Complex Reflection Group - List of Irreducible Complex Reflection Groups

List of Irreducible Complex Reflection Groups

There are a few duplicates in the first 3 lines of this list; see the previous section for details.

  • ST is the Shephard–Todd number of the reflection group.
  • Rank is the dimension of the complex vector space the group acts on.
  • Structure describes the structure of the group. The symbol * stands for a central product of two groups. For rank 2, the quotient by the (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (T = Alt(4), O = Sym(4), I = Alt(5), of orders 12, 24, 60), as stated in the table. For the notation 21+4, see extra special group.
  • Order is the number of elements of the group.
  • Reflections describes the number of reflections: 26412 means that there are 6 reflections of order 2 and 12 of order 4.
  • Degrees gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.
ST Rank Structure and names Order Reflections Degrees Codegrees
1 n−1 Symmetric group G(1,1,n) = Sym(n) n! 2n(n − 1)/2 2, 3, ...,n 0,1,...,n − 2
2 n G(m,p,n) m > 1, n > 1, p|m (G(2,2,2) is reducible) mnn!/p 2mn(n−1)/2,dnφ(d) (d|m/p, d > 1) m,2m,..,(n − 1)m; mn/p 0,m,..., (n − 1)m if p < m; 0,m,...,(n − 2)m, (n − 1)m − n if p = m
3 1 Cyclic group G(m,1,1) = Zm m dφ(d) (d|m, d > 1) m 0
4 2 Z2.T = 33 24 38 4,6 0,2
5 2 Z6.T = 33 72 316 6,12 0,6
6 2 Z4.T = 32 48 2638 4,12 0,8
7 2 Z12.T = 〈3,3,3〉2 144 26316 12,12 0,12
8 2 Z4.O = 44 96 26412 8,12 0,4
9 2 Z8.O = 42 192 218412 8,24 0,16
10 2 Z12.O = 43 288 26316412 12,24 0,12
11 2 Z24.O = 〈4,3,2〉12 576 218316412 24,24 0,24
12 2 Z2.O= GL2(F3) 48 212 6,8 0,10
13 2 Z4.O = 〈4,3,2〉2 96 218 8,12 0,16
14 2 Z6.O = 32 144 212316 6,24 0,18
15 2 Z12.O = 〈4,3,2〉6 288 218316 12,24 0,24
16 2 Z10.I = 55 600 548 20,30 0,10
17 2 Z20.I = 52 1200 230548 20,60 0,40
18 2 Z30.I = 53 1800 340548 30,60 0,30
19 2 Z60.I = 〈5,3,2〉30 3600 230340548 60,60 0,60
20 2 Z6.I = 33 360 340 12,30 0,18
21 2 Z12.I = 32 720 230340 12,60 0,48
22 2 Z4.I = 〈5,3,2〉2 240 230 12,20 0,28
23 3 W(H3) = Z2 × PSL2(5), Coxeter 120 215 2,6,10 0,4,8
24 3 W(J3(4)) = Z2 × PSL2(7), Klein 336 221 4,6,14 0,8,10
25 3 W(L3) = W(P3) = 31+2.SL2(3), Hessian 648 324 6,9,12 0,3,6
26 3 W(M3) =Z2 ×31+2.SL2(3), Hessian 1296 29 324 6,12,18 0,6,12
27 3 W(J3(5)) = Z2 ×(Z3.Alt(6)), Valentiner 2160 245 6,12,30 0,18,24
28 4 W(F4) = (SL2(3)* SL2(3)).(Z2 × Z2) Weyl 1152 212+12 2,6,8,12 0,4,6,10
29 4 W(N4) = (Z4*21 + 4).Sym(5) 7680 240 4,8,12,20 0,8,12,16
30 4 W(H4) = (SL2(5)*SL2(5)).Z2 Coxeter 14400 260 2, 12, 20,30 0,10,18,28
31 4 W(EN4) = W(O4) = (Z4*21 + 4).Sp4(2) 46080 260 8,12,20,24 0,12,16,28
32 4 W(L4) = Z3 × Sp4(3) 155520 380 12,18,24,30 0,6,12,18
33 5 W(K5) = Z2 ×Ω5(3) = Z2 × PSp4(3) = Z2 × PSU4(2) 51840 245 4,6,10,12,18 0,6,8,12,14
34 6 W(K6)= Z3.Ω−
6(3).Z2, Mitchell's group		 39191040 2126 6,12,18,24,30,42 0,12,18,24,30,36
35 6 W(E6) = SO5(3) = O−
6(2) = PSp4(3).Z2 = PSU4(2).Z2, Weyl		 51840 236 2,5,6,8,9,12 0,3,4,6,7,10
36 7 W(E7) = Z2 ×Sp6(2), Weyl 2903040 263 2,6,8,10,12,14,18 0,4,6,8,10,12,16
37 8 W(E8)= Z2.O+
8(2), Weyl		 696729600 2120 2,8,12,14,18,20,24,30 0,6,10,12,16,18,22,28

For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).

Read more about this topic:  Complex Reflection Group

Famous quotes containing the words list of, list, irreducible, complex, reflection and/or groups:

    “Every morning I woke in dread, waiting for the day nurse to go on her rounds and announce from the list of names in her hand whether or not I was for shock treatment, the new and fashionable means of quieting people and of making them realize that orders are to be obeyed and floors are to be polished without anyone protesting and faces are to be made to be fixed into smiles and weeping is a crime.”
    —Janet Frame (b. 1924)

    “I am opposed to writing about the private lives of living authors and psychoanalyzing them while they are alive. Criticism is getting all mixed up with a combination of the Junior F.B.I.- men, discards from Freud and Jung and a sort of Columnist peep- hole and missing laundry list school.... Every young English professor sees gold in them dirty sheets now. Imagine what they can do with the soiled sheets of four legal beds by the same writer and you can see why their tongues are slavering.”
    —Ernest Hemingway (1899–1961)

    “If an irreducible distinction between theatre and cinema does exist, it may be this: Theatre is confined to a logical or continuous use of space. Cinema ... has access to an alogical or discontinuous use of space.”
    —Susan Sontag (b. 1933)

    “In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “It seems certain, that though a man, in a flush of humour, after intense reflection on the many contradictions and imperfections of human reason, may entirely renounce all belief and opinion, it is impossible for him to persevere in this total scepticism, or make it appear in his conduct for a few hours.”
    —David Hume (1711–1776)

    “In properly organized groups no faith is required; what is required is simply a little trust and even that only for a little while, for the sooner a man begins to verify all he hears the better it is for him.”
    —George Gurdjieff (c. 1877–1949)