Complex reflection group
In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise. (Complex reflections are sometimes called pseudo reflections or unitary reflections or sometimes just reflections.)
Any finite complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces. So it is sufficient to classify the irreducible complex reflection groups.
The finite irreducible complex reflection groups were classified by G. C. Shephard and J. A. Todd (1954). They found an infinite family G(m,p,n) depending on 3 positive integer parameters (with p dividing m), and 34 exceptional cases, that they numbered from 4 to 37, listed below. The group G(m,p,n), of order mnn!/p, is the semidirect product of the abelian group of order mn/p whose elements are (θa1,θa2, ...,θan), by the symmetric group Sn acting by permutations of the coordinates, where θ is a primitive mth root of unity and Σai≡ 0 mod p; it is an index p subgroup of the generalized symmetric group
Special cases of G(m,p,n):
- G(1,1,n) is the Coxeter group An−1
- G(2,1,n) is the Coxeter group Bn = Cn
- G(2,2,n) is the Coxeter group Dn
- G(m,p,1) is a cyclic group of order m/p.
- G(m,m,2) is the Coxeter group I2(m) (and the Weyl group G2 when m = 6).
- The group G(m,p,n) acts irreducibly on Cn except in the cases m=1, n>1 (symmetric group) and G(2,2,2) (Klein 4 group), when Cn splits as a sum of irreducible representations of dimensions 1 and n−1.
- The only cases when two groups G(m,p,n) are isomorphic as complex reflection groups are that G(ma,pa,1) is isomorphic to G(mb,pb,1) for any positive integers a,b. However there are other cases when two such groups are isomorphic as abstract groups.
- The complex reflection group G(2,2,3) is isomorphic as a complex reflection group to G(1,1,4) restricted to a 3-dimensional space.
- The complex reflection group G(3,3,2) is isomorphic as a complex reflection group to G(1,1,3) restricted to a 2-dimensional space.
- The complex reflection group G(2p,p,1) is isomorphic as a complex reflection group to G(1,1,2) restricted to a 1-dimensional space.
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|
|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|
6(3).Z2, Mitchell's group
|35||6||W(E6) = SO5(3) = O−
6(2) = PSp4(3).Z2 = PSU4(2).Z2, Weyl
|36||7||W(E7) = Z2 ×Sp6(2), Weyl||2903040||263||2,6,8,10,12,14,18||0,4,6,8,10,12,16|
For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).
Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For being the rank of the reflection group, the degrees of the generators of the ring of invariants are called degrees of W and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:
- The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
- The order of a complex reflection group is the product of its degrees.
- The number of reflections is the sum of the degrees minus the rank.
- An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
- The degrees di satisfy the formula
For being the rank of the reflection group, the codegrees of W can be defined by
- For a real reflection group, the codegrees are the degrees minus 2.
- The number of reflection hyperplanes is the sum of the codegrees plus the rank.
Well-generated complex reflection groups
An irreducible complex reflection group of rank is generated by or by reflections. It is said to be well-generated if it is generated by reflections ; it is proved that this is equivalent to the property for all . For irreducible well-generated complex reflection groups, the Coxeter number is defined to be the largest degree, . A reducible complex reflection group is said to be well-generated if it is a product of irreducible well-generated complex reflection groups. Any finite real reflection group is well-generated.
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