Binary Cubic Groups

On this page:

  • Q8 - the quaternion group
  • 2T - the binary tetrahedral group
  • 2O - the binary octahedral group
  • 2I - the binary icosahedral group (unfinished)

See also the binary dihedral groups, or dicyclic groups.

We realize these groups as subgroups of the group Sp(1) of unit quaternions. Given a unit quaternion q, the map \(a\mapsto qa\bar q\) (where a is an imaginary quaternion) defines an element of SO(3), giving the well-known double cover Sp(1) → SO(3). (In particular the images of q and -q coincide.)

The binary cubic groups are the preimages under this double cover of the cubic groups T, O, and I. We also include the quaternion group Q8, which is in fact the double cover of the group of order 4 of rotations by \(\pi\) about each of the Cartesian axes. If G is the cubic group in question, one writes 2G for the binary version, and there is a short exact sequence,

$$\mathbb{Z}_2^c \longrightarrow 2G \longrightarrow G $$

where \(\mathbb{Z}_2^c =\{ 1\}\) is the centre of Sp(1). In particular, any representation of G gives rise to a representation of 2G, via the homomorphism 2GG.

Note that Sp(1) ≅ SU(2).

Q8

\(Q_8\) is the quaternion group and has order 8. It consists of the unit quaternions ±1, ±i, ±j, ±k. It is the double cover of the group D2, which is the subgoup of SO(3) consisting of the identity and the rotation by \(\pi\) about each of the x- y- and z-axes (so \(Q_8\) could be called the binary Klein group). The character table for its real representations is:

Q8 1 -1 {±i} {±j} {±k} notes
# 1 1 2 2 2 |Q8|=8
A0 1 1 1 1 1 trivial rep
A1 1 1 1 -1 -1
A2 1 1 -1 1 -1
A3 1 1 -1 -1 1
H 4 -4 0 0 0 quaternionic type
  • Notice that jij-1=-i, so i and -i are conjugate, as are j and -j, and k and -k.
  • The 1-dimensional reps are all induced from the reps of D2 = Z2×Z2.
  • The irreducible representation H is the natural action of Q8 on the space of all quaternions (hence its name). It is therefore of quaternionic type. It is also the underlying representation of the 2-dimensional complex irreducible representation given by Q8SU(2) (under the isomorphism of Sp(1) with SU(2)).
  • The complex character table has the same 1-dimensional characters, but H should be replaced by E whose character row is (2,-2,0,0,0) = \(\frac12H\).

2T

Here TA4 is the group of all rotational symmetries of the tetrahedron. The binary group 2T is therefore also denoted 2A4. It is (isomorphic to) SL(2; Z3). See Tony Phillips' pages and wikipedia for further details about this group. It contains Q8 as a normal subgroup. As unit quaternions, the elements are of the form

±1, ±i, ±j, ±k,   ½ (±1 ±i ±j ±k).

The elements of the form ½(1 ±i ±j ±k) are of order 6, while those of the form ½(-1 ±i ±j ±k) are of order 3. And of course ±i etc are of order 4, and -1 is of order 2.

Conjugation within the group permutes the coefficients of the 3 "imaginary" parts and changes an even number of signs, but does not change the real part of a quaternion. For example,

½(1 + i + j + k) is conjugate to ½(1 - i - j + k), and +i is conjugate to -i and to -k.

The conjugacy classes are therefore represented by:

1, -1, i, a = ½(1 + i + j + k), b = ½(1 + i + j - k), c = ½(-1 + i + j + k), d = ½(-1 + i + j - k), so a and b are of order 6 and c, d are of order 3.

Note that a² = d² = c, and b² = c² = d. Finally, Q8 is a normal subgroup.

2T 1 -1 i a b c d notes
# 1 1 6 4 4 4 4 |2T|=24
A0 1 1 1 1 1 1 1 trivial rep
E 2 2 2 -1 -1 -1 -1 complex type
T 3 3 -1 0 0 0 0  
G 4 -4 0 -1 -1 1 1 complex type
H 4 -4 0 2 2 -2 -2 quaternionic type
  • A0 E and T arise from reps of T (see A4) via the double covering.
  • A0 and T are absolutely irreducible (ie, of real type), E and G are of complex type and H is of quaternionic type
  • E, G and H split over C, and have characters:
    E = (1, 1, 1, ω, ω², ω², ω) ⊕ (1, 1, 1, ω², ω, ω, ω²), where ω = ½(-1+√3 i),
    G = (2, -2, 0, ω, ω², -ω², -ω) ⊕ (2, -2, 0, ω², ω, -ω, -ω²), and
    H = 2 (2, -2, 0, 1, 1, -1, -1) (2 copies of the same complex rep)
  • The elements of this group considered as points in the unit sphere in R4 (space of quaternions) are the vertices of the convex regular 24-cell. The permutation representation on this set of points is the regular representation, equal to A0 + E + 3T +2G + H.
  • H is the natural representation of this group acting on the quaternions.

