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:
“Feminism is an entire world view or gestalt, not just a laundry list of womens issues.”
—Charlotte Bunch (b. 1944)
“Modern tourist guides have helped raised tourist expectations. And they have provided the nativesfrom Kaiser Wilhelm down to the villagers of Chichacestenangowith 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)
“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)
“Specialization is a feature of every complex organization, be it social or natural, a school system, garden, book, or mammalian body.”
—Catharine R. Stimpson (b. 1936)
“The Americans ... have invented so wide a range of pithy and hackneyed phrases that they can carry on an amusing and animated conversation without giving a moments reflection to what they are saying and so leave their minds free to consider the more important matters of big business and fornication.”
—W. Somerset Maugham (18741965)
“screenwriter
Policemen so cherish their status as keepers of the peace and protectors of the public that they have occasionally been known to beat to death those citizens or groups who question that status.”
—David Mamet (b. 1947)