and their associated braid groups, ‘‘Representations of Groups ŽBanff, AB, 1994.,’’ CMS Conf. Proc. 16, pp. 1᎐13, Am. Math. Soc., Providence, RI, 1995. 4. R. Carter, Conjugacy classes in the Weyl groups, Compositio Math. 25 Ž1972., 1᎐59. ´ 5. A. Cohen, Finite complex reflection groups, Ann. Sci. Ecole Norm. Sup Ž 4 . Ž1972., 1᎐59. 6. L. Grove, The characters of the hecatonicosahedroidal group, J. Reine Angew. Math. 265 Ž1974., 160᎐169. 7. J. Humphreys, ‘‘Reflection Groups and Coxeter Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 8. G. Shephard, Unitary groups generated by reflections, Canad. J. Math. 5 Ž1953., 364᎐383. 9. G. Shephard and J. Todd, Finite unitary reflection groups, Canad. J. Math. 6 Ž1954., 274᎐304. 10. K. Shinoda, The conjugacy classes of the finite Ree groups of type F4 , J. Fac. Sci. Uni¨ . Tokyo 22 Ž1975., 1᎐15. 11. T. Springer, Regular elements of finite reflection groups, In¨ ent. Math. 25 Ž1974., 159᎐198.
Complex Reflection Subgroups of Real Reflection Groups J. Patrick Brewer Lebanon Valley College, Ann¨ ille, Pennsyl¨ ania 17003 E-mail: [email protected] Communicated by Michel Broue´ Received July 8, 1999
We find all irreducible rank n complex reflection subgroups of finite irreducible rank 2 n real reflection groups. 䊚 2000 Academic Press Key Words: real reflection groups; complex reflection groups.
1. INTRODUCTION In this paper a triple Ž V, G, W . denotes an n-dimesnional Ž n ) 1. complex vector space V, an irreducible rank n truly complex Žnot complexified. reflection group G in V, and a rank 2 n finite real reflection group W in R V, such that G ; W and W is minimal for G; i.e., there is no real reflection group W X such that G - W X - W. We classify such triples up to isomorphism Žsee Table 4., where two triples Ž V1 , G1 , W1 . and Ž V2 , G 2 , W2 . are isomorphic if there exists a complex linear isomorphism : V1 ª V2 such that G1 y1 s G 2 and W1 y1 s W2 . Hereafter, we will suppress the vector space and write simply G - W when referring to a triple Ž V, G, W .. If W is a real reflection group, ⌽ W denotes its root system and AutŽ ⌽ W . the orthogonal root system automorphisms. In w11x Springer defined regular elements of complex reflection groups and proved that their centralizers are again complex reflection groups. Using that result we show that if V is a finite dimensional real vector space, W is an irreducible reflection group in V , and g AutŽ ⌽ W . with characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n, then CW Ž . is a complex reflection group in V, where 2 d s e i r d and V is the complex vector space defined via Ž1. R V s V and Ž2. 2 d ¨ s Ž ¨ ., ¨ g V ŽTheorem 3.2.. 310 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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On the other hand, if G - W, then W is irreducible ŽProposition 3.1., roots of unity act as orthogonal transformations ŽProposition 3.2., and there exists a d g 2, 3, 4, 54 such that 2 d is an automorphism of ⌽ W ŽCorollary 3.1.. Thus CW Ž 2 d . is a complex reflection group that contains G, so the problem reduces to identifying such centralizers. To do that we rely heavily on the fact that if G - W has reflections of orders d and e, d - e, then d s 2 and e s 4 ŽTheorem 3.1.. Throughout this paper, both real and complex reflections are discussed. To distinguish them, real reflections are denoted by subscripted lowercase esses: s␣ , s , etc., while complex reflections are denoted with greek letters: , .
2. PRELIMINARIES 2.1. Real Reflection Groups w2, 7x are common references for real reflection groups. Here we gather together in tabular form a summary of facts needed later. Table 1 lists the finite irreducible real reflection groups encountered later together with their root systems and orthogonal root system automorphisms. The root systems are described with respect to an orthonormal basis. We will need to know which conjugacy classes in AutŽ ⌽ W . have characteristic polynomials of the form Ž x 2 y 2 cosŽ 2d . x q 1. n, where rankŽW . s 2 n. Table 2 lists those polynomials and the notation used by different authors to refer to the corresponding conjugacy classes. The notations yA32 and yA42 do not appear in w4x, but they refer to the conjugacy classes with representatives yg and yh, where g and h are representatives of A32 and A42 , respectively. 2.2. Complex Reflection Groups Let G be a complex reflection group in a complex vector space V. Then G is called complexified if there is a G-invariant R-subspace V0 of V such that the canonical map C m V0 ª V is bijective. We are concerned only with complex reflection groups that are truly complex, not complexified. The group G is called imprimiti¨ e if V is a direct sum V s V1 [ V2 [ ⭈⭈⭈ [ Vk of nontrivial proper linear subspaces Vi of V, 1 F i F k, such that V1 , V2 , . . . , Vk 4 is invariant under G; G is primiti¨ e if it is not imprimitive. Table 3 lists the truly complex irreducible primitive reflection groups as well as some imprimitive families encountered later. The notation of the
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TABLE 1 Some Irreducible Real Reflection Groups and Their Orthogonal Root System Automorphisms ⌽W
W B2 n
D2 n n) 2
D4
AutŽ ⌽ W .
"e i " e j
1 F i / j F 2n
'2 "e i "e i " e j
1 F i F 2n
'2 "e i " e j
'2 "e i " e j
E6
'2 "
E8
1
Ž e8 y e 7 y e6 q Ý5is1 i e i .
2'2 "e i " e j 1
1Fi/jF4
D4 i S3 s F4
1Fi-jF5
E6 i Z 2
i s "1, Ł 5is 1 i
s y1
1Fi/jF4
"e i 1 Ž . 2 "e1 " e 2 " e 3 " e 4 ŽŽ"1, 0, 0, 0..U
1FiF4
ŽŽ"b, " , " a, 0..
a s cos
ž /
Ž" 12 , " 12 , " 12 , " 12 .
b s cos
ž /
1 2
E8
Ł 8is 1 i s 1
'2
H4
B2 n
i s "1,
Ý8is 1 i e i
2'2 "e i " e j
F4
1 F i / j F 2n
1Fi-jF8
'2 "
B2 n
5 2
F4 i Z 2
H4 ,
5
* The double parentheses indicate all even permutations of the entries.
primitive groups is from w9x. The missing groups G 23 , G 28 , and G 30 are the complexified Coxeter groups H3 , F4 , and H4 , respectively. 3. CLASSIFICATION We begin with a series of simple results which ultimately restrict both G and W: W must be irreducible ŽProposition 3.1. and not isomorphic to A 2 n ŽProposition 3.7.; G may contain reflections of only one order, unless it has order 4 reflections Žmixing theorem.. The main tool used in this
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TABLE 2 Characteristic Polynomials of the Form Ž x 2 y 2 cosŽ 2d . x q 1. n W B2 n D4 D2 n E6 E8 F4
H4
Charpoly Ž x 2 q 1. n Ž x 2 q 1. 2 Ž x 2 q 1. n Ž x 2 y x q 1. 3 Ž x 2 q 1. 4 Ž x 2 y x q 1. 4 Ž x 2 q 1. 2 Ž x 2 q 1. 2 Ž x 2 y x q 1. 2 Ž x 2 y '2 x q 1. 2 Ž x 2 q 1. 2 Ž x 2 y x q 1. 2 Ž x 2 y 2 cosŽ 210 . x q 1. 2
Carter w4x
Grove w6x
Shinoda w10x
B2n D4 Ž a1 . B2n yA32 D4 Ž a1 . 2 yA42 D4 Ž a1 . ᎏ F4 Ž a1 .
ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ
ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ w8 ᎏ
ᎏ ᎏ ᎏ ᎏ
ᎏ C1 = C1 C1 = C4 C1 = C 8
ᎏ ᎏ ᎏ ᎏ
context is the fact that roots of unity act as orthogonal transformations ŽProposition 3.2.. We finish by finding all the complex reflection subgroups of F4 , B4 , D4 , H4 , E6 , E8 , and finally B2 n and D 2 n for n ) 2. 3.1. Tools Throughout, R denotes the set of real numbers and C the set of complex numbers. The symbol n refers to the set 1, 2, . . . , n4 , and G - W is an abbreviation for Ž V, G, W .. PROPOSITION 3.1.
If G - W, then W is irreducible.
Proof. It is enough to prove that if G - W, then G is a complexified Coxeter group or R V is RG-irreducible, so assume R V is RG-reducible and choose a simple RG-submodule V0 of R V. Since V is CG-irreducible, iV0 [ V0 s V and : C mR V0 ª iV0 [ V0 , [ ¨ ¬ ŽImŽ . ¨ , ReŽ . ¨ . is bijective, so G is complexified. PROPOSITION 3.2. mations.
If G - W, then roots of unit act as orthogonal transfor-
Proof. The result follows from the fact that End R G ŽR V . s C. Indeed, assume End R G ŽR V . s C and let Ž , . be a W-invariant inner product on ² , : be a G-invariant unitary inner product on V. Since Ž , . R V, and let and Re² , : are both G-invariant and non-degenerate, there exists a G-equivariant linear map such that Ž x, y . s Re² x, Ž y .:, and since End R G ŽR V . s C, Ž x, y . s Re² x, Ž y .: s ReŽ² x, y :. s ReŽ ² x, y :. for
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TABLE 3 Primitive Complex Reflection Groups and Some Imprimitive Families w5x Number of reflections of order G
RankŽ G .
< ZŽ G .<
2
3
4
5
G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G 20 G 21 G 22 G 24 G 25 G 26 G 27 G 29 G 31 G 32 G 33 G 34 GŽ4, 1, n. GŽ4, 2, n. GŽ m, m, n.
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 5 6 n n n
24 72 48 144 96 192 288 576 48 96 144 288 600 1200 1800 3600 360 720 240 336 648 1296 2160 7680 46080 155520 51,840 39,191,040 4 n n! 2 ⭈ 4 ny 1 n! m ny 1 n!
2 6 4 12 4 8 12 24 2 4 6 12 10 20 30 60 6 12 4 2 3 6 6 4 4 6 2 6 4 2 gcdŽ m, n.
0 0 6 6 6 18 6 18 12 18 12 18 0 30 0 30 0 30 30 21 0 9 45 40 60 0 45 126 nŽ2 n y 1. nŽ2 n y 1.
8 16 8 16 0 0 16 16 0 0 16 16 0 0 40 40 40 40 0 0 24 24 0 0 0 80 0 0 0 0 0
0 0 0 0 12 12 12 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2n 0 0
0 0 0 0 0 0 0 0 0 0 0 0 48 48 48 48 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m Ž n y 1. n
2
some g C. In particular, if is a root of unity, then Ž x, x . s ReŽ ² x, x :. s ReŽ ² x, x :. s ReŽ ² x, x :. s Ž x, x .. To prove End R G ŽR V . s C, put A s End R G ŽR V .. Since G and V are complex, C ; A, so dim C Ž A. G 1. Let g be a reflection and let V s V1 [ V be the decomposition of V into g-eigenspaces. Since V s ¨ ¬ g 2 ¨ y Ž q . g¨ q ¨ s 04 and q g R, AV s V . But the proof of Proposition 3.1 implies that A is an R-division ring, so C ¨ A ¨ End R Ž V . , M2 ŽR..
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315
If dim C Ž A. s 2 then A , M2 ŽR.. Since M2 ŽR. is not a division ring, dim C Ž A. s 1. Remark 3.1. Proposition 3.2 strongly links the reflections of G to certain rank 2 root subsystems of ⌽ W : the eigenspaces of nontrivial eigenvalues contain rank 2 root subsystems of ⌽ W . More specifically, let g G - W be an order d reflection and 2 d s e i r d. Choose ␣ ,  g ⌽ W such that s s␣ s w4, Lemma 2x. Then the angles between ␣ and " are d ⭈ n and y d ⭈ n, where n is an integer relatively prime to d, so the possible rank two root subsystems containing ␣ and  are limited. Indeed, L Žthe -eigenspace corresponding to the nontrivial eigenvalue. is a complex subspace, so ␣⌽ [ ␣ , 2 d ␣ , . . . , 2Ž2ddy1.␣ 4 ; L , and since 2 d acts orthogonally, ␣⌽ ; ⌽ W , so ␣ ⌽ is a rank 2 root subsystem of ⌽ W . Thus, since dimŽ V . ) 1, dimŽW . ) 2, so d g 2, 3, 4, 54 . When d / 2, ␣⌽ s ⌽ W l L and is determined by : ␣ ⌽
s ⌽ W l L
¡⌽ ¢⌽
A2 ,
if d s 3
B2 ,
if d s 4
H2 ,
if d s 5.
