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:

    “Modern tourist guides have helped raised tourist expectations. And they have provided the natives—from Kaiser Wilhelm down to the villagers of Chichacestenango—with a detailed and itemized list of what is expected of them and when. These are the up-to- date scripts for actors on the tourists’ stage.”
    —Daniel J. Boorstin (b. 1914)

    “We saw the machinery where murderers are now executed. Seven have been executed. The plan is better than the old one. It is quietly done. Only a few, at the most about thirty or forty, can witness [an execution]. It excites nobody outside of the list permitted to attend. I think the time for capital punishment has passed. I would abolish it. But while it lasts this is the best mode.”
    —Rutherford Birchard Hayes (1822–1893)

    “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)

    “The money complex is the demonic, and the demonic is God’s ape; the money complex is therefore the heir to and substitute for the religious complex, an attempt to find God in things.”
    —Norman O. Brown (b. 1913)

    “Fanny was not there! How she would have enjoyed the scene.... I could not but think of her, and in spite of my efforts to prevent, the unbidden tear would flow. Alas! I cannot feel the satisfaction some appear to do in the reflection that her eyes beheld the scene from the other world.”
    —Rutherford Birchard Hayes (1822–1893)

    “... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
    —Mary Barnett Gilson (1877–?)