2O

where O is the group of all rotational symmetries of the cube. O is isomorphic to S4, so 2O is often denoted 2S4, it has order 48. See wikipedia for details.

The binary tetrahedral group 2T is a normal subgroup, as is Q8. The remaining elements are all of the form (1/√2)(±1±i) etc.

The conjugacy classes are represented by 1, -1, i, and

a = ½(1 +i + j + k), c = ½(-1 +i + j + k),
e = (1/√2)(1+i), f = (1/√2)(-1+i), g = (1/√2)(i+j).
2O 1 -1 i a c e f g notes
# 1 1 6 8 8 6 6 12 |2O|=48
A0 1 1 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 1 -1 -1 -1  
E 2 2 2 -1 -1 0 0 0  
T1 3 3 -1 0 0 1 1 -1  
T2 3 3 -1 0 0 -1 -1 1  
H1 4 -4 0 2 -2 2√2 -2√2 0 action on quaternions
H2 4 -4 0 2 -2 -2√2 2√2 0
L 8 -8 0 -2 2 0 0 0
  • The reps A0, A1, E, T1 and T2 arise from the corresponding reps of the octahedral group; they are all absolutely irreducible (since they are for S4).
  • The reps H1, H2 and L are all irreducible of quaternionic type. The corresponding complex irreducible reps have simply half the character.

2I

where I is the group of all rotational symmetries of the icosahedron. 2I has order 120.

I is isomorphic to A5, so 2I is often denoted 2A5 (a double cover of A5); it is isomorphic to SL(2,F5). See wikipedia for details.

As unit quaternions, the elements of 2I are:

1, -1, \(\pm i, \, \pm j, \, \pm k\) and \(\frac12(\pm 1 \pm i \pm j \pm k)\) (16 of these) - these form the 24 Hurwitz units, and the subgroup 2T.

and then the 96 quaternions similar to \(\frac12(\pm 0 \pm i\pm \varphi^-j\pm\varphi^+ k)\) formed from even permutations of \(0,1,\varphi^-,\varphi^+\) and all combinations of signs, where \(\varphi^+=\frac12(1+\sqrt5)\) (the golden ratio) and \(\varphi^-=\frac12(1-\sqrt5)\).

Note that \(\varphi^+>0\) while \(\varphi^-<0\). Indeed, in descending order, $$\cos(\pi/5) = \tfrac12 \varphi^+,\quad \cos(2\pi/5) = -\tfrac12 \varphi^-,\quad \cos(3\pi/5) = \tfrac12 \varphi^-,\quad \cos(4\pi/5) = -\tfrac12 \varphi^+$$

Conjugacy classes:

These are classified entirely by the real part of the quaternion:

name Re(q) # order A5
1 1 1 1 e
-1 -1 1 2 e
a 0 30 4 C2C2
b \(-\frac12\) 20 3 C3
c \(\frac12\) 20 6 C3
d \(-\frac12\varphi^+\) 12 5 C5
e \(-\frac12\varphi^-\) 12 5 C5'
f \(\frac12\varphi^+\) 12 10 C5
g \(\frac12\varphi^-\) 12 10 C5'

The column marked A5 is the corresponding conjugacy class in A5 of the image of the given conjugacy class under the projection (and C5 and C5' denote the two different conjugacy classes of elements of order 5 in A5).

Note that the conjugacy classes satisfy \(a^2=-1\), \(b^2=b\), \(c^2=b\), \(d^2=e\), \(e^2=d\), \(f^2=e\) and \(g^2=d\).

Complex representation table:

( need to do real one)

2I 1 -1 a b c d e f g notes
# 1 1 30 20 20 12 12 12 12 |2I|=120
A0 1 1 1 1 1 1 1 1 1 trivial rep
E1 2 -2 0 -1 1 \(-\varphi^+\) \(-\varphi^- \) \(\varphi^+ \) \(\varphi^-\)  
E2 2 -2 0 -1 1 \(-\varphi^- \) \(-\varphi^+\) \(\varphi^-\) \(\varphi^+ \)  
T1 3 3 -1 0 0 \(\varphi^-\) \(\varphi^+ \) \(\varphi^+ \) \(\varphi^-\) lifted from A5
T2 3 3 -1 0 0 \(\varphi^+ \) \(\varphi^-\) \(\varphi^-\) \(\varphi^+ \) lifted from A5
G1 4 4 0 1 1 -1 -1 -1 -1 lifted from A5
G2 4 -4 0 1 -1 -1 -1 1 1
H 5 5 1 -1 -1 0 0 0 0 lifted from A5
J 6 -6 0 0 0 1 1 -1 -1
  • \(\varphi^+ =\frac12(1+\sqrt5)\) (the golden ratio)
  • \(\varphi^-=\frac12(1-\sqrt5)\) (the `rational conjugate' \(=-(\varphi^+)^{-1}\))
  • The rational irreducible representations of 2I are \(A_0, E_1+E_2, T_1+T_2, G_1,G_2,H\) and \(J\).
  • The labels are not standard.

See also https://people.maths.bris.ac.uk/~matyd/GroupNames/97/SL(2,5).html