s~⌽
If d s 2, then ␣⌽ s ⌽A 1=A 1, but ⌽ W l L may be ⌽A 1=A 1 of ⌽B 2 . If ⌽ W l L s ⌽A 1=A 1, then ␣ ⌽ again depends only on , but if ⌽ W l L s ⌽B 2 , it is possible that s␣ s s s s␥ s␦ and ␣ ⌽ , ␥ ⌽ ; ⌽ W l L , but ␣⌽ l␥ ⌽ s ⭋. PROPOSITION 3.3. Let G - W, 2 d s e i r d, ⌽d s ␣ g ⌽ W ¬ 2 d ␣ g ⌽ W 4 , and Wd s ² s␣ ¬ ␣ g ⌽d :. Then ⌽d is a root subsystem of ⌽w . Proof. Put ⌽d s w Ž  . ¬ w g Wd ,  g ⌽d 4 . Then ⌽d is a root system for Wd that contains ⌽d . In fact ⌽d s ⌽d . To see this, let ␣ g ⌽d . Then 2 d ␣ and ␣ span a rank 2 root subsystem, ⌽ 2 . Since 2 d acts orthogonally, 2 d Ž 2 d ␣ . g ⌽ 2 , so 2 d ⌽d s ⌽d . Then 2 d Wd y1 2 d s Wd which implies 2 d ⌽d s ⌽d , so ⌽d ; ⌽d . The following theorem is vital. It implies that if G - W, then all the reflections in G have the same order, unless G has order 4 reflections, in which case G has order 4 and order 2 reflections. THEOREM 3.1 Žmixing theorem.. If G - W has reflections of orders d and e, d - e, then d s 2 and e s 4. Proof. The theorem follows from the rank 2 case. Indeed, assume the rank 2 case is true and that rankŽ G . ) 2. Let GŽ d . and GŽ e . denote the subgroups of G generated by its order d and order e reflections, respec-
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tively. Since GŽ d . and GŽ e . are normal, they have full rank in G, so not all the roots of reflections in GŽ d . can be orthogonal to all those of GŽ e .. Choose reflections g GŽ d ., g GŽ e . corresponding to non-orthogonal roots. If GŽ d . and GŽ e . commute, is a reflection whose order must be 2, 3, 4, or 5, so d s 2 and e s 4. If and do not commute, then ² , : is an irreducible rank 2 reflection group, so d s 2 and e s 4 by assumption. Assume G has rank 2. If G is imprimitive, then G s GŽ m, p, 2., so let V1 , V2 4 be an imprimitivity system and s s␥ si␥ a reflection which interchanges V1 and V2 . Put ⌽X s V1 l ⌽ W , ⌽Y s V2 l ⌽ W . Then ⌽X and ⌽Y are Žisomorphic. irreducible rank 2 root subsystems. Choose ␣ g ⌽X such that Ž ␣ , ␥ . / 0 / Ž i ␣ , ␥ . and put  s Ž ␣ .. Then B [ Ž ␣ , i ␣ ,  , i  . is a real ordered basis for V. Let x s Ž␥ , ␣ . and y s Ž␥ , i ␣ .. Since Ž ␣ . s  and Ž␥ . s y␥ , ␥ s Ž x, y, yx, yy . B , so x 2 q y 2 s 12 . Since ␣ , ␥ 4 is contained in a root system, x, y g " 12 , " 1r '2 , " cosŽ 5 ., "cosŽ 25 .4 . But <Ž ␣ , ␥ .< 2 q <Ž  , ␥ .< 2 s 12 , so x, y g " 12 4 . Let ␥ 1 be the projection of ␥ along V1. Then 5 ␥ 1 5 s 1r '2 , and for ␦ g ⌽X , Ž ␦ , ␥ . s Ž ␣ , ␥ 1 .. If Ž ␦ , ␥ . / 0 / Ž i ␦ , ␥ ., then <Ž ␦ , ␥ 1 .< s 12 , so
C ␣ l ⌽W
¡⌽ s~⌽ ¢⌽
A2 ,
if e s 3
B2 ,
if e s 4
H2 ,
if e s 5,
so the theorem follows. If ⌽e / ⌽ W , then 2 e g AutŽ ⌽e ._W. Moreover, ⌽e is an irreducible rank 4 root system that is properly contained in another irreducible rank 4 root system. The possible pairs Ž ⌽e , ⌽ W . are Ž ⌽D , ⌽F ., Ž ⌽D , ⌽B ., Ž ⌽D , ⌽H ., Ž ⌽B , ⌽F ., and Ž ⌽A , ⌽H .. Now, Ž ⌽D , 4 4 4 4 4 4 4 4 4 4 4 ⌽F4 ., Ž ⌽B4 , ⌽F4 ., and Ž ⌽A 4 , ⌽H 4 . are ruled out since AutŽ ⌽D 4 ., AutŽ ⌽B4 . ; F4 , and AutŽ ⌽A 4 . ; H4 . Before considering Ž ⌽D 4 , ⌽B4 . and Ž ⌽D 4 , ⌽H 4 . individually, note that neither 4 nor 5 is an angle between roots of ⌽D 4 , so if ⌽e s ⌽D 4 , then e s 3 and d s 2.
COMPLEX REFLECTION SUBGROUPS
317
Assume Ž ⌽ 3 , ⌽ W . s Ž ⌽D 4 , ⌽B4 .. Since ⌽ 2 has rank 4 and ⌽ 2 l ⌽ 3 s ⭋, ⌽ 2 s ⌽A 1=A 1=A 1=A , so GŽ2. ; W2 s A1 = A1 = A1 = A1 is Abelian and therefore reducible. But GŽ2. reducible implies that G is imprimitive, a contradiction. Assume Ž ⌽ 3 , ⌽ W . s Ž ⌽D 4 , ⌽H 4 .. Let k be the order of 6 g AutŽ ⌽D 4 .rD4 , S3 . Then 6k g D4 and has characteristic polynomial Ž x 2 y2 cosŽ 26 k . x q 1. 2 . For k s 1 or k s 2, D4 does not have an element with the required characteristic polynomial, so k s 3. But order 3 outer automorphisms of ⌽D 4 ; ⌽H 4 are contained in a subgroup H3 F H4 , a contradiction. To see this let ⌫ be the Coxeter diagram ⌽D 4 ; ⌽H 4 . Since ⌽H 4 is a single root orbit, we may choose the middle vertex of ⌫ to be e4 , so an order 3 graph automorphism must fix e4 and permute "e1 , " e2 , " e34 . Now, ␣ g ⌽H 4 ¬ Ž ␣ , e4 . s 04 is a root system of type H3 , and the corresponding reflection group, H3 , contains all order 3 permutations of "e1 , " e2 , " e34 . Indeed, since s e1, s e 2 , and s e 3 are in H3 , it is enough to show that e1 ª e2 ª e3 ª e1 is in H3 . There are four elements in H3 which send e1 to e2 . Since these elements fix e4 , they must permute "e1 , " e2 , " e34 , so multiplying by members of s e1, s e 2 , s e 3 4 as necessary, there are two possibilities: e1 ª e 2 ª e1 ,
e3 ª e3
and
e1 ª e 2 ª e 3 ª e1 .
The first case would imply that s e1ye 2 g H4 , a contradiction, so e1 ª e2 ª e3 ª e1 is in H3 . COROLLARY 3.1. If G - W, then there exists d g 2, 3, 4, 54 such that ⌽ W s ⌽d . Therefore, if 2 d denotes the R-linear map ¨ ¬ e i r d ¨ , then 2 d g AutŽ ⌽ W .. Proof. Let G - W. If G has solely order d reflections, then G - Wd , so since W is minimal, W s Wd , and ⌽ W s ⌽d . If G has an order 4 reflection, then ⌽4 : ⌽ 2 , so G - W2 and ⌽ W s ⌽ 2 . Remark 3.2. Springer defined regular elements in complex reflection groups and proved that their centralizers are again complex reflection groups w11, Theorem 4.2x. Below, the fact that 2 d g AutŽ ⌽ W . Ž d g 2, 3, 4, 54 , as above. is used to prove that 1 m 2 d is a regular element in the complex reflection group 1 m W so that C1mW Ž1 m 2 d . is a complex reflection group. It then follows that CW Ž 2 d . is a complex reflection group, which reduces the problem to identifying such centralizers and determining their subgroups. LEMMA 3.1. Let W be an irreducible rank 2 n Ž n ) 1. real reflection group and d g 2, 3, 4, 54 . If g AutŽ ⌽ W . has characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n, then is regular.
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Proof. If W s E6 , E8 , F4 , or H4 , the result follows from w11, Sects. 5 and 6x via case-by-case inspection. If W s A 2 n , the result follows from the proof of Proposition 3.7 below. For W s B2 n , note that its elements have characteristic polynomials of the form ŁŽ x n i " 1., where ⌺n i s 2 n, so g AutŽ B2 n . s B2 n implies d s 2, in which case is regular w11, Sects. 5 and 6x. Finally, the case W s D 2 n follows from the above arguments since AutŽ ⌽D 4 . s F4 and AutŽ ⌽D 2 n . s B2 n for n ) 2. THEOREM 3.2. Let V be a 2 n-dimensional real ¨ ector space Ž n ) 1., let W be an irreducible rank 2 n reflection group in V , and let g AutŽ ⌽ W . with characteristic polynomial Ž x 2 y 2 cosŽ d . x q 1. n. Define a complex ¨ ector space V ¨ ia Ž1. R V s V and Ž2. 2 d ¨ s Ž ¨ ., where 2 d s e i r d. Then CW Ž . is a complex reflection group in V. Proof. Apply Springer’s theorem w11, Theorem 4.2x to the complex reflection group 1 m W in U [ C mR V. By the above lemma, 1 m is regular, so C1mW Ž1 m . is a complex reflection group in U , where U s U m U is the decomposition of U into 1 m eigenspaces w11, Theorem 4.2x. If 1 m g C1mW Ž1 m . is a complex reflection in U , then charpolyR V Ž . s charpolyU Ž1 m . s charpolyU˜Ž1 m .charpolyUŽ1 m . s Ž x y 1. ny 1 Ž x y .Ž x y 1. ny1 Ž x y ., where 2 n s dim R V and is the non-trivial eigenvalue of 1 m . Thus, fixes a complex hyperplane, and so is a complex reflection. The next four propositions are more specific than some previous results but help streamline the procedure for finding complex reflection subgroups. The first two concern the case when G is imprimitive, the third restricts W when G is primitive and has order four reflections, and the fourth states that A 2 n has no complex reflection subgroups of rank n. PROPOSITION 3.4. If GŽ m, p, n. - W, then W has an element with characteristic polynomial Ž o¨ er R. equal to Ž x 2 y 2 cosŽ 2m . x q 1. 2 Ž x y 1. 2 ny4 , and if m / p, then m g 4, 84 . Proof. The group G [ GŽ m, p, n. is subgroup of both GLnŽC. and W, so if g G is an order d reflection, it may be written as a diagonal matrix, s dŽ a1 , a2 , . . . , a n ., where one a i is a primitive dth root of unity and the others are 1, but since g W, it is also a product of real reflections in W, s s␣ s w4, Lemma 2x, where ␣ ,  g ⌽ W and Ž ␣ ,  . s "cosŽ d .. Let Vi ¬ i g n x be an imprimitivity system for G, and let m be a . primitive mth root of unity. Then [ dŽ m , y1 m , 1, . . . , 1 g G, and 2 2 2 2 ny4 charpoly R Ž . s Ž x y 2 cosŽ m . x q 1. Ž x y 1. , which proves the first statement. For the second, assume m / p and note that since G has order 2 reflections, the mixing theorem implies ⌽ W s ⌽ 2 . But G also has
COMPLEX REFLECTION SUBGROUPS
319
reflections of order mp , so mp s 2 if G has only order 2 reflections, while m p s 4 if G has order 4 reflections. In any case, m is even, so [ dŽy1, 1, . . . , 1. g G. Choose ␣ ,  g ⌽ W such that s s␣ s . Then Ž ␣ ,  . s 0, so ⌽ W l V1 s ⌽A 1=A 1 or ⌽ W l V1 s ⌽B 2 . Since g W, Ž ⌽ W . s ⌽ W , and Ž V1 . s mV1 s V1 , so Ž ⌽ W l V1 . s mŽ ⌽ W l V1 . s ⌽ W l V1. Therefore, since m acts orthogonally, m g 2, 4, 84 . But GŽ2, 1, n. and GŽ2, 2, n. are complexified Coxeter groups, so m / 2. Thus, m g 4, 84 . Proposition 3.5 is used below to limit the possible complex subgroups of H4 , E6 , and E8 . PROPOSITION 3.5. Let G s GŽ4, 2, n. and assume W is a real reflection group whose Coxeter diagram has no double bond. If G - W, then W s D 2 n . Proof. Remark 3.1 describes how roots are associated with complex reflections. The set of all such roots corresponding to the reflections in G forms the root system ⌽D 2 n, so the proposition follows from the minimality of W. Indeed, let Vk ¬ k g n x be an imprimitivity system for G. There are two types of reflections in Gᎏthose that act diagonally on Vk ¬ k g n4 and those that interchange a pair of subspaces in Vk ¬ k g n4 . But since the Coxeter diagram of W has no double bond and all the reflections in G have order 2, the root subsystems associated with them are type A1 = A1. To identify these roots choose an R-basis from the root system as follows: For each k g n, put ⌽ k s Vk l ⌽ W , and choose ␣ k g ⌽ k ; then ⌽ k s "␣ k , " i ␣ k 4 . Fix i, j g n and let g G be a reflection that interchanges Vi and Vj . Choose ␥ g ⌽ W such that s s␥ si␥ , and let Ž a1 , a2 , . . . , a2 n . be the coordinates of ␥ with respect to the ordered R-basis Ž ␣ 1 , i ␣ 1 , ␣ 2 , i ␣ 2 , . . . , ␣ n , i ␣ n .. The coordinates of ␥ are inner products of roots, for example, a1 s Ž␥ , ␣ ., and the Coxeter diagram of W has no double bond, so the a i are in 0, " 12 4 . Since fixes Vk , k / i, j, there are at most four non-zero coordinates for ␥ , which must be " 21 , so there are at most 16 such roots. In fact there are exactly 16, since there are four reflections that interchange Vi and Vj , each corresponding to four roots. Thus, the coordinates of the roots corresponding to reflections that interchange a pair of subspaces in Vk ¬ k g n4 have exactly four nonzero entries taken from " 12 4 . For example, the coordinates for the roots corresponding to the reflections that interchange V1 and V2 are Ž" 21 , " 21 , " 21 , " 21 , 0, . . . , 0.. The coordinates of the roots corresponding to the diagonal reflections are Žy1, y1, 0, . . . , 0., Ž0, 0, y1, y1, 0, . . . , 0., . . . , Ž0, . . . , 0, y1, y1.. Define a new ordered R-basis e i : i g 2 n4 via e2 ky1 s Ž ␣ k q i ␣ k .r2 and e2 k s Ž ␣ k y i ␣ k .r2, for k g n. With respect to this basis the roots corresponding to the reflections in G are "e i " e j : 1 F i, j F 2 n4 .
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Proposition 3.6 restricts possible primitive subgroups for it implies that if G - W is primitive, then both 4 and 8 are in AutŽ ⌽ W .. PROPOSITION 3.6. ⌽4 s ⌽ 2 s ⌽ W .
If G - W is primiti¨ e and has order 4 reflections, then
Proof. The last equality follows from the minimality of W. To prove the first, note that no primitive group of rank greater than 2 has an order 4 reflection, so assume G has rank 2. The mixing theorem implies that G has only order 2 and order 4 reflections; there are only two such groups: G 8 and G 9 ŽTable 3.. Since all its order 2 reflections are squares of order 4 reflections, G 8 satisfies the theorem. Now, G 9 has 18 order 2 reflections, and each corresponds to a root subsystem of type A1 = A1 or B2 , so ⌽ W must have at least 18 ⭈ 4 s 72 roots. But the largest rank 4 root system whose Coxeter diagram has a double bond is ⌽F4 , which has only 48 roots. PROPOSITION 3.7.
A 2 n has no complex reflection subgroups of rank n.
Proof. If G - A 2 n , then there exists a d g 2, 3, 4, 54 such that ⌽d s ⌽A 2 n, so 2 d g AutŽ ⌽A 2 n . s A 2 n i "14 . Elements of A 2 n have characteristic polynomials of the form Ž x 1 y 1.Ž x 2 y 1. ⭈⭈⭈ Ž x k y 1.rŽ x y 1., 1 q 2 q ⭈⭈⭈ q k s 2 n q 1, and charpoly R Ž 2 d . s Ž x 2 y 2 cosŽ d . x q 1. n. But Ž x y 1. divides all the x i y 1, while 1 is not a root of x 2 y 2 cosŽ d . x q 1, so k s 1; thus 2 d f A 2 n . The elements in AutŽ ⌽A 2 n ._ A 2 n can be expressed as y1V ⭈ , where y1V : V ª V, ¨ ¬ y¨ , and g A 2 n . Since 2 n is even, charpoly R Žy1V ⭈ . s pŽyx ., where pŽ x . s charpoly R Ž ., so the above argument applies again; thus 2 d f AutŽ ⌽A 2 n .. 3.2. Classification Now we describe a general method for finding complex and real reflection groups G and W that satisfy G - W. Remark 3.1 implies that the possible orders of reflections in any complex reflection subgroup of W are determined by the rank 2 root subsystems of ⌽ W : ⌽A 1=A 1 l order 2 ⌽A 2 l order 3 ⌽H 2 l order 5 ⌽B 2 l order 2 or 4.
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The mixing theorem states that if G - W, then all the reflections in G have the same order or that G has reflections of orders 2 and 4. Moreover, Corollary 3.1 says that if G has reflections of a single order d, then ⌽ W s ⌽d , while ⌽ W s ⌽ 2 otherwise, so if G - W, there exists d g 2, 3, 4, 54 such that 2 d g AutŽ ⌽ W . and G - CW Ž 2 d .. Since Theorem 3.2 states that centralizers of the form CW Ž 2 d . are complex reflection groups, to find the complex reflection subgroups of W it is enough to identify such centralizers and their reflection subgroups. To identify a centralizer CW Ž 2 d . we will usually try to decide: Ž1. whether CW Ž 2 d . is primitive or imprimitive, and Ž2. how many order d reflections it contains. For Ž1., note that if ⌽ W s ⌽d , d / 2, then CW Ž 2 d . contains only order d reflections so must be primitive, since imprimitive groups have order 2 reflections. If ⌽ W s ⌽ 2 , then CW Ž 2 d . has order 2 reflections but may or may not have order 4 reflections and may or may not be primitive. For Ž2., recall that an order d reflection s s␣ s corresponds to a rank 2 root subsystem ␣ ⌽ of ⌽ W . In fact , 2 , . . . , dy 1 all correspond to ␣⌽ . Moreover, any two order d reflections correspond to the same rank 2 root subsystem or to disjoint subsystems since the subsystems are contained in the eigenspaces for the nontrivial eigenvalues of the respective reflections. Therefore, Ž< ⌽ W
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1. 2 ŽTable 2., so the complex reflection group CF4Ž 6 . is unique up to isomorphism. Since CF4Ž 6 . has Ž< ⌽
The action of 4 partitions ⌽ into 12 A1 = A1 root subsystems of the form ␣ , 4 ␣ , 42 ␣ , 43 ␣ 4 , so CF4Ž 4 . has at most 12 order 2 reflections. There are less than 12 exactly when pairs of those subsystems together form B2 subsystems. Since orthogonal roots in a B2 root subsystem of F4 are in the same F4 orbit, C Ž 4o . has 12 order two reflections and C Ž 4i . has 6. Thus, C Ž 4o . , G12 ŽTable 3.. To identify C Ž 4i . note that the action of 8 is given by an element from the conjugacy class in AutŽ ⌽ . with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , and since AutŽ ⌽ . s F4 i Z 2 , we may and shall assume that 82 s 4i . Since CW Ž 8 . has 488 ⭈ 2 s 12 order 4 reflections, it is isomorphic to G 8 ŽTable 3., and CW Ž 8 . ; CW Ž 82 . s CW Ž 4i ., so CW Ž 4i . s CW Ž 8 . , G 8 . Neither G12 nor G 8 have primitive reflection subgroups ŽTable 3., and since F4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , Proposition 3.4 implies that the only possible imprimitive subgroups of G 8 or G12 are GŽ4, 1, 2. and GŽ4, 2, 2.. Since Ž 0i 0i . g GŽ4, 2, 2. F GŽ4, 1, 2. and < ZŽ G12 .< s 2, G12 has no imprimitive reflection subgroups. On the other hand, the 2-Sylow subgroup of G 8 is isomorphic to GŽ4, 1, 2., but F4 is not minimal for GŽ4, 1, 2.. To see this, let H be the subgroup of G 8 generated by its order 2 reflections. Then H is an irreducible rank 2 complex reflection group whose 6 reflections all have order 2, so H is imprimitive ŽTable 3., and since H has only order 2 reflections H s GŽ4, 2, 2.. Now H is a 2-group, so it is contained in a 2-Sylow subgroup P2 of G 8 , and since G 8 has 12 order four reflections and three 2-Sylow subgroups, each 2-Sylow subgroup must contain at least 4 order four reflections, so P2 , GŽ4, 1, 2.. Let i s s␣ i s i , i s 1, 2, . . . , 10,
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be the reflections in GŽ4, 1, 2., where 1 , 2 , 3 , 4 have order 4 and 5 , 6 are squares of order 4 reflections. Then 7 , 8 , 9 , 10 are conjugate in GŽ4, 1, 2., and since GŽ4, 1, 2. - CF4Ž 4 . , G 8 , where 4 acts via an inner automorphism of ⌽, ␣ i and i are in the same F4 root orbit for i G 7, so s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 . The same is true for 5 and 6 , so the roots ␣ 5 , 5 , . . . , ␣ 10 ,  10 can be chosen from the same F4 root orbit, and therefore, the roots ␣ i , i , i s 1, 2, . . . , 10, can be chosen so that D10 is1 i⌽ i is a root system of type B4 or C4 , so F4 is not minimal for GŽ4, 1, 2.. Remark 3.3. In the last paragraph we saw that GŽ4, 1, 2. ; B4 l C4 since i s s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 for i G 5. In particular, both B4 and C4 are minimal for GŽ4, 1, 2. - F4 . Something similar occurs for GŽ4, 2, 2. - F4 . Let 1 , . . . , 6 be the reflections in GŽ4, 2, 2. and 1 , 2 4 , 3 , 4 , 5 , 6 4 the conjugacy classes. Choose ␣ 1 ,  1 , . . . , ␣ 6 , 6 such that i s s␣ i s i . Since 3 , 4 , 5 , 6 4 are conjugate, we may assume that ␣ 3 ,  3 , . . . , ␣ 6 , 6 4 are in the same F4 root orbit; similarly, ␣ 1 ,  1 , ␣ 2 ,  2 4 are in the same orbit. Since each F4 root orbit is a D4 root subsystem, s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 , for i G 1. Thus, we may choose the ␣ i and i so that D6is1 ␣ i⌽ i is either of the F4 root orbits; so if D4 and D4X are the groups corresponding to the orbits, then GŽ4, 2, 2. ; D4 l D4X . B4 Let G - B4 . Possible orders of reflections in G are 2, 3, and 4, so ⌽ s ⌽ 2 or ⌽ s ⌽ 3 . ⌽ s ⌽ 3 . Here, 6 g AutŽ ⌽ ., but B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 2 , so there are no complex reflection subgroups in this case. 䢇
⌽ s ⌽ 2 . There is exactly one conjugacy class in B4 with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , so the complex reflection group CB4Ž 4 . is unique. Since D4 e B4 and has only one conjugacy class with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , 4 g D4 . 䢇
Suppose CB4Ž 4 . is primitive. If CB4Ž 4 . has an order 4 reflection, then Proposition 3.6 implies ⌽ s ⌽ 2 s ⌽4 , so 8 g AutŽ ⌽ .. But B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 . On the other hand, if CB4Ž 4 . has solely order 2 reflections, then CB4Ž 4 . has at most eight reflections since there are only 324 s 8 4-orbits on ⌽, but there is no such primitive group ŽTable 3..
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Therefore, CB4Ž 4 . is imprimitive, and since B4 does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 28 . x q 1. 2 , CB4Ž 4 . s GŽ4, 1, 2. or CB4Ž 4 . s GŽ4, 2, 2.. Direct computation shows that < CB4Ž 4 .< s 32, so CB4Ž 4 . s GŽ4, 1, 2.. Now, GŽ4, 2, 2. - GŽ4, 1, 2., but GŽ4, 2, 2. D4 . To see this let i s s␣ i s i be the six order 2 reflections in GŽ4, 2, 2. divided into conjugacy classes 1 , 2 , 3 , 4 4 and 5 , 6 4 . Since the corresponding reflections are conjugate, the 16 roots in D4is1 ␣ i⌽ i can be chosen from a single Žlong. B4-orbit. Assume i ) 4. If ␣ i and i are short, then i s s␣ i s i s sŽ ␣ iq  i .r62 sŽ ␣ iy  i .r62 . Therefore, the ␣ i and i can be chosen so that the 24 roots of D6is1 ␣ i⌽ i are contained in a single B4-orbit and hence form a D4 root subsystem. D4 Possible orders of reflections in G - D4 are 2 and 3, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . There is only one conjugacy class in AutŽ ⌽ . s F4 with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 3 , so the complex reflection group CD 4Ž 6 . is unique. Since CD 4Ž 6 . has 246 ⭈ 2 s 8 order 3 reflections, CD 4Ž 6 . , G4 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . The complex reflection group CD 4Ž 4 . is unique since there is only one conjugacy class in F4 with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 . Since there are 244 s 6 4-orbits on ⌽, there are at most six order 2 reflections, so CD 4Ž 4 . is not primitive ŽTable 3.. Therefore, CD 4Ž 4 . is imprimitive, and since there can be no order 4 reflections, CD 4Ž 4 . s GŽ4, 2, 2.. The only reflection subgroup of GŽ4, 2, 2. is GŽ4, 4, 2., and it is unique since it is the subgroup of GŽ4, 2, 2. generated by the non-diagonal reflections. 䢇
䢇
H4 Let G - H4 . Possible orders of reflections in G are 2, 3, and 5, so ⌽ s ⌽ 2 , ⌽ s ⌽ 3 , or ⌽ s ⌽5 . Moreover, AutŽ ⌽H 4 . s H4 w1, Appendixx and each conjugacy class in H4 has a distinct characteristic polynomial, so the complex reflection groups CH 4Ž 2 d ., d g 2, 3, 54 are unique. ⌽ s ⌽5 . Since CH 4Ž 10 . has 120 10 ⭈ 4 s 48 order 5 reflections, it is isomorphic to G16 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 3 . Since CH 4Ž 6 . has 120 6 ⭈ 2 s 40 order 3 reflections, it is isomorphic to G 20 ; the only possible subgroups are G4 and G5 . To find 䢇
䢇
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them note that Z [ ZŽ G 20 . has order 6 and that G 20rA , A5 Žthe simple group of order 60. w5x. Moreover, : G 20 ª A5 induces a bijective correspondence between the 3-Sylow subgroups of G 20 and A5 . Choose P1 and P2 , 3-Sylow subgroups of A5 , so that y1 Ž² P1 , P2 :. , G5 . Since G4 - G5 , existence of both G4 and G5 in CH 4Ž 6 . is proven; next consider uniqueness. Let 3 be an order 3 element of Z. Then G5 s G4 = ² 3 :, so G5 and G4 have the same 2-Sylow subgroups; a counting argument shows that G4 has only one. Let P2 be the 2-Sylow subgroup contained in G4 l G5 . Since w G 20 : G5 x s 5, NG Ž P2 . s G5 or NG Ž P2 . s G 20 , but P2 eG 20 would im20 20 ply that G 20 rP2 and hence G 20 is solvable, which is not possible since G 20 has a factor isomorphic to A5 . Therefore, G5 s NG 20Ž P2 ., so there is a unique isomorphism class of G5 in G 20 . More about G4 follows in Remark 3.4 below; for now note that G4 - D4 - H4 . ⌽ s ⌽ 2 . Since the Coxeter diagram of H4 has no double bond, CH 4Ž 4 . has 120 4 s 30 order 2 reflections and no order 4 reflections, so it is isomorphic to G 22 . The only other irreducible, rank 2, primitive, complex reflection groups generated by order 2 reflections are G12 and G13 , but < G13 < does not divide < G 22 <, so consider G12 . Suppose G12 - G 22 . Then they share at least one 2-Sylow subgroup, P2 . Since G 22 is irreducible, its center, Z, contains only scalars. But Z has order 4, so Z - P2 , which contradicts the fact that G12 has a center of order 2. Thus, G 22 has no primitive subgroups. In fact G 22 has no imprimitive subgroups for which H4 is minimal. 䢇
As above, let P2 be a 2-Sylow subgroup of G 22 . A counting argument Žusing the fact that G 22 rZŽ G 22 . , A5 . shows that there are at most five 2-Sylow subgroups of G 22 , so each must contain at least 305 s 6 order 2 reflections. Since the subgroup of P2 generated by its reflections must be imprimitive ŽTable 3., it follows that P2 , GŽ4, 2, 2., so by Proposition 3.5, it is contained in D4 . Remark 3.4. Ž1. The reflections in G5 account for 48 roots. Since < ⌽F < s 48, it is not surprising that G5 - F4 , but < ⌽H < s 120, and ⌽H has 4 4 4 no root subsystem containing 48 roots. Something similar happens with GŽ4, 4, n. in D 2 n and G 29 in E8 . Ž2. Assume G4 - H4 and let P2 be the unique 2-Sylow subgroup of G4 . Since G4 - NG 20Ž P2 . s G5 - G 20 , it is enough to consider G 20-conjugacy classes of G4 in G5 . From the discussion of G4 in F4 , there are at least two distinct copies of G4 in G5 . To see that there are at most two, let G4 and G4X be distinct subgroups of G5 ; then G4 l G4X s P2 , so they share
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no reflections. Since G4 has 8 reflections and there are only 16 in G5 , there can be at most two copies of G4 in G5 . Those two copies of G4 are not conjugate in G 20 . Let G4 , G4X - G5 with G4 / G4X . Choose a reflection g G4 _G4X . Since G4 = ² 3 : s G4 s G4X = ² 3 :, there exist g X g G4X and X3 g ² 3 : such that s g XX3 . Since g X has order 3, it is a reflection with detŽ g X . / detŽ .. Suppose G4 and G4X are conjugate by an element of G 20 . Since reflections in G4X of a given determinant are conjugate Žin G4 . there exists x g G 20 such that x xy1 s Ž g X . 2 . Multiplying x by a suitable element from ² 3 :, if necessary, detŽ x . s 1. Choose a basis so that is represented as
ž
3 0
0 . 1
/
Then corresponding to Ž g X . 2 and x are
ž
1 0
0 3
/
and
ž
0 ycy1
c . 0
/
In particular, x has order 4. Since xG4 xy1 s G4X , xP2 xy1 s P2 , so x g NG 20Ž P2 . s G5 . But G5 s G4 = ² 3 :, so x g G4 , a contradiction. E6 Possible orders of reflections in G - E6 are 2 or 3, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . There is only one conjugacy class in AutŽ ⌽ . with characteristic polynomial Ž x 2 y 2 cosŽ 26 . x q 1. 2 , so the complex reflection group CE 6Ž 6 . is unique. Since CE 6Ž 6 . has 726 ⭈ 2 s 24 order 3 reflections, CE 6Ž 6 . , G 25 and there are no irreducible reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . There are no complex subgroups in this case because AutŽ ⌽ . does not have an element with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 3. 䢇
䢇
E8 Possible orders of reflections in G - E8 are 2 and 3, so ⌽ s ⌽ 2 or ⌽ s ⌽ 3 . Moreover, AutŽ ⌽ . s E8 and there is only one conjugacy class in E8 corresponding to the characteristic polynomials Ž x 2 y 2 cosŽ 24 . x q 1. 4 and Ž x 2 y 2 cosŽ 26 . x q 1. 4 , so the complex reflection groups CE 8Ž 4 . and CE 8Ž 6 . are unique.
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⌽ s ⌽ 3 . The complex reflection group CE 8Ž 6 . has 240 6 ⭈ 2 s 80 order 3 reflections, so it is isomorphic to G 32 , and it has no reflection subgroups ŽTable 3.. ⌽ s ⌽ 2 . Since CE 8Ž 4 . has 240 4 s 60 order 2 reflections, it is isomorphic to G 31 , and G 29 is its only primitive subgroup w5, p. 409x. There are at most two isomorphism classes of G 29 in G 31. To see this choose generators 1 , 2 , 3 , 4 for G 29 corresponding to the presentation given in w3x. Choose ␣ i , i g ⌽ such that i s s␣ i s i , and construct a graph ⌫ with vertices in one-to-one correspondence with ␣ 1 ,  1 , ␣ 2 ,  2 , ␣ 3 ,  3 , ␣ 4 , 4 4 by joining two vertices with an edge if the corresponding roots are not orthogonal; label the edge with the inner product of the roots w4x. If the inner product is " 12 , the edge label is often omitted. Note that since i has order 2, Ž ␣ i , i . s 0. Since the Coxeter diagram of E8 has no double bond, the resulting graph is unique and is displayed in Fig. 1. 䢇
䢇
The numbered nodes correspond to a D6 root subsystem and there is only one up to conjugacy in E8 , so let e6 y e5 , e5 y e4 , e4 y e3 , e3 y e2 , e2 y e1 , e1 q e2 correspond to nodes 1, 2, 3, 4, 5, and 6, respectively. Then the constraints of the diagram require that up to sign change there are only four roots that can correspond to node a: ye4 " e 7 and ye4 " e8 . Similarly, node b may correspond to 12 Žye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., where s s "1. Since the roots corresponding to nodes a and b are orthogonal, there are four possibilities: Žye4 q e 7 , ye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., Žye4 q e8 , ye1 y e2 q e3 q e4 y e5 y e6 q e 7 q e8 ., Žye4 y e 7 , ye1 y e2 q e3 q e4 y e5 y e6 y e 7 y e8 ., Žye4 y e 7 , ye1 y e2 q e3 q e4 y e5 y e6 y e 7 y e8 .4 . These pairs of roots are all conjugate in E4 by elements which fix the chosen D6 root subsystem. Thus, there is a unique D 8-conjugacy class of G 29 in G 31. Assume G 29 , GX29 - G 31 Žs CE 8Ž 4 ... Put Z s ZŽ G 31 . s 1, 4 , 42 , 434 and choose g E8 such that G 29 y1 s GX29 . Since Z ; G 29 l GX29 , 4 y1 s 4 or 4 y1 s 43, there are at most two isomorphism classes of G 29 in G 31. In fact using a computer algebra system, it is possible to check that there is only one isomorphism class of G 29 in G 31. Finally, G 31 has no imprimitive subgroups for which E8 is minimal, for suppose GŽ m, p, 4. - G 31 - E8 . Then GŽ m, m, 4. - G 31 since GŽ m, m, 4. F GŽ m, p, 4.. But < GŽ m, m, 4.< s 3Ž2 m. 3, while < G 31 < s 2 10 ⭈ 3 ⭈ 5, so m F 4. Thus the only possibilities are GŽ4, 1, 4., GŽ4, 2, 4., and GŽ4, 4, 4.. Since
FIG. 1. Carter graph for G 29 .
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G 31 has no order 4 reflections, GŽ4, 1, 4. is eliminated, and by Proposition 3.5, GŽ4, 4, 4. - GŽ4, 2, 4. - D 8 . Remark 3.5. In w8x, Shephard described G 31 as the symmetry group of a four-dimensional complex polytope that has 240 vertices, and he listed the following coordinates for the vertices:
Ž 2 i a , 0, 0, 0 . Ž Ž 1 q i . i a , Ž 1 q i . i b , 0, 0 . Ž ia, ib, ic, id .
permuted, Ž a s 0, 1, 2, 3 . permuted, Ž a, b s 0, 1, 2, 3 .
Ž a q b q c q d s 0 Ž mod 2. . .
If Shephard’s complex basis is denoted e1 , e2 , e3 , and e4 , then with respect to the real ordered basis 1 s Ž1 q i . e1 , 2 s Ž1 y i . e1 , . . . , 7 s Ž1 q i . e4 , 8 s Ž1 y i . e4 , the coordinates are " i " j , i - j and 12 Ý " i Ževen number of plus signs., which is the usual realization of the E8 root system. B2 n Possible orders of reflections in G - B2 n are 2, 3, and 4, so ⌽ s ⌽ 2 or ⌽ s ⌽3. ⌽ s ⌽ 3 . The elements in B2 n have characteristic polynomials of the form ŁŽ x n i " 1., where ⌺n i s 2 n, so 6 f AutŽ ⌽ . s B2 n . ⌽ s ⌽ 2 . There is one conjugacy class in B2 n with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. 2 , so the complex reflection group CB 2 nŽ 4 . is unique. 䢇
䢇
Since every order two reflection requires four roots from ⌽, the rank 3, 4, and 5 primitive reflection groups have too many order 2 reflections to be contained in B2 n . Thus, CB 2 nŽ 4 . is imprimitive, so from w11, Theorem 4.2x, it follows that CB 2 nŽ 4 . s GŽ4, 1, n.. Now GŽ4, 2, n. is the subgroup of GŽ4, 1, n. generated by the order 2 reflections and GŽ4, 4, n. is the subgroup generated by the non-diagonal order 2 reflections, so there is only one isomorphism class of each. The argument used to show that GŽ4, 2, 2. - D4 can also be used here to prove that GŽ4, 4, n. - GŽ4, 2, n. - D 2 n . D2 n Since there is one conjugacy class in D 2 n with characteristic polynomial Ž x 2 y 2 cosŽ 24 . x q 1. n, the complex reflection group CD Ž 4 . is unique, 2n and it follows from the B2 n discussion that CD 2 nŽ 4 . s GŽ4, 2, n. and that the subgroup GŽ4, 4, n. is unique.
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TABLE 4 Complex Reflection Subgroups of Real Reflection Groups W F4
B4 D4
H4
G-W G12 , CF4Ž 4o . G 8 , CF4Ž 4i . G5 , CF4Ž 6o . GŽ4, 1, 2. , CB4Ž 4 . GŽ4, 2, 2. , CD 4Ž 4 . GŽ4, 4, 2. - GŽ4, 2, 2. G4 , CD 4Ž 6 . G 22 , CH 4Ž 4 . G 20 , CH Ž 6 . 4
E6 E8
B2 n D2 n
G5 - G 20 G16 , CH 4Ž 10 . G 25 , CE 6Ž 6 . G 31 , CE 8Ž 4 . G 29 - G 31 G 32 , CE 8Ž 6 . GŽ4, 1, n. , CB 2 nŽ 4 . GŽ4, 2, n. , CD 2 nŽ 6 . GŽ4, 4, n. - GŽ4, 2, n.
3.3. Summary Table 4 lists the rank 2 n Ž n ) 1. irreducible real reflection groups with their truly complex irreducible rank n reflection subgroups. When appropriate, the complex reflection groups are expressed as centralizers.
ACKNOWLEDGMENTS Much of this material first appeared in a dissertation written under the supervision of J. N. Spaltenstein at the University of Oregon. I gratefully acknowledge his support and encouragement. I also thank C. Curtis who suggested that w11x would be relevant, and I thank the referee, whose suggestions made this a better paper.
REFERENCES 1. E. Bannai, Automorphisms of Irreducible Weyl Groups, J. Fac. Sci. Uni¨ . Tokyo 16 Ž1969., 273᎐286.
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2. N. Bourbaki, ‘‘Groupes et Algebres de Lie,’’ Chaps. IV, V, VI, Hermann, Paris, 1968. ` 3. M. Broue, ´ G. Malle, and R. Roquier, On complex reflection groups and their associated braid groups, ‘‘Representations of Groups ŽBanff, AB, 1994.,’’ CMS Conf. Proc. 16, pp. 1᎐13, Am. Math. Soc., Providence, RI, 1995. 4. R. Carter, Conjugacy classes in the Weyl groups, Compositio Math. 25 Ž1972., 1᎐59. ´ 5. A. Cohen, Finite complex reflection groups, Ann. Sci. Ecole Norm. Sup Ž 4 . Ž1972., 1᎐59. 6. L. Grove, The characters of the hecatonicosahedroidal group, J. Reine Angew. Math. 265 Ž1974., 160᎐169. 7. J. Humphreys, ‘‘Reflection Groups and Coxeter Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 8. G. Shephard, Unitary groups generated by reflections, Canad. J. Math. 5 Ž1953., 364᎐383. 9. G. Shephard and J. Todd, Finite unitary reflection groups, Canad. J. Math. 6 Ž1954., 274᎐304. 10. K. Shinoda, The conjugacy classes of the finite Ree groups of type F4 , J. Fac. Sci. Uni¨ . Tokyo 22 Ž1975., 1᎐15. 11. T. Springer, Regular elements of finite reflection groups, In¨ ent. Math. 25 Ž1974., 159᎐198.